0.00/0.09 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.00/0.09 % Command : tptp2X_and_run_prover9 %d %s 0.08/0.28 % Computer : n025.cluster.edu 0.08/0.28 % Model : x86_64 x86_64 0.08/0.28 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.08/0.28 % Memory : 8042.1875MB 0.08/0.28 % OS : Linux 3.10.0-693.el7.x86_64 0.08/0.28 % CPULimit : 1440 0.08/0.28 % WCLimit : 180 0.08/0.28 % DateTime : Mon Jul 3 08:38:53 EDT 2023 0.13/0.28 % CPUTime : 2.34/2.72 ============================== Prover9 =============================== 2.34/2.72 Prover9 (32) version 2009-11A, November 2009. 2.34/2.72 Process 14999 was started by sandbox on n025.cluster.edu, 2.34/2.72 Mon Jul 3 08:38:55 2023 2.34/2.72 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 1440 -f /tmp/Prover9_14846_n025.cluster.edu". 2.34/2.72 ============================== end of head =========================== 2.34/2.72 2.34/2.72 ============================== INPUT ================================= 2.34/2.72 2.34/2.72 % Reading from file /tmp/Prover9_14846_n025.cluster.edu 2.34/2.72 2.34/2.72 set(prolog_style_variables). 2.34/2.72 set(auto2). 2.34/2.72 % set(auto2) -> set(auto). 2.34/2.72 % set(auto) -> set(auto_inference). 2.34/2.72 % set(auto) -> set(auto_setup). 2.34/2.72 % set(auto_setup) -> set(predicate_elim). 2.34/2.72 % set(auto_setup) -> assign(eq_defs, unfold). 2.34/2.72 % set(auto) -> set(auto_limits). 2.34/2.72 % set(auto_limits) -> assign(max_weight, "100.000"). 2.34/2.72 % set(auto_limits) -> assign(sos_limit, 20000). 2.34/2.72 % set(auto) -> set(auto_denials). 2.34/2.72 % set(auto) -> set(auto_process). 2.34/2.72 % set(auto2) -> assign(new_constants, 1). 2.34/2.72 % set(auto2) -> assign(fold_denial_max, 3). 2.34/2.72 % set(auto2) -> assign(max_weight, "200.000"). 2.34/2.72 % set(auto2) -> assign(max_hours, 1). 2.34/2.72 % assign(max_hours, 1) -> assign(max_seconds, 3600). 2.34/2.72 % set(auto2) -> assign(max_seconds, 0). 2.34/2.72 % set(auto2) -> assign(max_minutes, 5). 2.34/2.72 % assign(max_minutes, 5) -> assign(max_seconds, 300). 2.34/2.72 % set(auto2) -> set(sort_initial_sos). 2.34/2.72 % set(auto2) -> assign(sos_limit, -1). 2.34/2.72 % set(auto2) -> assign(lrs_ticks, 3000). 2.34/2.72 % set(auto2) -> assign(max_megs, 400). 2.34/2.72 % set(auto2) -> assign(stats, some). 2.34/2.72 % set(auto2) -> clear(echo_input). 2.34/2.72 % set(auto2) -> set(quiet). 2.34/2.72 % set(auto2) -> clear(print_initial_clauses). 2.34/2.72 % set(auto2) -> clear(print_given). 2.34/2.72 assign(lrs_ticks,-1). 2.34/2.72 assign(sos_limit,10000). 2.34/2.72 assign(order,kbo). 2.34/2.72 set(lex_order_vars). 2.34/2.72 clear(print_given). 2.34/2.72 2.34/2.72 % formulas(sos). % not echoed (1257 formulas) 2.34/2.72 2.34/2.72 ============================== end of input ========================== 2.34/2.72 2.34/2.72 % From the command line: assign(max_seconds, 1440). 2.34/2.72 2.34/2.72 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 2.34/2.72 2.34/2.72 % Formulas that are not ordinary clauses: 2.34/2.72 1 (all Z_1 all W hAPP(int,int,times_times(int,Z_1),W) = hAPP(int,int,times_times(int,W),Z_1)) # label(fact_87_zmult__commute) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 2 (all I_1 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),N_1)) -> I_1 = hAPP(nat,nat,minus_minus(nat,N_1),hAPP(nat,nat,minus_minus(nat,N_1),I_1)))) # label(fact_624_diff__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 3 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all X_2 all Y_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,X_2)),number_number_of(X_a,Y_2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X_2),Y_2)))))) # label(fact_31_le__number__of) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 4 (all M all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),hAPP(nat,nat,plus_plus(nat,M),J_1))))) # label(fact_679_trans__le__add2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 5 (all Z_1 ti(int,Z_1) = hAPP(int,int,times_times(int,one_one(int)),Z_1)) # label(fact_130_zmult__1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 6 (all X_a (number_ring(X_a) -> number_number_of(X_a,bit1(pls)) = one_one(X_a))) # label(fact_158_numeral__1__eq__1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 7 (all X_a (linord20386208strict(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,C),A_1)),hAPP(X_a,X_a,times_times(X_a,C),B)))))))) # label(fact_491_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 8 (all X_a (number_ring(X_a) -> (all V all W hAPP(X_a,X_a,minus_minus(X_a,number_number_of(X_a,V)),number_number_of(X_a,W)) = number_number_of(X_a,hAPP(int,int,minus_minus(int,V),W))))) # label(fact_369_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 9 (all X_a all X_b all X_c all B_1_1 all B_2_1 combc(X_a,X_b,X_c,B_1_1,B_2_1) = combc(X_a,X_b,X_c,ti(fun(X_a,fun(X_b,X_c)),B_1_1),B_2_1)) # label(tsy_c_COMBC_arg1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 10 (all X_a (linord20386208strict(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,times_times(X_a,A_1),B)))))))) # label(fact_500_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 11 (all P_2 all A_3 all B_2 (-((exists D_4 (-hBOOL(hAPP(nat,bool,P_2,D_4)) & A_3 = hAPP(nat,nat,plus_plus(nat,B_2),D_4))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_3),B_2)) & -hBOOL(hAPP(nat,bool,P_2,zero_zero(nat)))) <-> hBOOL(hAPP(nat,bool,P_2,hAPP(nat,nat,minus_minus(nat,A_3),B_2))))) # label(fact_711_nat__diff__split__asm) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 12 (all B_1_1 ti(int,bit0(B_1_1)) = bit0(B_1_1)) # label(tsy_c_Int_OBit0_res) # label(hypothesis) # label(non_clause). [assumption]. 2.34/2.72 13 (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]. 2.34/2.72 14 (all Z_1 hAPP(int,int,plus_plus(int,zero_zero(int)),Z_1) = ti(int,Z_1)) # label(fact_206_zadd__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 15 (all X_a (ordered_ring(X_a) -> (all A_3 all E all C_1 all B_2 all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_3),E)),C_1)),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,B_2),E)),D))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,minus_minus(X_a,A_3),B_2)),E)),C_1)),D)))))) # label(fact_545_less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 16 (all X_a (linord20386208strict(X_a) -> (all C all D_2 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),D_2)))))))))) # label(fact_532_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 17 (all X_a (linordered_semidom(X_a) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),one_one(X_a))))) # label(fact_508_zero__less__one) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 18 (all B_1_1 all B_2_1 multInv(B_1_1,B_2_1) = multInv(ti(int,B_1_1),B_2_1)) # label(tsy_c_Int2_OMultInv_arg1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 19 (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]. 2.34/2.72 20 (all X_a (order(X_a) -> (all X_1 all Y_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1)) -> (ti(X_a,Y_1) != ti(X_a,X_1) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))))))) # label(fact_332_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 21 (all Ma all K all N (Ma = N | K = zero_zero(nat) <-> hAPP(nat,nat,times_times(nat,N),K) = hAPP(nat,nat,times_times(nat,Ma),K))) # label(fact_652_mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 22 (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]. 2.34/2.72 23 (all N_1 all M ((zero_zero(nat) != M -> hAPP(nat,nat,times_times(nat,M),N_1) = hAPP(nat,nat,plus_plus(nat,N_1),hAPP(nat,nat,times_times(nat,hAPP(nat,nat,minus_minus(nat,M),one_one(nat))),N_1))) & (zero_zero(nat) = M -> hAPP(nat,nat,times_times(nat,M),N_1) = zero_zero(nat)))) # label(fact_437_mult__eq__if) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 24 (all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,bit1(bit0(bit1(pls))))),P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),P_1)) -> ti(int,A_1) = inv(P_1,inv(P_1,A_1))))))) # label(fact_817_inv__inv) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 25 (all X_a (semiring_div(X_a) -> (all A_1 all B hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),B) = hAPP(X_a,X_a,div_mod(X_a,A_1),B)))) # label(fact_927_mod__add__self2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 26 (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) | zero_zero(int) = ti(int,X_1)))) # label(fact_278_int__pos__lt__two__imp__zero__or__one) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 27 (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_444_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 28 (all W pls = hAPP(int,int,times_times(int,pls),W)) # label(fact_123_mult__Pls) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 29 (all K_1 all L_1 hAPP(int,int,times_times(int,bit1(K_1)),L_1) = hAPP(int,int,plus_plus(int,bit0(hAPP(int,int,times_times(int,K_1),L_1))),L_1)) # label(fact_161_mult__Bit1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 30 (all A_3 all P_3 (hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_3),hAPP(int,int,minus_minus(int,P_3),one_one(int))),one_one(int)),P_3)) <-> hBOOL(hAPP(int,bool,zcong(A_3,hAPP(int,int,minus_minus(int,P_3),one_one(int))),P_3)))) # label(fact_390_inv__not__p__minus__1__aux) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 31 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all X_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_2),pls)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),number_number_of(X_a,X_2)),zero_zero(X_a))))))) # label(fact_247_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 32 (all N_1 all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N_1),M)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),N_1),M)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),M),zero_zero(int))))) # label(fact_406_zdvd__bounds) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 33 (all B_1_1 all X_a (group_add(X_a) -> minus_minus(X_a,B_1_1) = ti(fun(X_a,X_a),minus_minus(X_a,B_1_1)))) # label(tsy_c_Groups_Ominus__class_Ominus_0_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 34 (all I_2 all K all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),J_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),hAPP(nat,nat,minus_minus(nat,J_2),K))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,I_2),K)),J_2))))) # label(fact_690_le__diff__conv2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 35 (all X_b (number(X_b) & ring(X_b) -> (all A_1 all B all V hAPP(X_b,X_b,minus_minus(X_b,hAPP(X_b,X_b,times_times(X_b,A_1),number_number_of(X_b,V))),hAPP(X_b,X_b,times_times(X_b,B),number_number_of(X_b,V))) = hAPP(X_b,X_b,times_times(X_b,hAPP(X_b,X_b,minus_minus(X_b,A_1),B)),number_number_of(X_b,V))))) # label(fact_377_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 36 (all X_a (ordered_ring(X_a) -> (all C all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,C),A_1)),hAPP(X_a,X_a,times_times(X_a,C),B)))))))) # label(fact_475_mult__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 37 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B all C (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),B)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B),C)))))) # label(fact_451_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 38 (all X_a (ordered_semiring(X_a) -> (all C all D_2 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),D_2)))))))))) # label(fact_474_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 39 (all X_a (semiring_div(X_a) -> (all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_3),B_2)) <-> hAPP(X_a,X_a,div_mod(X_a,B_2),A_3) = zero_zero(X_a))))) # label(fact_904_dvd__eq__mod__eq__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 40 (all I_1 all J_1 all K_1 hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,minus_minus(nat,I_1),J_1)),K_1) = hAPP(nat,nat,minus_minus(nat,I_1),hAPP(nat,nat,plus_plus(nat,J_1),K_1))) # label(fact_613_diff__diff__left) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 41 (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]. 2.34/2.72 42 (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))) -> (zero_zero(nat) != N_1 -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))))) # label(fact_526_divides__rev) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 43 (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,hAPP(int,int,times_times(int,V),W))),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))))) # label(fact_145_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 44 (all X_a (linordered_idom(X_a) -> (all X_2 all Y_2 (ti(X_a,X_2) != zero_zero(X_a) | ti(X_a,Y_2) != zero_zero(X_a) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X_2),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,X_a,power_power(X_a,Y_2),number_number_of(nat,bit0(bit1(pls))))))))))) # label(fact_268_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 45 (all X_b (number(X_b) & monoid_mult(X_b) -> (all W hAPP(X_b,X_b,times_times(X_b,number_number_of(X_b,W)),number_number_of(X_b,W)) = hAPP(nat,X_b,power_power(X_b,number_number_of(X_b,W)),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_10_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 46 (all Ma all K all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),N)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,times_times(nat,Ma),K)),hAPP(nat,nat,times_times(nat,N),K))))) # label(fact_714_mult__less__cancel2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 47 (all B_1_1 all X_a (dvd(X_a) & linorder(X_a) & plus(X_a) -> plus_plus(X_a,B_1_1) = ti(fun(X_a,X_a),plus_plus(X_a,B_1_1)))) # label(tsy_c_Groups_Oplus__class_Oplus_0_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 48 (all X_a (dvd(X_a) -> (all A_1 all B all K_1 (A_1 = hAPP(X_a,X_a,times_times(X_a,B),K_1) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B),A_1)))))) # label(fact_453_dvdI) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 49 (all X_a (ordere236663937imp_le(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,C),A_1)),hAPP(X_a,X_a,plus_plus(X_a,C),B))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)))))) # label(fact_784_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 50 (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_663_less__diff__conv) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 51 (all X_a (semiring_div(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,div_mod(X_a,A_1),zero_zero(X_a))))) # label(fact_943_mod__by__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 52 (all X_a (linord581940658strict(X_a) -> (all A_3 all B_2 all C_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),A_3)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,C_1),A_3)),hAPP(X_a,X_a,times_times(X_a,C_1),B_2)))))))) # label(fact_495_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 53 (all M all N_1 (M = hAPP(nat,nat,times_times(nat,M),N_1) -> one_one(nat) = N_1 | M = zero_zero(nat))) # label(fact_717_mult__eq__self__implies__10) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 54 (all X_2 all Y_2 (X_2 != Y_2 & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_2),Y_2)) <-> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_2),X_2)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_2),Y_2)))) # label(fact_601_dvd_Oless__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 55 (all K_1 bit1(K_1) = hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,one_one(int)),K_1)),K_1)) # label(fact_153_Bit1__def) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 56 (all K_1 all M all N_1 hAPP(nat,nat,times_times(nat,K_1),hAPP(nat,nat,minus_minus(nat,M),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))) # label(fact_631_diff__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 57 (all X_a (semiring_0(X_a) & power(X_a) -> (all W ((number_number_of(nat,W) != zero_zero(nat) -> hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),number_number_of(nat,W)) = zero_zero(X_a)) & (zero_zero(nat) = number_number_of(nat,W) -> one_one(X_a) = hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),number_number_of(nat,W))))))) # label(fact_350_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 58 (all L_1 all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),L_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,minus_minus(nat,L_1),N_1)),hAPP(nat,nat,minus_minus(nat,L_1),M)))))) # label(fact_574_diff__less__mono2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 59 (all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,minus_minus(int,P_1),one_one(int)))) -> inv(P_1,A_1) != ti(int,A_1))))) # label(fact_821_inv__distinct) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 60 (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]. 2.34/2.72 61 (all B_1_1 all X_a (number(X_a) -> ti(X_a,number_number_of(X_a,B_1_1)) = number_number_of(X_a,B_1_1))) # label(tsy_c_Int_Onumber__class_Onumber__of_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 62 (all A_1 all J_1 all K_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zcong(J_1,K_1),P_1)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),multInv(P_1,J_1)),hAPP(int,int,times_times(int,A_1),multInv(P_1,K_1))),P_1))))) # label(fact_828_MultInv__zcong__prop1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 63 (all K all Ma all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,K),Ma)),hAPP(nat,nat,plus_plus(nat,K),N))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),N)))) # label(fact_655_nat__add__left__cancel__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 64 (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_566_diffs0__imp__equal) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 65 (all U all Ma all N all J_2 all I_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_2),I_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,hAPP(nat,nat,minus_minus(nat,I_2),J_2)),U)),Ma)),N)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,I_2),U)),Ma)),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,J_2),U)),N)))))) # label(fact_738_nat__less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 66 (all Z_1 hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),Z_1) = hAPP(nat,nat,plus_plus(nat,Z_1),Z_1)) # label(fact_35_nat__mult__2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 67 (all X_2 all Y_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_2),Y_2)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_2),X_2)) | X_2 = Y_2 <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_2),Y_2)))) # label(fact_602_dvd_Ole__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 68 (all X_a (ordere216010020id_add(X_a) -> (all Y_2 all X_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y_2)) -> (zero_zero(X_a) = ti(X_a,Y_2) & ti(X_a,X_2) = zero_zero(X_a) <-> zero_zero(X_a) = hAPP(X_a,X_a,plus_plus(X_a,X_2),Y_2))))))) # label(fact_800_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 69 (all M zero_zero(nat) = hAPP(nat,nat,times_times(nat,M),zero_zero(nat))) # label(fact_649_mult__0__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 70 (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_198_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 71 (all B_1_1 all B_2_1 inv(B_1_1,B_2_1) = inv(ti(int,B_1_1),B_2_1)) # label(tsy_c_WilsonRuss_Oinv_arg1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 72 (all X_a (comm_semiring_1(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C))))))) # label(fact_427_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 73 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(K)),bit0(L))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)))) # label(fact_58_rel__simps_I33_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 74 (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_212_zdvd__reduce) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 75 (all B_3 all Q_4 all R_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B_3),Q_4)),R_3)),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_3)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_4),zero_zero(int))))))) # label(fact_358_q__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 76 (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,D) = ti(X_a,C_1) <-> ti(X_a,A_3) = ti(X_a,B_2)))))) # label(fact_765_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 77 (all Z_1 ti(int,Z_1) = hAPP(int,int,plus_plus(int,Z_1),zero_zero(int))) # label(fact_207_zadd__0__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 78 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K)),bit1(L))))) # label(fact_39_rel__simps_I17_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 79 (all C all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) -> (B = C -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C))))) # label(fact_593_dvd_Oord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 80 (all M all K_1 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,plus_plus(nat,N_1),M)),K_1))))) # label(fact_687_le__add__diff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 81 (all I_1 all K_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J_1)) -> hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,plus_plus(nat,I_1),J_1)),K_1) = hAPP(nat,nat,plus_plus(nat,I_1),hAPP(nat,nat,minus_minus(nat,J_1),K_1)))) # label(fact_693_diff__add__assoc) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 82 (all B_2 all P_3 all A_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_3)) -> (hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),wset(hAPP(int,int,minus_minus(int,A_3),one_one(int)),P_3))) -> hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),wset(A_3,P_3)))))) # label(fact_844_wset__subset) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 83 (all J_1 all A_1 all P_1 all K_1 (hBOOL(hAPP(int,bool,zcong(J_1,hAPP(int,int,times_times(int,A_1),multInv(P_1,K_1))),P_1)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,J_1),K_1),hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,A_1),multInv(P_1,K_1))),K_1)),P_1)))) # label(fact_820_aux______1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 84 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,A_1),B)),zero_zero(int))) -> hAPP(int,int,div_mod(int,A_1),B) = hAPP(int,int,plus_plus(int,A_1),B)))) # label(fact_961_mod__pos__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 85 (all X_a (one(X_a) -> (all X_2 (ti(X_a,X_2) = one_one(X_a) <-> ti(X_a,X_2) = one_one(X_a))))) # label(fact_764_one__reorient) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 86 (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_978_mod__if) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 87 (all Z_1 all W (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),W)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),Z_1)) -> ti(int,W) = ti(int,Z_1)))) # label(fact_25_zle__antisym) # label(axiom) # label(non_clause). [assumption]. 2.34/2.72 88 (all M M = hAPP(nat,nat,minus_minus(nat,M),zero_zero(nat))) # label(fact_564_minus__nat_Odiff__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 89 (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_686_le__diff__conv) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 90 (all N_1 all M hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),hAPP(nat,nat,plus_plus(nat,N_1),M)))) # label(fact_675_le__add1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 91 (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),hAPP(int,int,plus_plus(int,W_1),one_one(int))),Z_2)))) # label(fact_62_add1__zle__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 92 (all X_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),P_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),standardRes(P_1,X_1)),P_1)))) # label(fact_989_StandardRes__ubound) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 93 (all N_1 all M hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,plus_plus(nat,N_1),M)),N_1) = M) # label(fact_612_diff__add__inverse) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 94 (all X_a (linord20386208strict(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),B)),zero_zero(X_a)))))))) # label(fact_499_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 95 (all B_1_1 all X_a (power(X_a) -> ti(fun(X_a,X_a),times_times(X_a,B_1_1)) = times_times(X_a,B_1_1))) # label(tsy_c_Groups_Otimes__class_Otimes_1_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 96 (all X_a (cancel_semigroup_add(X_a) -> (all A_3 all B_2 all C_1 (hAPP(X_a,X_a,plus_plus(X_a,A_3),B_2) = hAPP(X_a,X_a,plus_plus(X_a,A_3),C_1) <-> ti(X_a,B_2) = ti(X_a,C_1))))) # label(fact_759_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 97 (all X_2 (zero_zero(real) = X_2 <-> -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,times_times(real,X_2),X_2))))) # label(fact_299_not__real__square__gt__zero) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 98 (all Z_1 hAPP(nat,nat,times_times(nat,Z_1),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,nat,plus_plus(nat,Z_1),Z_1)) # label(fact_36_nat__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 99 (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,B_2),E)),D) = 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,hAPP(X_a,X_a,minus_minus(X_a,A_3),B_2)),E)),C_1) = ti(X_a,D))))) # label(fact_514_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 100 (all X_2 all P_2 ((all A_5 ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_5)) -> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,minus_minus(int,A_5),one_one(int))))) -> hBOOL(hAPP(int,bool,P_2,A_5)))) -> hBOOL(hAPP(int,bool,P_2,X_2)))) # label(fact_851_d22set__induct__old) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 101 (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_272_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 102 (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)) -> (hAPP(int,int,minus_minus(int,A_1),one_one(int)) != ti(int,X_1) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),hAPP(int,int,minus_minus(int,A_1),one_one(int)))))))) # label(fact_385_Euler_Oaux1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 103 (all X_1 all Y_1 all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),M)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,standardRes(M,X_1)),standardRes(M,Y_1)),hAPP(int,int,times_times(int,X_1),Y_1)),M)))) # label(fact_973_StandardRes__prop4) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 104 (all X_1 all Y_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y_1),X_1)) | ti(int,Y_1) = ti(int,X_1) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),Y_1)))) # label(fact_23_zless__linear) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 105 (all B_1_1 bit1(B_1_1) = ti(int,bit1(B_1_1))) # label(tsy_c_Int_OBit1_res) # label(hypothesis) # label(non_clause). [assumption]. 2.34/2.73 106 (all Z_1 all X_1 all Y_1 (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) -> (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z_1),Y_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),Z_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Z_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z_1),X_1))))) # label(fact_577_dvd_Oless__trans) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 107 (all B_1_1 quadRes(B_1_1) = ti(fun(int,bool),quadRes(B_1_1))) # label(tsy_c_Residues_OQuadRes_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 108 t = one_one(int) -> (exists X exists Y hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,X),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,Y),number_number_of(nat,bit0(bit1(pls))))) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int))) # label(fact_1__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 109 (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]. 2.34/2.73 110 (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_788_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 111 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)) -> (M != N_1 -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1))))) # label(fact_667_le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 112 (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_796_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 113 (all X_a (number_ring(X_a) & ring_char_0(X_a) -> (all X_2 all Y_2 (number_number_of(X_a,Y_2) = number_number_of(X_a,X_2) <-> ti(int,X_2) = ti(int,Y_2))))) # label(fact_82_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 114 (all X_a (number_ring(X_a) -> (all W number_number_of(X_a,bit0(W)) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,plus_plus(X_a,one_one(X_a)),one_one(X_a))),number_number_of(X_a,W))))) # label(fact_162_double__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 115 (all X_a (number_ring(X_a) -> (all V all W hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,V)),number_number_of(X_a,W)) = number_number_of(X_a,hAPP(int,int,plus_plus(int,V),W))))) # label(fact_149_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 116 (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]. 2.34/2.73 117 (all B_1_1 all X_a (cancel_semigroup_add(X_a) -> plus_plus(X_a,B_1_1) = ti(fun(X_a,X_a),plus_plus(X_a,B_1_1)))) # label(tsy_c_Groups_Oplus__class_Oplus_2_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 118 (all A_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A_3)) <-> A_3 = zero_zero(nat))) # label(fact_868_gcd__lcm__complete__lattice__nat_Otop__unique) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 119 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)))) # label(fact_666_less__imp__le__nat) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 120 (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_40_less__eq__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 121 (all A_1 all B hAPP(nat,int,power_power(int,hAPP(int,int,plus_plus(int,A_1),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_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_7_zadd__power2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 122 (all A_3 all B_2 all N (zero_zero(nat) != N -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,A_3),N)),hAPP(nat,nat,power_power(nat,B_2),N))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_3),B_2))))) # label(fact_878_pow__divides__eq__nat) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 123 (all A_3 all B_2 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma)) -> (hBOOL(hAPP(int,bool,zcong(A_3,B_2),Ma)) <-> hAPP(int,int,div_mod(int,B_2),Ma) = hAPP(int,int,div_mod(int,A_3),Ma)))) # label(fact_954_zcong__zmod__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 124 (all M all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),X_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),M)) -> -hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),M))))) # label(fact_405_zcong__not__zero) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 125 (all X_a (real_normed_algebra(X_a) -> (all B zero_zero(X_a) = hAPP(X_a,X_a,times_times(X_a,zero_zero(X_a)),B)))) # label(fact_887_mult_Ozero__left) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 126 (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_880_divides__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 127 (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),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,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_4),B_3)),C)))))) # label(fact_947_mod__diff__cong) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 128 (all X_a (real_normed_algebra(X_a) -> (all Xa all X_1 all Y_1 hAPP(X_a,X_a,times_times(X_a,Xa),hAPP(X_a,X_a,minus_minus(X_a,X_1),Y_1)) = hAPP(X_a,X_a,minus_minus(X_a,hAPP(X_a,X_a,times_times(X_a,Xa),X_1)),hAPP(X_a,X_a,times_times(X_a,Xa),Y_1))))) # label(fact_893_mult__right_Odiff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 129 (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,N_1) = ti(int,M)))))) # label(fact_191_zdvd__antisym__nonneg) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 130 (all X_a (order(X_a) -> ord_less_eq(X_a) = ti(fun(X_a,fun(X_a,bool)),ord_less_eq(X_a)))) # label(tsy_c_Orderings_Oord__class_Oless__eq_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 131 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B)) -> ti(int,A_1) = hAPP(int,int,div_mod(int,A_1),B)))) # label(fact_957_mod__pos__pos__trivial) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 132 (all M all N_1 hAPP(nat,nat,plus_plus(nat,M),N_1) = hAPP(nat,nat,plus_plus(nat,N_1),M)) # label(fact_606_nat__add__commute) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 133 (all X_1 all M all Y_1 (hAPP(int,int,div_mod(int,X_1),M) = hAPP(int,int,div_mod(int,Y_1),M) -> hBOOL(hAPP(int,bool,zcong(X_1,Y_1),M)))) # label(fact_912_Residues_Oaux) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 134 (all A_3 all B_2 all C_1 (zero_zero(real) != C_1 -> (hAPP(real,real,times_times(real,B_2),C_1) = hAPP(real,real,times_times(real,A_3),C_1) <-> A_3 = B_2))) # label(fact_462_real__mult__right__cancel) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 135 (all N_1 N_1 = hAPP(nat,nat,times_times(nat,N_1),one_one(nat))) # label(fact_705_nat__mult__1__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 136 (all Z1 all Z2 all W hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,Z1),Z2)),W) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,Z1),W)),hAPP(int,int,times_times(int,Z2),W))) # label(fact_132_zadd__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 137 (all Ma all X_2 (zero_zero(int) = standardRes(Ma,X_2) <-> hBOOL(hAPP(int,bool,zcong(X_2,zero_zero(int)),Ma)))) # label(fact_986_StandardRes__eq__zcong) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 138 (all K_1 all L_1 all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),L_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,I_1),K_1)),hAPP(nat,nat,plus_plus(nat,J_1),L_1)))))) # label(fact_681_add__le__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 139 (all M all N_1 all I_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),I_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,power_power(nat,I_1),M)),hAPP(nat,nat,power_power(nat,I_1),N_1))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1))))) # label(fact_289_nat__power__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 140 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),bit1(K))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),K)))) # label(fact_307_rel__simps_I26_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 141 (all A_3 all B_2 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Ma),hAPP(int,int,minus_minus(int,A_3),B_2))) <-> hBOOL(hAPP(int,bool,zcong(A_3,B_2),Ma)))) # label(fact_383_zcong__def) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 142 -hBOOL(hAPP(int,bool,quadRes(hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int))),number_number_of(int,min))) -> legendre(number_number_of(int,min),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int))) != one_one(int) # label(fact_367__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_N) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 143 (all Q all P (-hBOOL(Q) | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q)) | -hBOOL(P))) # label(help_fconj_1_1_U) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 144 (all X_a (number_ring(X_a) -> (all V all W hAPP(X_a,X_a,times_times(X_a,number_number_of(X_a,V)),number_number_of(X_a,W)) = number_number_of(X_a,hAPP(int,int,times_times(int,V),W))))) # label(fact_147_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 145 (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_eq(int),zero_zero(int)),R_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_1),one_one(int))))))) # label(fact_281_self__quotient__aux2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 146 (all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A_1)) -> -(-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A_1))))) # label(fact_578_dvd_Oless__asym_H) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 147 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K)),min)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),min)))) # label(fact_310_rel__simps_I28_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 148 (all A_1 all B all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M),B)) -> hAPP(int,int,div_mod(int,A_1),M) = hAPP(int,int,div_mod(int,hAPP(int,int,div_mod(int,A_1),B)),M)))) # label(fact_953_zmod__zdvd__zmod) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 149 (all Ma all N (one_one(nat) = N & Ma = one_one(nat) <-> hAPP(nat,nat,times_times(nat,Ma),N) = one_one(nat))) # label(fact_704_nat__mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 150 (all K all Ma all N (Ma = N | zero_zero(nat) = K <-> hAPP(nat,nat,times_times(nat,K),Ma) = hAPP(nat,nat,times_times(nat,K),N))) # label(fact_651_mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 151 (all V_1 all K_1 all V ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls)) -> hAPP(nat,nat,times_times(nat,number_number_of(nat,V)),hAPP(nat,nat,times_times(nat,number_number_of(nat,V_1)),K_1)) = zero_zero(nat)) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls)) -> hAPP(nat,nat,times_times(nat,number_number_of(nat,hAPP(int,int,times_times(int,V),V_1))),K_1) = hAPP(nat,nat,times_times(nat,number_number_of(nat,V)),hAPP(nat,nat,times_times(nat,number_number_of(nat,V_1)),K_1))))) # label(fact_330_nat__number__of__mult__left) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 152 (all P_1 all M ((zero_zero(nat) != M -> hAPP(nat,nat,times_times(nat,P_1),hAPP(nat,nat,power_power(nat,P_1),hAPP(nat,nat,minus_minus(nat,M),one_one(nat)))) = hAPP(nat,nat,power_power(nat,P_1),M)) & (M = zero_zero(nat) -> hAPP(nat,nat,power_power(nat,P_1),M) = one_one(nat)))) # label(fact_438_power__eq__if) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 153 (all M all K_1 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,M),K_1)),N_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N_1)))) # label(fact_682_add__leD2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 154 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all X_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X_2),pls)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,X_2)),zero_zero(X_a))))))) # label(fact_249_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 155 (all Ma all D ((exists Q_2 hAPP(nat,nat,times_times(nat,D),Q_2) = Ma) <-> zero_zero(nat) = hAPP(nat,nat,div_mod(nat,Ma),D))) # label(fact_976_mod__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 156 (all N (zero_zero(nat) != N <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)))) # label(fact_635_neq0__conv) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 157 (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_142_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 158 (all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),P_1)) -> (hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),A_1),one_one(int)),P_1)) -> one_one(int) = ti(int,A_1) | hAPP(int,int,minus_minus(int,P_1),one_one(int)) = ti(int,A_1)))))) # label(fact_394_zcong__square__zless) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 159 (all A_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),one_one(nat)),A_1))) # label(fact_873_gcd__lcm__complete__lattice__nat_Obot__least) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 160 (all X_a (linordered_idom(X_a) -> (all X_2 all Y_2 (zero_zero(X_a) = ti(X_a,Y_2) & zero_zero(X_a) = ti(X_a,X_2) <-> zero_zero(X_a) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X_2),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,X_a,power_power(X_a,Y_2),number_number_of(nat,bit0(bit1(pls))))))))) # label(fact_258_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 161 (all X_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zcong(X_1,number_number_of(int,min)),P_1)) -> -hBOOL(hAPP(int,bool,zcong(X_1,one_one(int)),P_1))))) # label(fact_364_zcong__neg__1__impl__ne__1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 162 (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]. 2.34/2.73 163 (all M zero_zero(nat) = hAPP(nat,nat,minus_minus(nat,M),M)) # label(fact_565_diff__self__eq__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 164 (all N_1 -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),zero_zero(nat)))) # label(fact_557_less__zeroE) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 165 (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)))))) <-> zero_zero(X_a) != ti(X_a,A_3))))) # label(fact_257_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 166 (all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),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))))) # label(fact_595_dvd_Ole__neq__trans) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 167 (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_528_mult__le__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 168 (all X_a (semiri456707255roduct(X_a) -> (all C_1 all D all A_3 all B_2 (ti(X_a,D) != ti(X_a,C_1) & ti(X_a,B_2) != ti(X_a,A_3) <-> hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_3),D)),hAPP(X_a,X_a,times_times(X_a,B_2),C_1)) != hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_3),C_1)),hAPP(X_a,X_a,times_times(X_a,B_2),D)))))) # label(fact_111_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 169 (all Z_1 hAPP(int,int,times_times(int,Z_1),one_one(int)) = ti(int,Z_1)) # label(fact_129_zmult__1__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 170 (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_920_zmod__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 171 (all X_a (comm_semiring_1(X_a) -> (all A_1 hAPP(X_a,X_a,times_times(X_a,A_1),zero_zero(X_a)) = zero_zero(X_a)))) # label(fact_199_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 172 (all X_a (linordered_semidom(X_a) -> (all N_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),one_one(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),hAPP(nat,X_a,power_power(X_a,A_1),N_1))),hAPP(nat,X_a,power_power(X_a,A_1),N_1)))))))) # label(fact_232_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 173 (all X_a (monoid_mult(X_a) -> (all A_1 all M all N_1 hAPP(nat,X_a,power_power(X_a,A_1),hAPP(nat,nat,plus_plus(nat,M),N_1)) = hAPP(X_a,X_a,times_times(X_a,hAPP(nat,X_a,power_power(X_a,A_1),M)),hAPP(nat,X_a,power_power(X_a,A_1),N_1))))) # label(fact_261_power__add) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 174 (all K_1 all L_1 hAPP(int,int,minus_minus(int,bit1(K_1)),bit1(L_1)) = bit0(hAPP(int,int,minus_minus(int,K_1),L_1))) # label(fact_379_diff__bin__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 175 (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_716_nat__0__less__mult__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 176 (all A_3 all B_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),B_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),A_3)) -> hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),d22set(A_3)))))) # label(fact_860_d22set__mem) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 177 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all X_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X_2),bit1(pls))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,X_2)),one_one(X_a))))))) # label(fact_106_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 178 (all X_a (comm_semiring_1(X_a) -> (all A_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),zero_zero(X_a)))))) # label(fact_423_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 179 (all K_1 all L_1 bit0(K_1) != bit1(L_1)) # label(fact_119_rel__simps_I49_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 180 (all K_1 min != bit0(K_1)) # label(fact_295_rel__simps_I45_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 181 (all N_1 all M hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),hAPP(nat,nat,plus_plus(nat,M),N_1)))) # label(fact_674_le__add2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 182 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,div_mod(nat,M),N_1)),N_1)))) # label(fact_975_mod__less__divisor) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 183 (all X_1 all Y_1 all Z_1 hAPP(nat,nat,plus_plus(nat,X_1),hAPP(nat,nat,plus_plus(nat,Y_1),Z_1)) = hAPP(nat,nat,plus_plus(nat,Y_1),hAPP(nat,nat,plus_plus(nat,X_1),Z_1))) # label(fact_607_nat__add__left__commute) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 184 (all X_1 all Y_1 (Y_1 = X_1 -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)))) # label(fact_598_dvd_Oeq__refl) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 185 (all K_1 all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,times_times(nat,I_1),K_1)),hAPP(nat,nat,times_times(nat,J_1),K_1)))))) # label(fact_713_mult__less__mono1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 186 (all P_2 all K all I_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),I_2)) -> (hBOOL(hAPP(int,bool,P_2,K)) -> ((all I (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),I)) -> (hBOOL(hAPP(int,bool,P_2,I)) -> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,plus_plus(int,I),one_one(int))))))) -> hBOOL(hAPP(int,bool,P_2,I_2)))))) # label(fact_852_int__ge__induct) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 187 (all X_a (linord581940658strict(X_a) -> (all C all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),C)))))))) # label(fact_489_mult__strict__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 188 (all X_a (semiring_div(X_a) -> (all A_1 all C all B hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,div_mod(X_a,A_1),C)),B)),C)))) # label(fact_933_zmod__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 189 (all X_a (ring_11004092258visors(X_a) -> (all A_3 (ti(X_a,A_3) = zero_zero(X_a) <-> zero_zero(X_a) = hAPP(nat,X_a,power_power(X_a,A_3),number_number_of(nat,bit0(bit1(pls)))))))) # label(fact_252_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 190 (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]. 2.34/2.73 191 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),N_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))))) # label(fact_877_dvd__pos__nat) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 192 (all B_1_1 bit0(ti(int,B_1_1)) = bit0(B_1_1)) # label(tsy_c_Int_OBit0_arg1) # label(hypothesis) # label(non_clause). [assumption]. 2.34/2.73 193 (all P_2 all K all I_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),I_2)) -> (hBOOL(hAPP(int,bool,P_2,hAPP(int,int,plus_plus(int,K),one_one(int)))) -> ((all I (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),I)) -> (hBOOL(hAPP(int,bool,P_2,I)) -> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,plus_plus(int,I),one_one(int))))))) -> hBOOL(hAPP(int,bool,P_2,I_2)))))) # label(fact_853_int__gr__induct) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 194 (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]. 2.34/2.73 195 (all X_a (linordered_semiring(X_a) -> (all A_1 all C all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),C))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))))))) # label(fact_534_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 196 (all X_a (monoid_mult(X_a) -> (all A_1 all M all N_1 hAPP(nat,X_a,power_power(X_a,hAPP(nat,X_a,power_power(X_a,A_1),M)),N_1) = hAPP(nat,X_a,power_power(X_a,A_1),hAPP(nat,nat,times_times(nat,M),N_1))))) # label(fact_263_power__mult) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 197 (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),bit1(K)),pls)))) # label(fact_92_rel__simps_I12_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 198 (all W hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),W))) # label(fact_20_zle__refl) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 199 (all K_1 all L_1 bit0(hAPP(int,int,times_times(int,K_1),L_1)) = hAPP(int,int,times_times(int,bit0(K_1)),L_1)) # label(fact_124_mult__Bit0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 200 (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]. 2.34/2.73 201 (all P_5 all P_2 all X_2 ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2)) -> (hBOOL(P_2) <-> hBOOL(P_5))) -> (hBOOL(P_5) & 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_eq(int),zero_zero(int)),X_2)) & hBOOL(P_2)))) # label(fact_742_conj__le__cong) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 202 (all X_1 all Y_1 (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) -> -(hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1))))) # label(fact_584_dvd_Oless__imp__not__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 203 (all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,minus_minus(int,P_1),one_one(int)))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),inv(P_1,A_1))))))) # label(fact_824_inv__g__1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 204 (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]. 2.34/2.73 205 (all X_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_1)) -> hBOOL(hAPP(int,bool,zcong(multInv(P_1,multInv(P_1,X_1)),hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,X_1),multInv(P_1,X_1))),multInv(P_1,multInv(P_1,X_1)))),P_1)))))) # label(fact_838_Int2_Oaux____1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 206 (all B all Q_1 all R_1 all B_3 all Q_4 all R_3 (hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B_3),Q_4)),R_3) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),Q_1)),R_1) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B_3),Q_4)),R_3))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_3),B_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_3),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_1),Q_4))))))))) # label(fact_356_zdiv__mono2__lemma) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 207 (all X_a (semiring_div(X_a) -> (all C all A_1 all B hAPP(X_a,X_a,times_times(X_a,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,times_times(X_a,C),A_1)),hAPP(X_a,X_a,times_times(X_a,C),B))))) # label(fact_938_mod__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 208 (all X_a (comm_semiring_1(X_a) -> (all A_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),one_one(X_a)),A_1))))) # label(fact_460_one__dvd) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 209 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(K1)),bit0(K2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)))) # label(fact_57_less__eq__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 210 (all X_1 all Y_1 all Z_1 hAPP(nat,int,power_power(int,X_1),hAPP(nat,nat,times_times(nat,Y_1),Z_1)) = hAPP(nat,int,power_power(int,hAPP(nat,int,power_power(int,X_1),Y_1)),Z_1)) # label(fact_28_zpower__zpower) # label(axiom) # label(non_clause). [assumption]. 2.34/2.73 211 (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,A_1),hAPP(X_a,X_a,div_mod(X_a,B),C))),C)))) # label(fact_935_mod__mult__right__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 212 (all X_a (linordered_semidom(X_a) -> (all A_1 all N_1 all N_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),N_3)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),one_one(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N_3)),hAPP(nat,X_a,power_power(X_a,A_1),N_1))))))))) # label(fact_215_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 213 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K)),min)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),min)))) # label(fact_318_rel__simps_I11_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 214 (all N_1 ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),N_1),zero_zero(int))) -> one_one(int) = zfact(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)))) # label(fact_754_zfact_Osimps) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 215 (all X_a (linordered_semidom(X_a) -> (all N_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,power_power(X_a,A_1),N_1))))))) # label(fact_208_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 216 (all L_1 all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,minus_minus(nat,L_1),N_1)),hAPP(nat,nat,minus_minus(nat,L_1),M))))) # label(fact_622_diff__le__mono2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 217 (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]. 2.34/2.74 218 (all X_a (semiring_div(X_a) -> (all A_1 all C all B hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,div_mod(X_a,A_1),B)),C) = 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))))) # label(fact_939_mod__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 219 (all X_a (monoid_mult(X_a) -> (all A_1 all 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)))) = 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_103_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 220 (all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> wset(hAPP(int,int,minus_minus(int,P_3),number_number_of(int,bit0(bit1(pls)))),P_3) = d22set(hAPP(int,int,minus_minus(int,P_3),number_number_of(int,bit0(bit1(pls))))))) # label(fact_855_d22set__eq__wset) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 221 (all X_1 all Y_1 all Z_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Z_1)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),Y_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,Z_1),X_1)),hAPP(real,real,times_times(real,Z_1),Y_1)))))) # label(fact_523_real__mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 222 (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),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_502_mult__less__cancel__left__disj) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 223 (all A_3 all B_2 all Ma (hBOOL(hAPP(int,bool,zcong(A_3,B_2),Ma)) <-> hBOOL(hAPP(int,bool,zcong(B_2,A_3),Ma)))) # label(fact_326_zcong__sym) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 224 (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),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(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)))))) # label(fact_724_nat__less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 225 (all X_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_1)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,X_1),multInv(P_1,X_1))),multInv(P_1,multInv(P_1,X_1))),X_1),P_1)))))) # label(fact_837_Int2_Oaux____2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 226 (all X_a (linordered_semidom(X_a) -> (all M all N_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,A_1),M)),hAPP(nat,X_a,power_power(X_a,A_1),N_1))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1))))))) # label(fact_178_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 227 (all K_1 all L_1 all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_1),L_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,I_1),K_1)),hAPP(nat,nat,plus_plus(nat,J_1),L_1)))))) # label(fact_659_add__less__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 228 (all X_a (linordered_semidom(X_a) -> (all N_1 all M (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),M)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),N_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(X_a,X_a,times_times(X_a,M),N_1)))))))) # label(fact_510_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 229 (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_331_mult__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 230 (all A_1 all B hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,A_1),B)),hAPP(int,int,minus_minus(int,A_1),B)) = hAPP(int,int,minus_minus(int,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,B),number_number_of(nat,bit0(bit1(pls)))))) # label(fact_395_zspecial__product) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 231 (all B hAPP(int,int,div_mod(int,zero_zero(int)),B) = zero_zero(int)) # label(fact_922_zmod__zero) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 232 (all A_1 all B hAPP(nat,int,power_power(int,hAPP(int,int,minus_minus(int,A_1),B)),number_number_of(nat,bit1(bit1(pls)))) = hAPP(int,int,minus_minus(int,hAPP(int,int,plus_plus(int,hAPP(int,int,minus_minus(int,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit1(bit1(pls))))),hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,number_number_of(int,bit1(bit1(pls)))),hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls)))))),B))),hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,number_number_of(int,bit1(bit1(pls)))),A_1)),hAPP(nat,int,power_power(int,B),number_number_of(nat,bit0(bit1(pls))))))),hAPP(nat,int,power_power(int,B),number_number_of(nat,bit1(bit1(pls)))))) # label(fact_403_zdiff__power3) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 233 (all I_1 all J_1 all K_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,I_1),J_1)),K_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),K_1)))) # label(fact_661_add__lessD1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 234 (all U all Ma all N all J_2 all I_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_2),I_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,I_2),U)),Ma)),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,J_2),U)),N))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,hAPP(nat,nat,minus_minus(nat,I_2),J_2)),U)),Ma)),N))))) # label(fact_732_nat__le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 235 (all M all N_1 (hAPP(int,int,times_times(int,M),N_1) = one_one(int) -> number_number_of(int,min) = ti(int,M) | one_one(int) = ti(int,M))) # label(fact_319_pos__zmult__eq__1__iff__lemma) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 236 (all M all N_1 all K_1 hAPP(nat,nat,times_times(nat,M),hAPP(nat,nat,times_times(nat,N_1),K_1)) = hAPP(nat,nat,times_times(nat,hAPP(nat,nat,times_times(nat,M),N_1)),K_1)) # label(fact_628_nat__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 237 (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,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)))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,W))) -> hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit1(pls)))),hAPP(int,int,div_mod(int,number_number_of(int,V)),number_number_of(int,W)))),one_one(int)) = hAPP(int,int,div_mod(int,number_number_of(int,bit1(V))),number_number_of(int,bit0(W)))))) # label(fact_897_zmod__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 238 (all N_1 hAPP(nat,nat,plus_plus(nat,zero_zero(nat)),N_1) = N_1) # label(fact_642_plus__nat_Oadd__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 239 (all X_a (ordere216010020id_add(X_a) -> (all B all C all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)))))))) # label(fact_801_add__increasing) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 240 (all M all N_1 all K_1 hAPP(nat,nat,times_times(nat,hAPP(nat,nat,plus_plus(nat,M),N_1)),K_1) = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,M),K_1)),hAPP(nat,nat,times_times(nat,N_1),K_1))) # label(fact_696_add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 241 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,minus_minus(nat,M),N_1)),M))))) # label(fact_637_diff__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 242 (all X_a (semiring_div(X_a) & number(X_a) -> (all X_2 all Y_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),number_number_of(X_a,X_2)),number_number_of(X_a,Y_2))) <-> hAPP(X_a,X_a,div_mod(X_a,number_number_of(X_a,Y_2)),number_number_of(X_a,X_2)) = zero_zero(X_a))))) # label(fact_951_dvd__eq__mod__eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 243 (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,B) = ti(int,A_1))))))) # label(fact_351_zcong__zless__imp__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 244 (all X_a (cancel146912293up_add(X_a) -> (all A_1 all B all C (hAPP(X_a,X_a,plus_plus(X_a,A_1),C) = hAPP(X_a,X_a,plus_plus(X_a,A_1),B) -> ti(X_a,C) = ti(X_a,B))))) # label(fact_762_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 245 (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,times_times(X_a,B),C))),B) = hAPP(X_a,X_a,div_mod(X_a,A_1),B)))) # label(fact_903_mod__mult__self2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 246 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),zero_zero(X_a)))))))) # label(fact_807_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 247 (all X_a (dvd(X_a) -> dvd_dvd(X_a) = ti(fun(X_a,fun(X_a,bool)),dvd_dvd(X_a)))) # label(tsy_c_Rings_Odvd__class_Odvd_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 248 (all X_a (plus(X_a) & linorder(X_a) & dvd(X_a) -> (all D all Sa exists Z all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Z)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),hAPP(X_a,X_a,plus_plus(X_a,X),Sa))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),hAPP(X_a,X_a,plus_plus(X_a,X),Sa)))))))) # label(fact_865_minf_I9_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 249 (all X_1 all M all Y_1 hAPP(int,int,div_mod(int,hAPP(int,int,minus_minus(int,X_1),Y_1)),M) = hAPP(int,int,div_mod(int,hAPP(int,int,minus_minus(int,hAPP(int,int,div_mod(int,X_1),M)),Y_1)),M)) # label(fact_946_zdiff__zmod__left) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 250 (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_685_diff__diff__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 251 (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_380_diff__bin__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 252 (all A_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),hAPP(int,int,minus_minus(int,B),one_one(int)))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B)))) # label(fact_752_norR__mem__unique__aux) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 253 (all X_a (number_ring(X_a) -> hAPP(X_a,X_a,plus_plus(X_a,one_one(X_a)),one_one(X_a)) = number_number_of(X_a,bit0(bit1(pls))))) # label(fact_18_one__add__one__is__two) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 254 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx all Ry hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,Lx),Ly)),hAPP(X_a,X_a,times_times(X_a,Rx),Ry)) = hAPP(X_a,X_a,times_times(X_a,Lx),hAPP(X_a,X_a,times_times(X_a,Ly),hAPP(X_a,X_a,times_times(X_a,Rx),Ry)))))) # label(fact_68_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 255 (all U all Ma all N all I_2 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,hAPP(nat,nat,minus_minus(nat,J_2),I_2)),U)),N))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,I_2),U)),Ma)),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,J_2),U)),N)))))) # label(fact_735_nat__le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 256 (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]. 2.34/2.74 257 (all V_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),V_2)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),number_number_of(nat,V_2))))) # label(fact_323_less__0__number__of) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 258 (all X_2 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X_2)) | zero_zero(nat) = N <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,power_power(nat,X_2),N))))) # label(fact_286_nat__zero__less__power__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 259 (all X_1 all M hBOOL(hAPP(int,bool,zcong(X_1,standardRes(M,X_1)),M))) # label(fact_988_StandardRes__prop1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 260 (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_78_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 261 (all X_a (semiring_div(X_a) -> (all N_1 all K_1 all M (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_1),M)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_1),N_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_1),hAPP(X_a,X_a,div_mod(X_a,M),N_1)))))))) # label(fact_924_dvd__mod) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 262 (all X_a (semiring_div(X_a) -> (all K_1 all M all N_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_1),hAPP(X_a,X_a,div_mod(X_a,M),N_1))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_1),N_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_1),M))))))) # label(fact_923_dvd__mod__imp__dvd) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 263 (all X_a (ordere236663937imp_le(X_a) -> (all A_1 all C all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)),hAPP(X_a,X_a,plus_plus(X_a,B),C))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)))))) # label(fact_776_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 264 (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]. 2.34/2.74 265 (all M all K_1 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,M),K_1)),N_1)) -> -(hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N_1))))) # label(fact_684_add__leE) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 266 (all D_2 all C all A_1 all B all M (hBOOL(hAPP(int,bool,zcong(A_1,B),M)) -> (ti(int,B) = ti(int,C) -> (hBOOL(hAPP(int,bool,zcong(C,D_2),M)) -> hBOOL(hAPP(int,bool,zcong(A_1,D_2),M)))))) # label(fact_391_zcong__eq__trans) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 267 (all A_1 all N_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),hAPP(nat,int,power_power(int,A_1),N_1))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),A_1))))) # label(fact_220_zprime__zdvd__power) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 268 (all X_1 all Y_1 hAPP(nat,nat,times_times(nat,hAPP(nat,nat,plus_plus(nat,X_1),Y_1)),hAPP(nat,nat,minus_minus(nat,X_1),Y_1)) = hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,power_power(nat,X_1),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,nat,power_power(nat,Y_1),number_number_of(nat,bit0(bit1(pls)))))) # label(fact_439_diff__square) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 269 (all X_a (ordere453448008miring(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),B)),zero_zero(X_a)))))))) # label(fact_481_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 270 (all X_a (semiring_div(X_a) -> (all A_1 all B hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,div_mod(X_a,A_1),B)),B) = hAPP(X_a,X_a,div_mod(X_a,A_1),B)))) # label(fact_918_mod__mod__trivial) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 271 (all I_1 all K_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J_1)) -> hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,minus_minus(nat,J_1),K_1)),I_1) = hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,plus_plus(nat,J_1),I_1)),K_1))) # label(fact_695_diff__add__assoc2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 272 (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),hAPP(nat,nat,times_times(nat,I_1),K_1)),hAPP(nat,nat,times_times(nat,J_1),K_1))))) # label(fact_701_mult__le__mono1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 273 (all X_2 all Y_2 all Z_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Z_2)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),Y_2)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,Z_2),X_2)),hAPP(real,real,times_times(real,Z_2),Y_2)))))) # label(fact_521_real__mult__le__cancel__iff2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 274 (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_72_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 275 (all M all N_1 (hAPP(nat,nat,plus_plus(nat,M),N_1) = M -> N_1 = zero_zero(nat))) # label(fact_639_add__eq__self__zero) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 276 (all P_2 all I_2 all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_2),K)) -> (hBOOL(hAPP(int,bool,P_2,hAPP(int,int,minus_minus(int,K),one_one(int)))) -> ((all I (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I),K)) -> (hBOOL(hAPP(int,bool,P_2,I)) -> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,minus_minus(int,I),one_one(int))))))) -> hBOOL(hAPP(int,bool,P_2,I_2)))))) # label(fact_850_int__less__induct) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 277 (all B all Q_4 all R_3 all Q_1 all R_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),Q_4)),R_3)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),Q_1)),R_1))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_1),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_3)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_1),Q_4))))))) # label(fact_355_unique__quotient__lemma__neg) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 278 (all X_1 all M all Y_1 hAPP(int,int,div_mod(int,hAPP(nat,int,power_power(int,hAPP(int,int,div_mod(int,X_1),M)),Y_1)),M) = hAPP(int,int,div_mod(int,hAPP(nat,int,power_power(int,X_1),Y_1)),M)) # label(fact_917_zpower__zmod) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 279 (all X_a (dvd(X_a) & comm_ring(X_a) -> (all Ta all D all D_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),D_3)) -> (all X all K_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,minus_minus(X_a,X),hAPP(X_a,X_a,times_times(X_a,K_2),D_3))),Ta))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),hAPP(X_a,X_a,plus_plus(X_a,X),Ta))))))))) # label(fact_746_inf__period_I3_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 280 (all L (bit0(L) = pls <-> pls = ti(int,L))) # label(fact_121_rel__simps_I38_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 281 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 all C (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)))))))) # label(fact_802_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 282 (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_402_zdiff__power2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 283 (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_433_real__mult__commute) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 284 (all K_1 all A_1 all B all M (hBOOL(hAPP(int,bool,zcong(A_1,B),M)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,K_1),A_1),hAPP(int,int,times_times(int,K_1),B)),M)))) # label(fact_338_zcong__scalar2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 285 (all K_1 all A_1 all B all M (hBOOL(hAPP(int,bool,zcong(A_1,B),M)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),K_1),hAPP(int,int,times_times(int,B),K_1)),M)))) # label(fact_339_zcong__scalar) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 286 (all X_1 all Y_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) -> Y_1 = X_1))) # label(fact_591_dvd_Oantisym) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 287 (all X_a (linordered_semidom(X_a) -> (all B all C all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)))))))) # label(fact_505_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 288 (all X_2 all P_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_3),X_2)) <-> hBOOL(hAPP(int,bool,zcong(X_2,zero_zero(int)),P_3)))) # label(fact_407_zcong__eq__zdvd__prop) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 289 (all X_a (ring_n68954251visors(X_a) -> (all A_3 all B_2 (zero_zero(X_a) = hAPP(X_a,X_a,times_times(X_a,A_3),B_2) <-> zero_zero(X_a) = ti(X_a,A_3) | ti(X_a,B_2) = zero_zero(X_a))))) # label(fact_442_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 290 (all X_a (comm_semiring_1(X_a) -> (all M all A_1 hAPP(X_a,X_a,plus_plus(X_a,M),hAPP(X_a,X_a,times_times(X_a,A_1),M)) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),one_one(X_a))),M)))) # label(fact_141_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 291 (all V_2 (zero_zero(nat) = number_number_of(nat,V_2) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),V_2),pls)))) # label(fact_325_eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 292 (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_889_mult_Oadd__left) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 293 (all X_a (ordere236663937imp_le(X_a) -> (all A_3 all C_1 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_3),C_1)),hAPP(X_a,X_a,plus_plus(X_a,B_2),C_1))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)))))) # label(fact_778_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 294 (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,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))) & (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)))))) # label(fact_52_add__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 295 (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_914_zcong__zmod) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 296 (all X_2 all Y_2 (Y_2 = X_2 <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_2),Y_2)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_2),X_2)))) # label(fact_603_dvd_Oeq__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 297 (all Ma all K all N (hAPP(nat,nat,plus_plus(nat,Ma),K) = hAPP(nat,nat,plus_plus(nat,N),K) <-> Ma = N)) # label(fact_610_nat__add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 298 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,A_1),B)))))))) # label(fact_806_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 299 (all Q_1 all B all R_1 all C (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_1),zero_zero(int))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,B),C)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),hAPP(int,int,div_mod(int,Q_1),C))),R_1))))))) # label(fact_966_zmult2__lemma__aux1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 300 (all M all N_1 (N_1 = M -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)))) # label(fact_618_eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 301 (all V_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),V_2),pls)) <-> number_number_of(nat,V_2) = zero_zero(nat))) # label(fact_324_eq__number__of__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 302 (all C all A_1 all B (A_1 = B -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),C)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C))))) # label(fact_594_dvd_Oord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 303 (all Q_1 all B all R_1 all C (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_1),zero_zero(int))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),hAPP(int,int,div_mod(int,Q_1),C))),R_1)),zero_zero(int))))))) # label(fact_967_zmult2__lemma__aux2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 304 (all V all W hAPP(int,int,times_times(int,number_number_of(int,V)),number_number_of(int,W)) = number_number_of(int,hAPP(int,int,times_times(int,V),W))) # label(fact_131_times__numeral__code_I5_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 305 (all A_1 all M all B hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),M),hAPP(int,int,times_times(int,B),M)),M))) # label(fact_340_zcong__zmult__self) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 306 (all X_a (comm_semiring_1(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,times_times(X_a,B),C))))))) # label(fact_456_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 307 (all P_5 all P_2 all X_2 ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2)) -> (hBOOL(P_5) <-> hBOOL(P_2))) -> ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2)) -> hBOOL(P_5)) <-> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2)) -> hBOOL(P_2))))) # label(fact_741_imp__le__cong) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 308 (all X_a (mult_zero(X_a) -> (all A_1 hAPP(X_a,X_a,times_times(X_a,A_1),zero_zero(X_a)) = zero_zero(X_a)))) # label(fact_443_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 309 (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),C) = hAPP(X_a,X_a,div_mod(X_a,B_3),C) -> hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_4),B_3)),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),C)))))) # label(fact_934_mod__add__cong) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 310 (all X_a (idom(X_a) -> (all C_1 all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,times_times(X_a,C_1),A_3)),hAPP(X_a,X_a,times_times(X_a,C_1),B_2))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_3),B_2)) | zero_zero(X_a) = ti(X_a,C_1))))) # label(fact_512_dvd__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 311 (all X_a (linordered_semidom(X_a) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),one_one(X_a))))) # label(fact_506_zero__le__one) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 312 (all C all A_1 all B (A_1 = B -> (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),C),B)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),C)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),C),A_1))))) # label(fact_589_dvd_Oord__eq__less__trans) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 313 (all L (min = ti(int,L) <-> bit1(L) = min)) # label(fact_291_rel__simps_I43_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 314 (all X_2 all Y_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_2),X_2)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_2),Y_2)) <-> X_2 = Y_2)) # label(fact_431_divides__antisym) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 315 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all X_2 all Y_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_2),Y_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),number_number_of(X_a,X_2)),number_number_of(X_a,Y_2))))))) # label(fact_30_less__number__of) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 316 (all X_a (ab_semigroup_add(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),C) = hAPP(X_a,X_a,plus_plus(X_a,A_1),hAPP(X_a,X_a,plus_plus(X_a,B),C))))) # label(fact_758_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 317 (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]. 2.34/2.74 318 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),bit1(K))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K)))) # label(fact_97_rel__simps_I22_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 319 (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_731_nat__mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 320 (all A_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),zero_zero(nat)))) # label(fact_870_gcd__lcm__complete__lattice__nat_Otop__greatest) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 321 (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_432_real__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 322 (all L_1 hAPP(int,int,minus_minus(int,min),bit1(L_1)) = bit0(hAPP(int,int,minus_minus(int,min),L_1))) # label(fact_389_diff__bin__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 323 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all Y_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),Y_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),number_number_of(X_a,Y_2))))))) # label(fact_246_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 324 (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)) -> zero_zero(int) = ti(int,A_1))))) # label(fact_352_zcong__zless__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 325 (all P_2 all I_2 all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_2),K)) -> (hBOOL(hAPP(int,bool,P_2,K)) -> ((all I (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I),K)) -> (hBOOL(hAPP(int,bool,P_2,I)) -> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,minus_minus(int,I),one_one(int))))))) -> hBOOL(hAPP(int,bool,P_2,I_2)))))) # label(fact_849_int__le__induct) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 326 (all X_a (linord20386208strict(X_a) -> (all A_1 all C all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),C))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))))))) # label(fact_535_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 327 (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_94_rel__simps_I16_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 328 (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_769_add__0__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 329 (all X_a (number_ring(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,pls)),A_1)))) # label(fact_143_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 330 (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]. 2.34/2.74 331 (all X_a (monoid_mult(X_a) -> (all A_1 hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),A_1)),A_1) = hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,bit1(bit1(pls))))))) # label(fact_163_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 2.34/2.74 332 (all Y_1 all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,plus_plus(int,X_1),Y_1)))))) # label(fact_346_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 333 (all K_1 all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,times_times(nat,K_1),M)),hAPP(nat,nat,times_times(nat,K_1),N_1))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),N_1))))) # label(fact_719_dvd__mult__cancel) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 334 (all K all Ma all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),N)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,K),Ma)),hAPP(nat,nat,plus_plus(nat,K),N))))) # label(fact_677_nat__add__left__cancel__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 335 (all X_a (ordere779506340up_add(X_a) -> (all C all D_2 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_2)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)),hAPP(X_a,X_a,plus_plus(X_a,B),D_2)))))))) # label(fact_775_add__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 336 (all X_a (comm_ring_1(X_a) -> (all Z_1 all X_1 all Y_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X_1),Y_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X_1),Z_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X_1),hAPP(X_a,X_a,minus_minus(X_a,Y_1),Z_1)))))))) # label(fact_461_dvd__diff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 337 (all X_a (one(X_a) -> ti(X_a,one_one(X_a)) = one_one(X_a))) # label(tsy_c_Groups_Oone__class_Oone_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 338 (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_620_le__antisym) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 339 (all X_a (linordered_semidom(X_a) -> (all A_1 all N_1 all N_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),N_3)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N_1)),hAPP(nat,X_a,power_power(X_a,A_1),N_3)))))))) # label(fact_271_power__increasing) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 340 (all A_1 all B hBOOL(hAPP(int,bool,zcong(A_1,B),one_one(int)))) # label(fact_336_zcong__1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 341 (all Y_1 all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,div_mod(int,X_1),Y_1)))))) # label(fact_908_Divides_Otransfer__nat__int__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 342 (all N_1 all F all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),F),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),F),N_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),F),hAPP(int,int,div_mod(int,M),N_1)))))) # label(fact_916_zdvd__zmod) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 343 (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]. 2.34/2.75 344 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),zero_zero(X_a)))))))) # label(fact_812_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 345 (all B_1_1 all X_a (power(X_a) -> power_power(X_a,ti(X_a,B_1_1)) = power_power(X_a,B_1_1))) # label(tsy_c_Power_Opower__class_Opower_arg1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 346 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),zero_zero(X_a)))))))) # label(fact_803_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 347 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1)) -> N_1 != zero_zero(nat))) # label(fact_633_gr__implies__not0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 348 (all X_a (dvd(X_a) & comm_ring(X_a) -> (all Ta all D all D_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),D_3)) -> (all X all K_2 (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),hAPP(X_a,X_a,plus_plus(X_a,X),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),hAPP(X_a,X_a,times_times(X_a,K_2),D_3))),Ta))))))))) # label(fact_745_inf__period_I4_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 349 (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_73_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 350 (all K (ti(int,K) = pls <-> pls = bit0(K))) # label(fact_120_rel__simps_I44_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 351 (all N all P_2 (-hBOOL(hAPP(nat,bool,P_2,zero_zero(nat))) -> (hBOOL(hAPP(nat,bool,P_2,N)) -> (exists K_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_2),N)) & hBOOL(hAPP(nat,bool,P_2,K_2)) & (all I (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),K_2)) -> -hBOOL(hAPP(nat,bool,P_2,I))))))))) # label(fact_856_ex__least__nat__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 352 (all N all Ma (hAPP(nat,nat,times_times(nat,N),Ma) = one_one(nat) <-> one_one(nat) = Ma & one_one(nat) = N)) # label(fact_467_nat__mult__eq__one) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 353 (all N_1 all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N_1),M))))) # label(fact_708_nat__dvd__not__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 354 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,A_1),B)))))))) # label(fact_816_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 355 (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),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))) | 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),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,times_times(X_a,A_3),B_2))))))) # label(fact_486_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 356 (all X_a (ordered_ring(X_a) -> (all A_3 all E all C_1 all B_2 all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,minus_minus(X_a,B_2),A_3)),E)),D))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_3),E)),C_1)),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,B_2),E)),D))))))) # label(fact_544_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 357 (all X_a (ring_1(X_a) -> (all X_1 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))) = 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))))) # label(fact_547_real__squared__diff__one__factored) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 358 (all X_a all X_c all X_b all P all Q all R hAPP(X_b,X_c,P,hAPP(X_a,X_b,Q,R)) = hAPP(X_a,X_c,combb(X_b,X_c,X_a,P,Q),R)) # label(help_COMBB_1_1_U) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 359 (all B_1_1 all X_a (monoid_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_1_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 360 (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),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_eq(X_a),A_3),B_2)))))) # label(fact_810_le__iff__diff__le__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 361 (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_47_rel__simps_I31_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 362 (all N_1 all A_1 all B all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),B)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,P_1),N_1)),hAPP(int,int,times_times(int,A_1),B))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,P_1),N_1)),A_1)))))) # label(fact_234_zprime__power__zdvd__cancel__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 363 (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_615_diff__cancel2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 364 (all X_a (linordered_semidom(X_a) -> (all N_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,power_power(X_a,A_1),N_1))))))) # label(fact_210_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 365 (all X_a (linord581940658strict(X_a) -> (all X_2 all Y_2 (ti(X_a,Y_2) != zero_zero(X_a) | ti(X_a,X_2) != zero_zero(X_a) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,X_2),X_2)),hAPP(X_a,X_a,times_times(X_a,Y_2),Y_2)))))))) # label(fact_239_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 366 (all X_a (linord20386208strict(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,times_times(X_a,B),A_1))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B))))))) # label(fact_496_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 367 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,A_1),B)))))))) # label(fact_799_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 368 (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]. 2.34/2.75 369 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all Y_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),number_number_of(X_a,Y_2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),Y_2)))))) # label(fact_248_le__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 370 (all X_a (ordere453448008miring(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,B),A_1)),zero_zero(X_a)))))))) # label(fact_482_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 371 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,K)),number_number_of(int,L))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)))) # label(fact_48_less__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 372 (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_86_zmult__assoc) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 373 (all K_1 hAPP(int,int,plus_plus(int,K_1),K_1) = bit0(K_1)) # label(fact_128_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 374 (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,ti(fun(X_a,fun(X_b,X_c)),B_1_1),B_2_1)) # label(tsy_c_COMBS_arg1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 375 (all A_1 zero_zero(int) = hAPP(int,int,div_mod(int,A_1),number_number_of(int,min))) # label(fact_952_zmod__minus1__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 376 (all B_1_1 d22set(B_1_1) = d22set(ti(int,B_1_1))) # label(tsy_c_IntFact_Od22set_arg1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 377 (all K_1 all L_1 bit0(hAPP(int,int,plus_plus(int,K_1),L_1)) = hAPP(int,int,plus_plus(int,bit0(K_1)),bit0(L_1))) # label(fact_127_add__Bit0__Bit0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 378 (all C all D_2 all A_1 all B all M (hBOOL(hAPP(int,bool,zcong(A_1,B),M)) -> (hBOOL(hAPP(int,bool,zcong(C,D_2),M)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),C),hAPP(int,int,times_times(int,B),D_2)),M))))) # label(fact_337_zcong__zmult) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 379 (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_317_rel__simps_I25_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 380 (all X_a (ordere223160158up_add(X_a) -> (all C all D_2 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_2)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)),hAPP(X_a,X_a,plus_plus(X_a,B),D_2)))))))) # label(fact_782_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 381 (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_733_nat__diff__add__eq1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 382 (all X_a (comm_semiring_1(X_a) -> (all X_1 all N_1 hAPP(nat,X_a,power_power(X_a,X_1),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),N_1)) = hAPP(X_a,X_a,times_times(X_a,hAPP(nat,X_a,power_power(X_a,X_1),N_1)),hAPP(nat,X_a,power_power(X_a,X_1),N_1))))) # label(fact_15_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 383 (all X_a (linordered_idom(X_a) -> (all Y_1 all X_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_1),one_one(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,Y_1),X_1)),X_1)))))))) # label(fact_540_mult__left__le__one__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 384 (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),B)),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)),hAPP(X_a,X_a,plus_plus(X_a,B),D_2))))) # label(fact_74_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 385 (all X_a (comm_semiring_1(X_a) -> (all A_1 hAPP(X_a,X_a,times_times(X_a,one_one(X_a)),A_1) = ti(X_a,A_1)))) # label(fact_114_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 386 (all X_a (semiring_div(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,div_mod(X_a,A_1),C)),B)),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),C)))) # label(fact_930_mod__add__left__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 387 (all X_a (ring_div(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,minus_minus(X_a,hAPP(X_a,X_a,div_mod(X_a,A_1),C)),B)),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,minus_minus(X_a,A_1),B)),C)))) # label(fact_949_mod__diff__left__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 388 (all Y_1 all X_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_1)) -> (-hBOOL(hAPP(int,bool,zcong(Y_1,zero_zero(int)),P_1)) -> -hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,X_1),Y_1),zero_zero(int)),P_1)))))) # label(fact_413_zcong__zmult__prop3) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 389 (all P_3 ((all M_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M_2),P_3)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M_2)) -> ti(int,M_2) = one_one(int) | ti(int,M_2) = ti(int,P_3))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),P_3)) <-> hBOOL(hAPP(int,bool,zprime,P_3)))) # label(fact_280_zprime__def) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 390 (all B all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),B),zero_zero(int)),P_1)) -> hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_1)) | hBOOL(hAPP(int,bool,zcong(B,zero_zero(int)),P_1)))))) # label(fact_414_zcong__zprime__prod__zero) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 391 (all B_1_1 all X_a (plus(X_a) & dvd(X_a) & linorder(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]. 2.34/2.75 392 (all I_1 all U_1 all J_1 all K_1 hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,hAPP(nat,nat,plus_plus(nat,I_1),J_1)),U_1)),K_1) = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,I_1),U_1)),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,J_1),U_1)),K_1))) # label(fact_726_left__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 393 (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_972_zmod__eq__0D) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 394 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)) -> hAPP(nat,nat,minus_minus(nat,M),N_1) = zero_zero(nat))) # label(fact_646_diff__is__0__eq_H) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 395 (all N all K all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),Ma)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),N)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,minus_minus(nat,Ma),K)),hAPP(nat,nat,minus_minus(nat,N),K))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),N)))))) # label(fact_669_less__diff__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 396 (all X_a (comm_semiring_1(X_a) -> (all A_1 all C hAPP(X_a,X_a,plus_plus(X_a,A_1),C) = hAPP(X_a,X_a,plus_plus(X_a,C),A_1)))) # label(fact_81_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 397 (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_697_add__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 398 (all X_a (ring(X_a) -> (all X_1 all Y_1 all A_1 all B hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,X_1),hAPP(X_a,X_a,minus_minus(X_a,Y_1),B))),hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,minus_minus(X_a,X_1),A_1)),B)) = hAPP(X_a,X_a,minus_minus(X_a,hAPP(X_a,X_a,times_times(X_a,X_1),Y_1)),hAPP(X_a,X_a,times_times(X_a,A_1),B))))) # label(fact_516_mult__diff__mult) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 399 (all K_1 all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,I_1),K_1)),hAPP(nat,nat,plus_plus(nat,J_1),K_1))))) # label(fact_658_add__less__mono1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 400 (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_387_diff__bin__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 401 (all U_1 all M all N_1 all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_1)) -> hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,I_1),U_1)),M)),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,J_1),U_1)),N_1)) = hAPP(nat,nat,minus_minus(nat,M),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,hAPP(nat,nat,minus_minus(nat,J_1),I_1)),U_1)),N_1)))) # label(fact_736_nat__diff__add__eq2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 402 (all L_1 all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,minus_minus(nat,M),L_1)),hAPP(nat,nat,minus_minus(nat,N_1),L_1))))) # label(fact_623_diff__le__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 403 (all Ma all D (zero_zero(int) = hAPP(int,int,div_mod(int,Ma),D) <-> (exists Q_2 ti(int,Ma) = hAPP(int,int,times_times(int,D),Q_2)))) # label(fact_910_zmod__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 404 (all X_a (linord893533164strict(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,C),A_1)),hAPP(X_a,X_a,times_times(X_a,C),B)))))))) # label(fact_490_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 405 (all X_a (zero(X_a) -> (all X_2 (zero_zero(X_a) = ti(X_a,X_2) <-> ti(X_a,X_2) = zero_zero(X_a))))) # label(fact_756_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 406 (all X_a (semiring_div(X_a) -> (all B all A_1 hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,times_times(X_a,B),A_1)),B) = zero_zero(X_a)))) # label(fact_899_mod__mult__self1__is__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 407 (all Y_1 all X_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_1)) -> (-hBOOL(hAPP(int,bool,zcong(Y_1,zero_zero(int)),P_1)) -> (hBOOL(hAPP(int,bool,zcong(multInv(P_1,X_1),multInv(P_1,Y_1)),P_1)) -> hBOOL(hAPP(int,bool,zcong(X_1,Y_1),P_1)))))))) # label(fact_829_MultInv__prop5) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 408 (all K_1 all M all N_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_1),hAPP(int,int,div_mod(int,M),N_1))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_1),N_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_1),M))))) # label(fact_915_zdvd__zmod__imp__zdvd) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 409 (all P_2 all Ma all N ((hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),N)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_2,N),Ma))) -> ((N = Ma -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_2,N),Ma))) -> ((hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),Ma)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_2,N),Ma))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_2,N),Ma)))))) # label(fact_567_nat__less__cases) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 410 (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(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B_3),Q_4)),R_3)),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_1),B)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_3),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_4),Q_1))))))))) # label(fact_354_zdiv__mono2__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 411 (all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(nat,real,power_power(real,number_number_of(real,bit0(bit1(pls)))),N_1)))) # label(fact_550_two__realpow__ge__one) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 412 (all X_a (comm_semiring_1(X_a) -> (all X_1 all Y_1 all Q_1 hAPP(X_a,X_a,times_times(X_a,hAPP(nat,X_a,power_power(X_a,X_1),Q_1)),hAPP(nat,X_a,power_power(X_a,Y_1),Q_1)) = hAPP(nat,X_a,power_power(X_a,hAPP(X_a,X_a,times_times(X_a,X_1),Y_1)),Q_1)))) # label(fact_115_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 413 (all X_a (linord20386208strict(X_a) -> (all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,times_times(X_a,A_1),B))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B))))))) # label(fact_497_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 414 (all X_a (number_semiring(X_a) -> (all Z_1 hAPP(X_a,X_a,times_times(X_a,Z_1),number_number_of(X_a,bit0(bit1(pls)))) = hAPP(X_a,X_a,plus_plus(X_a,Z_1),Z_1)))) # label(fact_167_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 415 (all W all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W),Z_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,W),one_one(int))),Z_1)))) # label(fact_61_zless__imp__add1__zle) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 416 (all N_1 all P_1 all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M)) -> (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),hAPP(int,int,times_times(int,M),N_1))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),M)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),N_1)))))) # label(fact_360_zprime__zdvd__zmult) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 417 (all B_1_1 all B_2_1 legendre(B_1_1,ti(int,B_2_1)) = legendre(B_1_1,B_2_1)) # label(tsy_c_Residues_OLegendre_arg2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 418 (all X_a (linordered_semidom(X_a) -> (all A_1 all N_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N_1)),hAPP(nat,X_a,power_power(X_a,B),N_1))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))))))) # label(fact_214_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 419 (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_709_add__gr__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 420 (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_971_pos__zmod__mult__2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 421 (all M hBOOL(hAPP(int,bool,zcong(M,zero_zero(int)),M))) # label(fact_396_zcong__id) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 422 (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_363_neg__one__power__eq__mod__m) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 423 (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_793_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 424 (all X_a (linord20386208strict(X_a) -> (all C all D_2 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),D_2)))))))))) # label(fact_531_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 425 (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,B_2),E)),D) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_3),E)),C_1) <-> ti(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))))) # label(fact_515_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 426 (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_804_zero__less__double__add__iff__zero__less__single__add) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 427 (all X_a (monoid_add(X_a) -> (all A_1 hAPP(X_a,X_a,plus_plus(X_a,zero_zero(X_a)),A_1) = ti(X_a,A_1)))) # label(fact_766_add__0__left) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 428 (all B_1_1 all B_2_1 ti(int,multInv(B_1_1,B_2_1)) = multInv(B_1_1,B_2_1)) # label(tsy_c_Int2_OMultInv_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 429 (all N_1 all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M)) -> hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,minus_minus(nat,M),N_1)),N_1) = hAPP(nat,nat,div_mod(nat,M),N_1))) # label(fact_980_le__mod__geq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 430 (all K_1 all I_1 all J_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_1),J_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),J_1),K_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_1),K_1))))) # label(fact_24_zle__trans) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 431 (all X_1 all Y_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) -> Y_1 != X_1)) # label(fact_582_dvd_Oless__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 432 (all X_b (number(X_b) & semiring(X_b) -> (all V all B all C hAPP(X_b,X_b,times_times(X_b,number_number_of(X_b,V)),hAPP(X_b,X_b,plus_plus(X_b,B),C)) = hAPP(X_b,X_b,plus_plus(X_b,hAPP(X_b,X_b,times_times(X_b,number_number_of(X_b,V)),B)),hAPP(X_b,X_b,times_times(X_b,number_number_of(X_b,V)),C))))) # label(fact_139_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 433 (all Z_1 all X_1 all Y_1 all P_1 (hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,X_1),Y_1),one_one(int)),P_1)) -> hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,X_1),hAPP(nat,nat,times_times(nat,Y_1),Z_1)),one_one(int)),P_1)))) # label(fact_211_zcong__zpower__zmult) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 434 (all C all A_1 all B all M (hBOOL(hAPP(int,bool,zcong(A_1,B),M)) -> (hBOOL(hAPP(int,bool,zcong(B,C),M)) -> hBOOL(hAPP(int,bool,zcong(A_1,C),M))))) # label(fact_328_zcong__trans) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 435 (all J_1 all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (-hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_1)) -> (-hBOOL(hAPP(int,bool,zcong(J_1,zero_zero(int)),P_1)) -> (-hBOOL(hAPP(int,bool,quadRes(P_1),A_1)) -> -hBOOL(hAPP(int,bool,zcong(J_1,hAPP(int,int,times_times(int,A_1),multInv(P_1,J_1))),P_1)))))))) # label(fact_818_MultInvPair__distinct) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 436 (all K_1 all L_1 all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),L_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,times_times(nat,I_1),K_1)),hAPP(nat,nat,times_times(nat,J_1),L_1)))))) # label(fact_699_mult__le__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 437 (all X_2 all N (one_one(nat) = X_2 | zero_zero(nat) = N <-> one_one(nat) = hAPP(nat,nat,power_power(nat,X_2),N))) # label(fact_527_exp__eq__1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 438 (all X_1 all Y_1 (X_1 != Y_1 -> (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_1),Y_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Y_1),X_1))))) # label(fact_571_linorder__neqE__nat) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 439 (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_100_rel__simps_I27_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 440 (all K_1 all I_1 all J_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_1),J_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,K_1),I_1)),hAPP(int,int,plus_plus(int,K_1),J_1))))) # label(fact_51_zadd__left__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.75 441 (all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,minus_minus(int,P_1),one_one(int)))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),inv(P_1,A_1)),hAPP(int,int,minus_minus(int,P_1),one_one(int)))))))) # label(fact_825_inv__less__p__minus__1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 442 (all V_2 all V_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),number_number_of(nat,V_2)),number_number_of(nat,V_3))) <-> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),V_2),V_3)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),V_2),pls))))) # label(fact_240_le__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 443 (all M all N_1 (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1)) -> M = hAPP(nat,nat,plus_plus(nat,N_1),hAPP(nat,nat,minus_minus(nat,M),N_1)))) # label(fact_662_add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 444 (all X_2 all W_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X_2)) | number_number_of(nat,W_1) = 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_2),number_number_of(nat,W_1)))))) # label(fact_288_zero__less__power__nat__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 445 (all X_a (linord581940658strict(X_a) -> (all X_2 all Y_2 (zero_zero(X_a) = ti(X_a,X_2) & ti(X_a,Y_2) = zero_zero(X_a) <-> hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,X_2),X_2)),hAPP(X_a,X_a,times_times(X_a,Y_2),Y_2)) = zero_zero(X_a))))) # label(fact_218_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 446 (all K all Ma all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,times_times(nat,K),Ma)),hAPP(nat,nat,times_times(nat,K),N))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),N)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K)))) # label(fact_715_mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 447 (all A_1 -(-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),one_one(nat)),A_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),one_one(nat))))) # label(fact_872_gcd__lcm__complete__lattice__nat_Onot__less__bot) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 448 (all P_2 all N all K (hBOOL(hAPP(int,bool,P_2,hAPP(int,int,div_mod(int,N),K))) <-> (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))) & hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,K),I)),J) = ti(int,N) -> hBOOL(hAPP(int,bool,P_2,J))))) & (hBOOL(hAPP(int,bool,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_2,J))))) & (ti(int,K) = zero_zero(int) -> hBOOL(hAPP(int,bool,P_2,N))))) # label(fact_963_split__zmod) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 449 (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_932_zmod__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 450 (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),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,minus_minus(X_a,A_3),B_2)),zero_zero(X_a))))))) # label(fact_811_less__iff__diff__less__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 451 (all M (hBOOL(hAPP(int,bool,zprime,hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),M)),one_one(int)))) -> legendre(number_number_of(int,min),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),M)),one_one(int))) = one_one(int))) # label(fact_417_Legendre__1mod4) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 452 (all B_1_1 (hBOOL(B_1_1) <-> hBOOL(ti(bool,B_1_1)))) # label(tsy_c_hBOOL_arg1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 453 (all X_a (linordered_idom(X_a) -> (all A_1 all N_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,power_power(X_a,A_1),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),N_1))))))) # label(fact_269_zero__le__even__power_H) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 454 (all K_1 all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,times_times(nat,K_1),I_1)),hAPP(nat,nat,times_times(nat,K_1),J_1))))) # label(fact_700_mult__le__mono2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 455 (all K_1 all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),hAPP(nat,nat,minus_minus(nat,M),N_1))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),M)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),N_1)))))) # label(fact_672_dvd__diffD1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 456 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),pls)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K)),pls)))) # label(fact_95_rel__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 457 (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]. 2.34/2.76 458 (all X_a (ordere236663937imp_le(X_a) -> (all A_1 all C all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)),hAPP(X_a,X_a,plus_plus(X_a,B),C))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)))))) # label(fact_783_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 459 (all A_1 hAPP(int,int,div_mod(int,A_1),A_1) = zero_zero(int)) # label(fact_921_zmod__self) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 460 (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_154_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 461 (all X_a (idom(X_a) -> (all A_3 all C_1 all 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),A_3),B_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,times_times(X_a,A_3),C_1)),hAPP(X_a,X_a,times_times(X_a,B_2),C_1))))))) # label(fact_513_dvd__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 462 (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]. 2.34/2.76 463 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M)))) # label(fact_617_nat__le__linear) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 464 (all X_a (linordered_semidom(X_a) -> (all N_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(X_a,X_a,times_times(X_a,A_1),hAPP(nat,X_a,power_power(X_a,A_1),N_1)))))))) # label(fact_177_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 465 (all P_2 all X_2 all Y_2 (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_2),X_2)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_2),Y_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_2),X_2)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_2),Y_2)) -> hBOOL(P_2)))) # label(fact_581_dvd_Oless__imp__triv) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 466 (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]. 2.34/2.76 467 (all X_a (linord581940658strict(X_a) -> (all A_3 all C_1 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_3),C_1)),hAPP(X_a,X_a,times_times(X_a,B_2),C_1))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),A_3)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a))) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1)))))) # label(fact_503_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 468 (all C all D_2 all A_1 all B all M (hBOOL(hAPP(int,bool,zcong(A_1,B),M)) -> (hBOOL(hAPP(int,bool,zcong(C,D_2),M)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,plus_plus(int,A_1),C),hAPP(int,int,plus_plus(int,B),D_2)),M))))) # label(fact_341_zcong__zadd) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 469 (all K_1 all I_1 all J_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),I_1),J_1)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),J_1),K_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),I_1),K_1))))) # label(fact_561_real__le__trans) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 470 (all N all K all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),Ma)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),N)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),N)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,minus_minus(nat,Ma),K)),hAPP(nat,nat,minus_minus(nat,N),K))))))) # label(fact_627_le__diff__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 471 (all X_a (linord20386208strict(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,B),A_1)),zero_zero(X_a)))))))) # label(fact_498_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 472 (all X_c all X_a all B_1_1 all B_2_1 ti(X_c,hAPP(X_a,X_c,B_1_1,B_2_1)) = hAPP(X_a,X_c,B_1_1,B_2_1)) # label(tsy_c_hAPP_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 473 (all X_a (number_ring(X_a) -> (all Z_1 hAPP(X_a,X_a,times_times(X_a,number_number_of(X_a,bit0(bit1(pls)))),Z_1) = hAPP(X_a,X_a,plus_plus(X_a,Z_1),Z_1)))) # label(fact_166_mult__2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 474 (all K_1 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),N_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N_1))))) # label(fact_718_dvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 475 (all C all A_1 all B all M (hBOOL(hAPP(int,bool,zcong(A_1,B),M)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,plus_plus(int,A_1),C),hAPP(int,int,plus_plus(int,B),C)),M)))) # label(fact_399_zcong__shift) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 476 (all N_1 all K_1 all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),M)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N_1)) -> hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,minus_minus(nat,M),K_1)),hAPP(nat,nat,minus_minus(nat,N_1),K_1)) = hAPP(nat,nat,minus_minus(nat,M),N_1)))) # label(fact_626_Nat_Odiff__diff__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 477 (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_21_zle__linear) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 478 (all M all N_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,div_mod(nat,M),N_1)),M))) # label(fact_985_mod__less__eq__dividend) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 479 (all K_1 all L_1 hAPP(int,int,minus_minus(int,bit0(K_1)),bit0(L_1)) = bit0(hAPP(int,int,minus_minus(int,K_1),L_1))) # label(fact_371_diff__bin__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 480 (all X_a (ordered_ring(X_a) -> (all C all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),C)))))))) # label(fact_476_mult__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 481 (all X_1 all Y_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) -> Y_1 != X_1)) # label(fact_587_dvd_Oless__imp__neq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 482 (all B_1_1 all X_a (no_zero_divisors(X_a) -> times_times(X_a,B_1_1) = times_times(X_a,ti(X_a,B_1_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_5_arg1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 483 (all X_a (comm_semiring_1(X_a) -> (all A_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),A_1))))) # label(fact_426_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 484 (all X_a (semiri456707255roduct(X_a) -> (all W_1 all Y_2 all X_2 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_2),Y_2)) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,W_1),Y_2)),hAPP(X_a,X_a,times_times(X_a,X_2),Z_2)) <-> ti(X_a,X_2) = ti(X_a,W_1) | ti(X_a,Y_2) = ti(X_a,Z_2))))) # label(fact_108_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 485 (all X_a (linordered_semidom(X_a) -> (all X_1 all Y_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,X_1),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,X_a,power_power(X_a,Y_1),number_number_of(nat,bit0(bit1(pls)))))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))))))) # label(fact_264_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 486 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,times_times(X_a,A_1),B)))))) # label(fact_458_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 487 (all X_a (linordered_semidom(X_a) -> (all A_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),hAPP(X_a,X_a,plus_plus(X_a,A_1),one_one(X_a))))))) # label(fact_511_less__add__one) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 488 (all N_1 hAPP(nat,nat,times_times(nat,one_one(nat)),N_1) = N_1) # label(fact_707_nat__mult__1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 489 (all X_a (linord20386208strict(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,C),A_1)),hAPP(X_a,X_a,times_times(X_a,C),B))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))))))) # label(fact_537_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 490 (all X_2 all Y_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_2),Y_2)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_2),X_2)) <-> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_2),X_2)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_2),Y_2)))) # label(fact_600_dvd_Oless__le__not__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 491 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),K)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),bit0(K))))) # label(fact_96_rel__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 492 (all B_1_1 all B_2_1 ti(int,legendre(B_1_1,B_2_1)) = legendre(B_1_1,B_2_1)) # label(tsy_c_Residues_OLegendre_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 493 (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_44_rel__simps_I14_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 494 (all Z_1 hAPP(real,real,times_times(real,one_one(real)),Z_1) = Z_1) # label(fact_434_real__mult__1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 495 (all X1 all X2 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma)) -> (hBOOL(hAPP(int,bool,zcong(X1,X2),Ma)) <-> standardRes(Ma,X2) = standardRes(Ma,X1)))) # label(fact_994_StandardRes__prop2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 496 (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,number_number_of(X_a,bit0(bit1(pls)))),Z_1)))) # label(fact_165_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 497 (all A_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_3),one_one(nat))) <-> A_3 = one_one(nat))) # label(fact_875_gcd__lcm__complete__lattice__nat_Obot__unique) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 498 (all X_a (semiring_div(X_a) -> (all A_1 hAPP(X_a,X_a,div_mod(X_a,A_1),one_one(X_a)) = zero_zero(X_a)))) # label(fact_901_mod__by__1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 499 (all X_a (real_normed_algebra(X_a) -> (all X_1 all Y_1 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_1),Ya)) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,minus_minus(X_a,X_1),Y_1)),Ya)))) # label(fact_895_mult__left_Odiff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 500 (all M all N_1 hAPP(nat,nat,times_times(nat,N_1),M) = hAPP(nat,nat,times_times(nat,M),N_1)) # label(fact_629_nat__mult__commute) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 501 (all X_a (comm_semiring_1(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_113_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 502 (all B_1_1 ti(int,zfact(B_1_1)) = zfact(B_1_1)) # label(tsy_c_IntFact_Ozfact_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 503 (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_93_less__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 504 (all X_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_1)) -> hBOOL(hAPP(int,bool,zcong(multInv(P_1,multInv(P_1,X_1)),X_1),P_1)))))) # label(fact_830_MultInv__prop4) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 505 (all X_1 -(hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),X_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),X_1)))) # label(fact_604_dvd_Oless__irrefl) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 506 (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),ord_less_eq(nat),A_1),B)) | zero_zero(nat) = B)) # label(fact_517_divides__ge) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 507 (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_382_xzgcda__linear__aux1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 508 (all V all W hAPP(int,int,div_mod(int,number_number_of(int,bit0(V))),number_number_of(int,bit0(W))) = hAPP(int,int,times_times(int,number_number_of(int,bit0(bit1(pls)))),hAPP(int,int,div_mod(int,number_number_of(int,V)),number_number_of(int,W)))) # label(fact_962_zmod__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 509 (all X_a (linordered_semidom(X_a) -> (all N_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N_1)),hAPP(nat,X_a,power_power(X_a,B),N_1)))))))) # label(fact_209_power__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 510 (all X_1 all Y_1 all Z_1 hAPP(int,int,times_times(int,hAPP(nat,int,power_power(int,X_1),Y_1)),hAPP(nat,int,power_power(int,X_1),Z_1)) = hAPP(nat,int,power_power(int,X_1),hAPP(nat,nat,plus_plus(nat,Y_1),Z_1))) # label(fact_34_zpower__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 511 (all X_a (ring_div(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,minus_minus(X_a,A_1),hAPP(X_a,X_a,div_mod(X_a,B),C))),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,minus_minus(X_a,A_1),B)),C)))) # label(fact_950_mod__diff__right__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 512 (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]. 2.34/2.76 513 (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)) -> number_number_of(X_a,hAPP(int,int,plus_plus(int,V),V_1)) = hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,V)),number_number_of(X_a,V_1))))))) # label(fact_136_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 514 (all Ma all N (one_one(nat) = hAPP(nat,nat,times_times(nat,Ma),N) <-> one_one(nat) = N & one_one(nat) = Ma)) # label(fact_706_nat__1__eq__mult__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 515 (all K_1 all M all N_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_1),hAPP(int,int,minus_minus(int,M),N_1))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_1),N_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_1),M))))) # label(fact_375_zdvd__zdiffD) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 516 (all N_1 all B all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,P_1),N_1)),hAPP(int,int,times_times(int,A_1),B))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,P_1),N_1)),B)))))) # label(fact_233_zprime__power__zdvd__cancel__left) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 517 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,Lx),Ly)),Rx) = hAPP(X_a,X_a,times_times(X_a,Lx),hAPP(X_a,X_a,times_times(X_a,Ly),Rx))))) # label(fact_70_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 518 (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_384_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 519 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),bit1(K))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),K)))) # label(fact_301_rel__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 520 (all X_a (linord1278240602ring_1(X_a) -> (all V all U_1 all Y_1 all X_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_1),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),U_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),V)) -> (one_one(X_a) = hAPP(X_a,X_a,plus_plus(X_a,U_1),V) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,U_1),X_1)),hAPP(X_a,X_a,times_times(X_a,V),Y_1))),A_1)))))))))) # label(fact_548_convex__bound__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 521 (all X_a (linord20386208strict(X_a) -> (all A_1 all C all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),C))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))))))) # label(fact_538_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 522 (all X_a (linordered_idom(X_a) -> (all X_1 all Y_1 (ti(X_a,Y_1) != ti(X_a,X_1) -> (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_1),X_1))))))) # label(fact_425_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 523 (all Ma all N (zero_zero(nat) = Ma | zero_zero(nat) = N <-> zero_zero(nat) = hAPP(nat,nat,times_times(nat,Ma),N))) # label(fact_650_mult__is__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 524 (all X_a (comm_semiring_1(X_a) -> (all B all A_1 all C (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,times_times(X_a,B),C))))))) # label(fact_455_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 525 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,times_times(int,A_1),B))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))))) # label(fact_229_pos__zmult__pos) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 526 (all K_1 all L_1 hAPP(int,int,minus_minus(int,bit1(K_1)),bit0(L_1)) = bit1(hAPP(int,int,minus_minus(int,K_1),L_1))) # label(fact_378_diff__bin__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 527 (all X_a all X_2 all A_6 (hBOOL(hAPP(X_a,bool,A_6,X_2)) <-> hBOOL(hAPP(fun(X_a,bool),bool,member(X_a,X_2),A_6)))) # label(fact_75_mem__def) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 528 (all X_a (linordered_semidom(X_a) -> (all X_2 all Y_2 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),B_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X_2),Y_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,B_2),X_2)),hAPP(nat,X_a,power_power(X_a,B_2),Y_2)))))))) # label(fact_179_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 529 (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_260_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 530 (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_27_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 531 (all Ma (Ma = one_one(nat) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),one_one(nat))))) # label(fact_698_nat__dvd__1__iff__1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 532 (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_69_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 533 (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,minus_minus(nat,J_2),I_2) = K <-> hAPP(nat,nat,plus_plus(nat,K),I_2) = J_2))) # label(fact_692_le__imp__diff__is__add) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 534 (all A_3 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Ma),A_3)) <-> hBOOL(hAPP(int,bool,zcong(A_3,zero_zero(int)),Ma)))) # label(fact_408_zcong__zero__equiv__div) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 535 (all X_2 (hBOOL(hAPP(int,bool,twoSqu658283162sum2sq,X_2)) <-> (exists A_5 exists B_4 ti(int,X_2) = twoSqu1929807760sum2sq(product_Pair(int,int,A_5,B_4))))) # label(fact_740_is__sum2sq__def) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 536 (all X_a (comm_semiring_1(X_a) -> (all X_1 all P_1 all Q_1 hAPP(nat,X_a,power_power(X_a,hAPP(nat,X_a,power_power(X_a,X_1),P_1)),Q_1) = hAPP(nat,X_a,power_power(X_a,X_1),hAPP(nat,nat,times_times(nat,P_1),Q_1))))) # label(fact_26_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 537 (all X_1 all Y_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zcong(X_1,Y_1),P_1)) -> hBOOL(hAPP(int,bool,zcong(multInv(P_1,X_1),multInv(P_1,Y_1)),P_1))))) # label(fact_826_MultInv__prop1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 538 (all K_1 all I_1 all J_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),J_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,K_1),I_1)),hAPP(int,int,times_times(int,K_1),J_1)))))) # label(fact_230_zmult__zless__mono2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 539 (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,N_1),hAPP(nat,nat,minus_minus(nat,M),N_1)))) # label(fact_688_le__add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 540 (all X_a (linordered_semidom(X_a) -> (all A_1 all N_1 all N_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),N_3)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N_1)),hAPP(nat,X_a,power_power(X_a,A_1),N_3)))))))) # label(fact_275_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 541 (all B_1_1 all B_2_1 product_Pair(int,int,B_1_1,ti(int,B_2_1)) = product_Pair(int,int,B_1_1,B_2_1)) # label(tsy_c_Product__Type_OPair_arg2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 542 (all M all N_1 (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1)) -> hAPP(nat,nat,div_mod(nat,M),N_1) = hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,minus_minus(nat,M),N_1)),N_1))) # label(fact_977_mod__geq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.76 543 (all X_b (ring(X_b) & number(X_b) -> (all V all B all C hAPP(X_b,X_b,times_times(X_b,number_number_of(X_b,V)),hAPP(X_b,X_b,minus_minus(X_b,B),C)) = hAPP(X_b,X_b,minus_minus(X_b,hAPP(X_b,X_b,times_times(X_b,number_number_of(X_b,V)),B)),hAPP(X_b,X_b,times_times(X_b,number_number_of(X_b,V)),C))))) # label(fact_376_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 544 (all M all N_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),hAPP(int,int,times_times(int,M),N_1))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),M)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),N_1))))) # label(fact_409_zprime__zdvd__zmult__better) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 545 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),bit0(K))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),K)))) # label(fact_304_rel__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 546 (all A_2 all B_1 all Q_3 all Y_1 (ti(int,A_2) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B_1),Q_3)),Y_1) -> ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y_1),B_1)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_1))) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_1),Y_1)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Y_1),zero_zero(int)))) -> (zero_zero(int) != ti(int,B_1) -> ti(int,Y_1) = hAPP(int,int,div_mod(int,A_2),B_1))))) # label(fact_968_divmod__int__rel__mod__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 547 (all X_a (ordere216010020id_add(X_a) -> (all B all C all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)))))))) # label(fact_814_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 548 (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_450_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 549 (all K_1 all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_1),K_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),K_1))))) # label(fact_619_le__trans) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 550 (all X_a undefined(X_a) = ti(X_a,undefined(X_a))) # label(tsy_c_HOL_Oundefined_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 551 (all M hAPP(nat,nat,plus_plus(nat,M),zero_zero(nat)) = M) # label(fact_641_Nat_Oadd__0__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 552 (all X_a (group_add(X_a) -> (all A_1 hAPP(X_a,X_a,minus_minus(X_a,A_1),A_1) = zero_zero(X_a)))) # label(fact_787_diff__self) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 553 (all C all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,times_times(nat,C),A_1)),hAPP(nat,nat,times_times(nat,C),B))))) # label(fact_465_divides__mul__l) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 554 (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]. 2.34/2.77 555 (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_937_mod__mult__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 556 -(all S_1 -(hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),S_1)) & 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(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)))))) # label(fact_183__096_B_Bthesis_O_A_I_B_Bs_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 557 (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,N),K) = hAPP(nat,nat,minus_minus(nat,Ma),K))))) # label(fact_625_eq__diff__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 558 (all Z_1 all X_1 all Y_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),Y_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,Z_1),X_1)),hAPP(real,real,plus_plus(real,Z_1),Y_1))))) # label(fact_435_real__add__left__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 559 (all M all N_1 all 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)) = hAPP(nat,nat,times_times(nat,hAPP(nat,nat,minus_minus(nat,M),N_1)),K_1)) # label(fact_630_diff__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 560 (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_876_gcd__lcm__complete__lattice__nat_Ole__bot) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 561 (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_368_Int2_Oaux1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 562 (all X_a (linordered_semidom(X_a) -> -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),zero_zero(X_a))))) # label(fact_509_not__one__less__zero) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 563 (all X_a (dvd(X_a) & linorder(X_a) & plus(X_a) -> (all D all Sa exists Z all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Z)) -> (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),hAPP(X_a,X_a,plus_plus(X_a,X),Sa))) <-> -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),hAPP(X_a,X_a,plus_plus(X_a,X),Sa)))))))) # label(fact_863_minf_I10_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 564 (all X_a (semiring_div(X_a) -> (all A_1 zero_zero(X_a) = hAPP(X_a,X_a,div_mod(X_a,zero_zero(X_a)),A_1)))) # label(fact_944_mod__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 565 (all X_a (ordered_ring(X_a) -> (all A_3 all E all C_1 all B_2 all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_3),E)),C_1)),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,B_2),E)),D))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,minus_minus(X_a,A_3),B_2)),E)),C_1)),D)))))) # label(fact_543_le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 566 (all X_a (ordere453448008miring(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,times_times(X_a,A_1),B)))))))) # label(fact_484_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 567 (all N_1 all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),M)) -> N_1 != M)) # label(fact_569_less__not__refl2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 568 (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_896_neg__zmod__mult__2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 569 (all X_a (ordere453448008miring(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),B)),zero_zero(X_a)))))))) # label(fact_483_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 570 (all A_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A_1)) -> A_1 = zero_zero(nat))) # label(fact_867_gcd__lcm__complete__lattice__nat_Otop__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 571 (all X_a (comm_semiring_1(X_a) -> (all N_1 all M all X_1 all Y_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X_1),Y_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,power_power(X_a,X_1),N_1)),hAPP(nat,X_a,power_power(X_a,Y_1),M)))))))) # label(fact_187_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 572 (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),hAPP(X_a,X_a,plus_plus(X_a,C_1),A_3)),hAPP(X_a,X_a,plus_plus(X_a,C_1),B_2))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)))))) # label(fact_779_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 573 (all X_a (linordered_semiring(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,C),A_1)),hAPP(X_a,X_a,times_times(X_a,C),B))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))))))) # label(fact_536_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 574 (all Z1 all Z2 all Z3 hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,Z1),Z2)),Z3) = hAPP(int,int,plus_plus(int,Z1),hAPP(int,int,plus_plus(int,Z2),Z3))) # label(fact_89_zadd__assoc) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 575 (all Z_1 all X_1 all Y_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),Z_1)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z_1),X_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Z_1))))) # label(fact_579_dvd_Oless__le__trans) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 576 (all X_a (ab_group_add(X_a) -> (all A_3 all B_2 (ti(X_a,A_3) = ti(X_a,B_2) <-> zero_zero(X_a) = hAPP(X_a,X_a,minus_minus(X_a,A_3),B_2))))) # label(fact_786_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 577 (all X_a (linord219039673up_add(X_a) -> (all A_3 (zero_zero(X_a) = ti(X_a,A_3) <-> hAPP(X_a,X_a,plus_plus(X_a,A_3),A_3) = zero_zero(X_a))))) # label(fact_203_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 578 (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_982_mod__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 579 (all Z_1 all X_1 all Y_1 all M (hBOOL(hAPP(int,bool,zcong(X_1,Y_1),M)) -> hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,X_1),Z_1),hAPP(nat,int,power_power(int,Y_1),Z_1)),M)))) # label(fact_400_zcong__zpower) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 580 (all M hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,times_times(nat,M),M)))) # label(fact_703_le__square) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 581 (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),Ma)) <-> 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)))))))) # label(fact_926_dvd__mod__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 582 (all X_1 hAPP(nat,int,power_power(int,hAPP(nat,int,power_power(int,X_1),number_number_of(nat,bit0(bit1(pls))))),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,int,power_power(int,X_1),number_number_of(nat,bit0(bit0(bit1(pls)))))) # label(fact_164_quartic__square__square) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 583 (all X_a (zero_neq_one(X_a) -> zero_zero(X_a) != one_one(X_a))) # label(fact_445_zero__neq__one) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 584 (all S all T_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),S),T_3)) -> S != T_3)) # label(fact_568_less__not__refl3) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 585 (all C all D_2 all A_1 all B all M (hBOOL(hAPP(int,bool,zcong(A_1,B),M)) -> (hBOOL(hAPP(int,bool,zcong(C,D_2),M)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,minus_minus(int,A_1),C),hAPP(int,int,minus_minus(int,B),D_2)),M))))) # label(fact_374_zcong__zdiff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 586 (all W_1 all Z_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W_1),Z_2)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),hAPP(int,int,plus_plus(int,Z_2),one_one(int)))))) # label(fact_63_zle__add1__eq__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 587 (all A_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,div_mod(int,A_1),B)),B)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,div_mod(int,A_1),B))))) # label(fact_956_pos__mod__conj) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 588 (all Y_1 all X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,times_times(real,X_1),Y_1)))))) # label(fact_522_real__mult__order) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 589 (all X_a (linordered_semidom(X_a) -> (all X_2 all Y_2 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),B_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,B_2),X_2)),hAPP(nat,X_a,power_power(X_a,B_2),Y_2))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_2),Y_2))))))) # label(fact_273_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 590 (all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),P_1)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),inv(P_1,A_1)),one_one(int)),P_1)))))) # label(fact_832_inv__is__inv) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 591 (all X_a (semiring_div(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,div_mod(X_a,A_1),C)),hAPP(X_a,X_a,div_mod(X_a,B),C))),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),C)))) # label(fact_931_mod__add__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 592 (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_56_rel__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 593 (all X_2 all N all Y_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N),hAPP(int,int,minus_minus(int,X_2),Y_2))) <-> hAPP(int,int,div_mod(int,X_2),N) = hAPP(int,int,div_mod(int,Y_2),N))) # label(fact_911_zmod__eq__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 594 (all X_a (number_ring(X_a) -> number_number_of(X_a,pls) = zero_zero(X_a))) # label(fact_222_number__of__Pls) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 595 (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_983_mod__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 596 (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_101_rel__simps_I21_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 597 (all X_a (number_semiring(X_a) -> hAPP(X_a,X_a,plus_plus(X_a,one_one(X_a)),one_one(X_a)) = number_number_of(X_a,bit0(bit1(pls))))) # label(fact_169_semiring__one__add__one__is__two) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 598 (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_59_less__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 599 (all M all N_1 all K_1 all L_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_1),L_1)) -> (hAPP(nat,nat,plus_plus(nat,K_1),N_1) = hAPP(nat,nat,plus_plus(nat,M),L_1) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1))))) # label(fact_660_less__add__eq__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 600 (all Q_1 all B all R_1 all C (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_1),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),hAPP(int,int,div_mod(int,Q_1),C))),R_1)),hAPP(int,int,times_times(int,B),C))))))) # label(fact_965_zmult2__lemma__aux4) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 601 (all X_a (number_ring(X_a) -> (all V hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,V)),one_one(X_a)) = number_number_of(X_a,hAPP(int,int,plus_plus(int,V),bit1(pls)))))) # label(fact_17_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 602 (all X_a (power(X_a) & zero_neq_one(X_a) & no_zero_divisors(X_a) & mult_zero(X_a) -> (all A_3 all W_1 (hAPP(nat,X_a,power_power(X_a,A_3),number_number_of(nat,W_1)) = zero_zero(X_a) <-> ti(X_a,A_3) = zero_zero(X_a) & number_number_of(nat,W_1) != zero_zero(nat))))) # label(fact_343_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 603 (all A_1 hAPP(int,int,times_times(int,A_1),hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls))))) = hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit1(bit1(pls))))) # label(fact_11_cube__square) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 604 (all X_a (number_ring(X_a) -> one_one(X_a) = number_number_of(X_a,bit1(pls)))) # label(fact_159_semiring__norm_I110_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 605 (all K all Ma all N (zero_zero(nat) = K | N = Ma <-> hAPP(nat,nat,times_times(nat,K),N) = hAPP(nat,nat,times_times(nat,K),Ma))) # label(fact_725_nat__mult__eq__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 606 (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_805_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 607 (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_16_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 608 (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]. 2.34/2.77 609 (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_936_mod__mult__left__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 610 (all X_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),X_1))) # label(fact_556_dvd_Oorder__refl) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 611 (all M all N_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,minus_minus(nat,M),N_1)),M))) # label(fact_621_Nat_Odiff__le__self) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 612 (all N all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),N)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,minus_minus(nat,N),Ma))))) # label(fact_638_zero__less__diff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 613 (all K_1 all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,div_mod(int,M),K_1)),M)))) # label(fact_909_zmod__le__nonneg__dividend) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 614 (all X_a (comm_monoid_add(X_a) -> (all A_1 hAPP(X_a,X_a,plus_plus(X_a,A_1),zero_zero(X_a)) = ti(X_a,A_1)))) # label(fact_770_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 615 (all K all P_2 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D)) -> ((all X (hBOOL(hAPP(int,bool,P_2,X)) -> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,minus_minus(int,X),D))))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) -> (all X (hBOOL(hAPP(int,bool,P_2,X)) -> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,minus_minus(int,X),hAPP(int,int,times_times(int,K),D)))))))))) # label(fact_748_decr__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 616 (all B_1_1 all X_a (mult_zero(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_4_arg1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 617 (all B all Q_4 all R_3 all Q_1 all R_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),Q_4)),R_3)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),Q_1)),R_1))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_3),B)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_1),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_4),Q_1))))))) # label(fact_357_unique__quotient__lemma) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 618 (all X_a (comm_monoid_add(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,plus_plus(X_a,zero_zero(X_a)),A_1)))) # label(fact_767_add__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 619 (all K all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),hAPP(nat,nat,plus_plus(nat,N),K))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),N)))) # label(fact_671_dvd__reduce) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 620 (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,A_1) = ti(X_a,B)))))))) # label(fact_189_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 621 (all X_a (linordered_idom(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_3),A_3)),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a))))))) # label(fact_217_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 622 (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_91_zadd__commute) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 623 (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 (hAPP(nat,real,power_power(real,X),N_1) = A_1 & (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y)) & A_1 = hAPP(nat,real,power_power(real,Y),N_1) -> Y = X)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))))))) # label(fact_552_realpow__pos__nth__unique) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 624 (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]. 2.34/2.77 625 (all N_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),N_1))) # label(fact_616_le__refl) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 626 (all J_1 all I_1 -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,J_1),I_1)),I_1))) # label(fact_654_not__add__less2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 627 (all X_a (linordered_idom(X_a) -> (all X_1 all Y_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X_1),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,X_a,power_power(X_a,Y_1),number_number_of(nat,bit0(bit1(pls)))))))))) # label(fact_265_sum__power2__ge__zero) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 628 (all N_1 -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),N_1))) # label(fact_570_less__irrefl__nat) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 629 (all P_3 sr(P_3) = collect(int,combs(int,bool,bool,combb(bool,fun(bool,bool),int,fconj,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int))),combc(int,int,bool,ord_less(int),P_3)))) # label(fact_883_SR__def) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 630 (all A_1 all B hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit1(bit1(pls))))),hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,number_number_of(int,bit1(bit1(pls)))),hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls)))))),B))),hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,number_number_of(int,bit1(bit1(pls)))),A_1)),hAPP(nat,int,power_power(int,B),number_number_of(nat,bit0(bit1(pls))))))),hAPP(nat,int,power_power(int,B),number_number_of(nat,bit1(bit1(pls))))) = hAPP(nat,int,power_power(int,hAPP(int,int,plus_plus(int,A_1),B)),number_number_of(nat,bit1(bit1(pls))))) # label(fact_8_zadd__power3) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 631 (all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1)) -> (exists P_4 (hBOOL(hAPP(int,bool,zprime,P_4)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_4),A_1)))))) # label(fact_750_zprime__factor__exists) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 632 (all A_1 -(-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),zero_zero(nat))) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A_1)))) # label(fact_871_gcd__lcm__complete__lattice__nat_Onot__top__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 633 (all B_1_1 all B_2_1 standardRes(B_1_1,B_2_1) = standardRes(ti(int,B_1_1),B_2_1)) # label(tsy_c_Residues_OStandardRes_arg1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 634 (all M all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),hAPP(nat,nat,plus_plus(nat,M),J_1))))) # label(fact_657_trans__less__add2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 635 (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(int),R_1),A_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),Q_1)))))) # label(fact_282_self__quotient__aux1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 636 (all X_a (linordered_ring(X_a) -> (all A_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,times_times(X_a,A_1),A_1)))))) # label(fact_487_zero__le__square) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 637 (all X_a (cancel_semigroup_add(X_a) -> (all B all A_1 all C (hAPP(X_a,X_a,plus_plus(X_a,B),A_1) = hAPP(X_a,X_a,plus_plus(X_a,C),A_1) -> ti(X_a,C) = ti(X_a,B))))) # label(fact_763_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 638 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all Y_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),number_number_of(X_a,Y_2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),Y_2)))))) # label(fact_107_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 639 (all A_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,div_mod(int,A_1),B)),B)))) # label(fact_907_pos__mod__bound) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 640 (all K_1 all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),hAPP(nat,nat,minus_minus(nat,M),N_1))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),N_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),M)))))) # label(fact_673_dvd__diffD) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 641 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K)),bit1(L))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)))) # label(fact_60_rel__simps_I15_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 642 (all X_a (comm_semiring_1(X_a) -> (all X_1 all P_1 all Q_1 hAPP(X_a,X_a,times_times(X_a,hAPP(nat,X_a,power_power(X_a,X_1),P_1)),hAPP(nat,X_a,power_power(X_a,X_1),Q_1)) = hAPP(nat,X_a,power_power(X_a,X_1),hAPP(nat,nat,plus_plus(nat,P_1),Q_1))))) # label(fact_33_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 643 (all B_3 all Q_4 all R_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B_3),Q_4)),R_3))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_3),B_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_3)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Q_4)))))) # label(fact_359_q__pos__lemma) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 644 (all X_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),P_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),standardRes(P_1,X_1))))) # label(fact_993_StandardRes__lbound) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 645 (all X_a all P_2 ti(fun(X_a,bool),P_2) = collect(X_a,P_2)) # label(fact_76_Collect__def) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 646 (all Z_1 all N_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Z_1),N_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),N_1))))) # label(fact_196_zdvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 647 (all X_a (ordere223160158up_add(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,C),A_1)),hAPP(X_a,X_a,plus_plus(X_a,C),B))))))) # label(fact_781_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 648 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K)),bit1(L))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)))) # label(fact_99_rel__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 649 (all N_1 hAPP(nat,nat,times_times(nat,zero_zero(nat)),N_1) = zero_zero(nat)) # label(fact_648_mult__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 650 (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_262_power__one) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 651 (all N_1 all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N_1),M)) -> zero_zero(nat) = 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)) | N_1 = M)) # label(fact_419_divides__cases) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 652 -(all S1 -hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,S1),number_number_of(nat,bit0(bit1(pls)))),number_number_of(int,min)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int))))) # label(fact_284__096_B_Bthesis_O_A_I_B_Bs1_O_A_091s1_A_094_A2_A_061_A_N1_093_A_Imod_A4_) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 653 (all X_2 all Y_2 (zero_zero(real) = hAPP(real,real,plus_plus(real,hAPP(nat,real,power_power(real,X_2),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,real,power_power(real,Y_2),number_number_of(nat,bit0(bit1(pls))))) <-> zero_zero(real) = Y_2 & zero_zero(real) = X_2)) # label(fact_421_realpow__two__sum__zero__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 654 (all X_a (comm_semiring_1(X_a) -> (all A_1 hAPP(X_a,X_a,plus_plus(X_a,A_1),zero_zero(X_a)) = ti(X_a,A_1)))) # label(fact_201_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 655 (all M all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),hAPP(nat,nat,plus_plus(nat,J_1),M))))) # label(fact_656_trans__less__add1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 656 (all X_1 all Y_1 all Z_1 hAPP(int,int,plus_plus(int,X_1),hAPP(int,int,plus_plus(int,Y_1),Z_1)) = hAPP(int,int,plus_plus(int,Y_1),hAPP(int,int,plus_plus(int,X_1),Z_1))) # label(fact_90_zadd__left__commute) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 657 (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_79_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 658 (all T all A ti(T,ti(T,A)) = ti(T,A)) # label(help_ti_idem) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 659 (all X_2 all Y_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),number_number_of(int,X_2)),number_number_of(int,Y_2))) <-> hAPP(int,int,div_mod(int,number_number_of(int,Y_2)),number_number_of(int,X_2)) = zero_zero(int))) # label(fact_898_zdvd__iff__zmod__eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 660 (all K_1 all N_1 all M hAPP(nat,nat,div_mod(nat,M),N_1) = hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,K_1),N_1)),M)),N_1)) # label(fact_979_mod__mult__self3) # label(axiom) # label(non_clause). [assumption]. 2.34/2.77 661 (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,ti(fun(X_a,X_b),B_2_1)) = combs(X_a,X_b,X_c,B_1_1,B_2_1)) # label(tsy_c_COMBS_arg2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 662 (all N_1 hAPP(nat,nat,minus_minus(nat,zero_zero(nat)),N_1) = zero_zero(nat)) # label(fact_563_diff__0__eq__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 663 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,div_mod(nat,M),N_1)),N_1)))) # label(fact_974_mod__le__divisor) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 664 (all X_a (linordered_ring(X_a) -> (all X_1 all Y_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,X_1),X_1)),hAPP(X_a,X_a,times_times(X_a,Y_1),Y_1))))))) # label(fact_235_sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 665 (all B_1_1 all X_a (mult_zero(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_4_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 666 (all Z1 all Z2 all W hAPP(real,real,times_times(real,hAPP(real,real,plus_plus(real,Z1),Z2)),W) = hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,Z1),W)),hAPP(real,real,times_times(real,Z2),W))) # label(fact_468_real__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 667 (all Ma all N (hAPP(nat,nat,plus_plus(nat,Ma),N) = zero_zero(nat) <-> zero_zero(nat) = N & Ma = zero_zero(nat))) # label(fact_640_add__is__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 668 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K1)),bit1(K2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)))) # label(fact_38_less__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 669 (all X_a (comm_semiring_1(X_a) -> (all A_1 hAPP(X_a,X_a,plus_plus(X_a,zero_zero(X_a)),A_1) = ti(X_a,A_1)))) # label(fact_200_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 670 (all B_1_1 all X_a (power(X_a) -> power_power(X_a,B_1_1) = ti(fun(nat,X_a),power_power(X_a,B_1_1)))) # label(tsy_c_Power_Opower__class_Opower_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 671 (all K_1 ti(int,K_1) = hAPP(int,int,plus_plus(int,pls),K_1)) # label(fact_126_add__Pls) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 672 (all X_a (linord20386208strict(X_a) -> (all C all D_2 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),D_2)))))))))) # label(fact_533_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 673 (all C all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_1),B)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),A_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,minus_minus(nat,A_1),C)),hAPP(nat,nat,minus_minus(nat,B),C)))))) # label(fact_670_diff__less__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 674 (all B_2 all A_3 all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,minus_minus(int,P_3),one_one(int)))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),B_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),A_3)) -> hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),wset(A_3,P_3)))))))) # label(fact_848_wset__mem) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 675 (all X_a (real_normed_algebra(X_a) -> (all A_1 all A_4 all B hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,minus_minus(X_a,A_1),A_4)),B) = hAPP(X_a,X_a,minus_minus(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),B)),hAPP(X_a,X_a,times_times(X_a,A_4),B))))) # label(fact_894_mult_Odiff__left) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 676 (all Q_1 all B all R_1 all C (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_1),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),hAPP(int,int,div_mod(int,Q_1),C))),R_1))))))) # label(fact_964_zmult2__lemma__aux3) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 677 (all X_a (linord219039673up_add(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,A_3),A_3))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3)))))) # label(fact_797_zero__le__double__add__iff__zero__le__single__add) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 678 (all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,plus_plus(int,one_one(int)),Z_1))))) # label(fact_250_le__imp__0__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 679 (all B_1_1 twoSqu1929807760sum2sq(B_1_1) = twoSqu1929807760sum2sq(ti(product_prod(int,int),B_1_1))) # label(tsy_c_TwoSquares__Mirabelle__lldtdwpyrp_Osum2sq_arg1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 680 (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]. 2.34/2.78 681 (all V_2 all V_3 ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_2),V_3)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),V_3))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_2),V_3)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),number_number_of(nat,V_2)),number_number_of(nat,V_3))))) # label(fact_237_less__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 682 (all X_2 all P_3 (hBOOL(hAPP(fun(int,bool),bool,member(int,X_2),sr(P_3))) -> standardRes(P_3,X_2) = ti(int,X_2))) # label(fact_990_StandardRes__SR__prop) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 683 (all X_a (semiring_div(X_a) -> (all A_1 hAPP(X_a,X_a,div_mod(X_a,A_1),A_1) = zero_zero(X_a)))) # label(fact_942_mod__self) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 684 (all X_a (number_ring(X_a) -> (all Z_1 hAPP(X_a,X_a,times_times(X_a,Z_1),number_number_of(X_a,bit0(bit1(pls)))) = hAPP(X_a,X_a,plus_plus(X_a,Z_1),Z_1)))) # label(fact_168_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 685 (all X_2 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,power_power(nat,X_2),N))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X_2)) | zero_zero(nat) = N)) # label(fact_287_zero__less__power__nat__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 686 (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]. 2.34/2.78 687 (all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (ti(int,P_1) != number_number_of(int,bit0(bit1(pls))) -> (number_number_of(int,bit1(bit1(pls))) != ti(int,P_1) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,bit1(bit0(bit1(pls))))),P_1)))))) # label(fact_172_prime__g__5) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 688 (all X_a (comm_semiring_1(X_a) -> (all N_1 all X_1 all Y_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X_1),Y_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,power_power(X_a,X_1),N_1)),hAPP(nat,X_a,power_power(X_a,Y_1),N_1))))))) # label(fact_193_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 689 (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_1),B)),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,times_times(X_a,A_4),B_3)),C)))))) # label(fact_941_mod__mult__cong) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 690 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B all C (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),B)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C)))))) # label(fact_452_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 691 (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)) & N != Ma)) # label(fact_664_nat__less__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 692 (all X_a (real_normed_algebra(X_a) -> (all Xa all X_1 all Y_1 hAPP(X_a,X_a,times_times(X_a,Xa),hAPP(X_a,X_a,plus_plus(X_a,X_1),Y_1)) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,Xa),X_1)),hAPP(X_a,X_a,times_times(X_a,Xa),Y_1))))) # label(fact_890_mult__right_Oadd) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 693 (all A_1 all J_1 all K_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,zcong(K_1,zero_zero(int)),P_1)) -> (-hBOOL(hAPP(int,bool,zcong(J_1,zero_zero(int)),P_1)) -> (hBOOL(hAPP(int,bool,zcong(J_1,hAPP(int,int,times_times(int,A_1),multInv(P_1,K_1))),P_1)) -> hBOOL(hAPP(int,bool,zcong(K_1,hAPP(int,int,times_times(int,A_1),multInv(P_1,J_1))),P_1)))))))) # label(fact_835_MultInv__zcong__prop2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 694 (all K all L (ti(int,K) = ti(int,L) <-> bit1(K) = bit1(L))) # label(fact_84_rel__simps_I51_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 695 (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]. 2.34/2.78 696 (all X_a (semiri456707255roduct(X_a) -> (all B_2 all A_3 (ti(X_a,A_3) = zero_zero(X_a) <-> hAPP(X_a,X_a,plus_plus(X_a,B_2),A_3) = ti(X_a,B_2))))) # label(fact_202_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 697 (all X_a (number_ring(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,times_times(X_a,A_1),number_number_of(X_a,bit1(pls)))))) # label(fact_156_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 698 (all X_1 all Y_1 all N_1 (zero_zero(nat) != N_1 -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,X_1),N_1)),Y_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1))))) # label(fact_525_divides__exp2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 699 (all X_a (linord581940658strict(X_a) -> (all C all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,C),A_1)),hAPP(X_a,X_a,times_times(X_a,C),B)))))))) # label(fact_488_mult__strict__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 700 (all K_1 ti(int,K_1) = number_number_of(int,K_1)) # label(fact_88_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 701 (all N_1 -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),zero_zero(nat)))) # label(fact_634_less__nat__zero__code) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 702 (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),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),A_3),B_2))))))) # label(fact_501_mult__less__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 703 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K1)),bit0(K2))))) # label(fact_43_less__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 704 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all X_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_2),bit1(pls))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),number_number_of(X_a,X_2)),one_one(X_a))))))) # label(fact_104_less__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 705 (all Z_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_2),zero_zero(int))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,one_one(int)),Z_2)),Z_2)),zero_zero(int))))) # label(fact_245_odd__less__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 706 (all M all N_1 M = hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,plus_plus(nat,M),N_1)),N_1)) # label(fact_611_diff__add__inverse2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 707 (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(nat,X_a,power_power(X_a,X_1),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))))) # label(fact_436_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 708 (all X_a (linordered_semidom(X_a) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,one_one(X_a)),one_one(X_a)))))) # label(fact_542_zero__less__two) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 709 (all X_a (comm_semiring_1(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,plus_plus(X_a,B),C)))))))) # label(fact_459_dvd__add) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 710 (all K_1 pls != bit1(K_1)) # label(fact_116_rel__simps_I46_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 711 (all M all N_1 (M = 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_668_less__or__eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 712 (all Y_1 all N_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),hAPP(nat,int,power_power(int,Y_1),N_1))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),Y_1)))))) # label(fact_416_zpower__zdvd__prop2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 713 (all Ma all N ((exists K_2 N = hAPP(nat,nat,plus_plus(nat,Ma),K_2)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),N)))) # label(fact_676_le__iff__add) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 714 (all I_1 all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,I_1),M)),hAPP(nat,nat,power_power(nat,I_1),N_1))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),I_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1))))) # label(fact_312_power__dvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 715 (all K_1 all L_1 bit0(L_1) != bit1(K_1)) # label(fact_118_rel__simps_I50_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 716 (all Z_1 all X_1 all Y_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) -> (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z_1),Y_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),Z_1)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z_1),X_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Z_1))))) # label(fact_588_dvd_Ole__less__trans) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 717 (all X_a (comm_semiring_1(X_a) -> (all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),zero_zero(X_a)),A_1)) -> ti(X_a,A_1) = zero_zero(X_a))))) # label(fact_449_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 718 (all X_a (number_ring(X_a) -> (all N_1 one_one(X_a) = hAPP(nat,X_a,power_power(X_a,number_number_of(X_a,min)),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),N_1))))) # label(fact_342_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 719 (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]. 2.34/2.78 720 (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_798_double__add__le__zero__iff__single__add__le__zero) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 721 (all P_2 all A_3 all B_2 (hBOOL(hAPP(nat,bool,P_2,hAPP(nat,nat,minus_minus(nat,A_3),B_2))) <-> (all D_4 (hAPP(nat,nat,plus_plus(nat,B_2),D_4) = A_3 -> hBOOL(hAPP(nat,bool,P_2,D_4)))) & (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_3),B_2)) -> hBOOL(hAPP(nat,bool,P_2,zero_zero(nat)))))) # label(fact_710_nat__diff__split) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 722 (all X_a (linordered_idom(X_a) -> (all A_1 all K_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,A_1),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),K_1))),zero_zero(X_a))) -> zero_zero(X_a) = ti(X_a,A_1))))) # label(fact_279_even__power__le__0__imp__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 723 (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]. 2.34/2.78 724 (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_224_bin__less__0__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 725 (all X_a (ordered_ring(X_a) -> (all A_3 all E all C_1 all B_2 all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,minus_minus(X_a,B_2),A_3)),E)),D))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_3),E)),C_1)),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,B_2),E)),D))))))) # label(fact_546_less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 726 (all W_1 all Z_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),Z_2)) | ti(int,W_1) = ti(int,Z_2) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),hAPP(int,int,plus_plus(int,Z_2),one_one(int)))))) # label(fact_102_zless__add1__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 727 (all X_a (ordere223160158up_add(X_a) -> (all C all D_2 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_2)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)),hAPP(X_a,X_a,plus_plus(X_a,B),D_2)))))))) # label(fact_809_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 728 (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_892_mult_Odiff__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 729 (all X_a (number_semiring(X_a) -> (all X_1 all Y_1 hAPP(nat,X_a,power_power(X_a,hAPP(X_a,X_a,plus_plus(X_a,X_1),Y_1)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X_1),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,X_a,power_power(X_a,Y_1),number_number_of(nat,bit0(bit1(pls)))))),hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,number_number_of(X_a,bit0(bit1(pls)))),X_1)),Y_1))))) # label(fact_9_power2__sum) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 730 (all M all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),hAPP(nat,nat,plus_plus(nat,J_1),M))))) # label(fact_678_trans__le__add1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 731 (all X_a (ordere223160158up_add(X_a) -> (all C all D_2 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_2)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)),hAPP(X_a,X_a,plus_plus(X_a,B),D_2)))))))) # label(fact_808_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 732 (all P_1 all Y_1 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),Y_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),hAPP(nat,int,power_power(int,Y_1),N_1)))))) # label(fact_411_zpower__zdvd__prop1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 733 (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]. 2.34/2.78 734 (all K (bit1(K) = min <-> min = ti(int,K))) # label(fact_290_rel__simps_I47_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 735 (all X_a (no_zero_divisors(X_a) -> (all A_1 all B (zero_zero(X_a) = hAPP(X_a,X_a,times_times(X_a,A_1),B) -> zero_zero(X_a) = ti(X_a,B) | ti(X_a,A_1) = zero_zero(X_a))))) # label(fact_440_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 736 (all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),A_1),one_one(int)),P_1)) -> hBOOL(hAPP(int,bool,zcong(A_1,hAPP(int,int,minus_minus(int,P_1),one_one(int))),P_1)) | hBOOL(hAPP(int,bool,zcong(A_1,one_one(int)),P_1)))))) # label(fact_393_zcong__square) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 737 (all X_a (mult_zero(X_a) & zero_neq_one(X_a) & no_zero_divisors(X_a) & power(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_185_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 738 (all M all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),M)) -> (hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),M)) -> ti(int,X_1) = zero_zero(int))))) # label(fact_410_Int2_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 739 (all X_a (number_semiring(X_a) -> zero_zero(X_a) = number_number_of(X_a,pls))) # label(fact_223_semiring__numeral__0__eq__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 740 (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,bit1(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)))))) # label(fact_739_number__of1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 741 (all X_1 all Y_1 (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) -> -(-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1))))) # label(fact_576_dvd_Oless__asym) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 742 (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]. 2.34/2.78 743 (all X_a (real_normed_algebra(X_a) -> (all X_1 all Y_1 all A_1 all B hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,minus_minus(X_a,X_1),A_1)),hAPP(X_a,X_a,minus_minus(X_a,Y_1),B))),hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,minus_minus(X_a,X_1),A_1)),B))),hAPP(X_a,X_a,times_times(X_a,A_1),hAPP(X_a,X_a,minus_minus(X_a,Y_1),B))) = hAPP(X_a,X_a,minus_minus(X_a,hAPP(X_a,X_a,times_times(X_a,X_1),Y_1)),hAPP(X_a,X_a,times_times(X_a,A_1),B))))) # label(fact_882_mult_Oprod__diff__prod) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 744 (all Ma all N (hAPP(nat,nat,minus_minus(nat,Ma),N) = zero_zero(nat) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),N)))) # label(fact_647_diff__is__0__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 745 (all X_a (ordere223160158up_add(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)),hAPP(X_a,X_a,plus_plus(X_a,B),C))))))) # label(fact_780_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 746 (all X_a (dvd(X_a) & linorder(X_a) & plus(X_a) -> (all D all Sa exists Z all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),X)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),hAPP(X_a,X_a,plus_plus(X_a,X),Sa))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),hAPP(X_a,X_a,plus_plus(X_a,X),Sa)))))))) # label(fact_862_pinf_I9_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 747 (all X_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_1)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,X_1),multInv(P_1,X_1)),one_one(int)),P_1)))))) # label(fact_840_MultInv__prop2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 748 (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]. 2.34/2.78 749 (all X_a (comm_monoid_mult(X_a) -> (all A_1 hAPP(X_a,X_a,times_times(X_a,one_one(X_a)),A_1) = ti(X_a,A_1)))) # label(fact_792_mult__1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 750 (all X_a (monoid_mult(X_a) -> (all A_1 all N_1 hAPP(X_a,X_a,times_times(X_a,hAPP(nat,X_a,power_power(X_a,A_1),N_1)),A_1) = hAPP(X_a,X_a,times_times(X_a,A_1),hAPP(nat,X_a,power_power(X_a,A_1),N_1))))) # label(fact_259_power__commutes) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 751 (all Y_1 all X_1 (hBOOL(hAPP(int,bool,twoSqu658283162sum2sq,X_1)) -> (hBOOL(hAPP(int,bool,twoSqu658283162sum2sq,Y_1)) -> hBOOL(hAPP(int,bool,twoSqu658283162sum2sq,hAPP(int,int,times_times(int,X_1),Y_1)))))) # label(fact_65_is__mult__sum2sq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 752 (all X_a all X_c all X_b all P all Q all R hAPP(X_a,X_c,combs(X_a,X_b,X_c,P,Q),R) = hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),P,R),hAPP(X_a,X_b,Q,R))) # label(help_COMBS_1_1_U) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 753 (all X_a (real_normed_algebra(X_a) -> (all X_1 all Y_1 all Ya hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,X_1),Ya)),hAPP(X_a,X_a,times_times(X_a,Y_1),Ya)) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,plus_plus(X_a,X_1),Y_1)),Ya)))) # label(fact_888_mult__left_Oadd) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 754 (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_150_number__of__add) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 755 (all X_a (group_add(X_a) -> (all A_1 all B ti(X_a,A_1) = hAPP(X_a,X_a,minus_minus(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),B)))) # label(fact_795_add__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 756 (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_231_odd__nonzero) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 757 (exists X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) & hBOOL(hAPP(int,bool,zcong(s1,X),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int)))) & (all Y (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y)) & hBOOL(hAPP(int,bool,zcong(s1,Y),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int)))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int)))) -> ti(int,X) = ti(int,Y))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int)))))) # label(fact_182__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 758 (all N_1 -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),zero_zero(nat)))) # label(fact_636_not__less0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 759 (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_549_divides__div__not) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 760 (all A_3 all B_2 all C_1 (zero_zero(real) != C_1 -> (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_463_real__mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 761 (all C all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,times_times(nat,A_1),C)),hAPP(nat,nat,times_times(nat,B),C))))) # label(fact_466_divides__mul__r) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 762 (all X_a (ordere779506340up_add(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,C),A_1)),hAPP(X_a,X_a,plus_plus(X_a,C),B))))))) # label(fact_774_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 763 (all X_a (linord581940658strict(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,times_times(X_a,A_1),B)))))))) # label(fact_493_mult__neg__neg) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 764 (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_138_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 765 (all A_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),one_one(nat)),A_3)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_3),one_one(nat))) <-> A_3 != one_one(nat))) # label(fact_874_gcd__lcm__complete__lattice__nat_Obot__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 766 (all X_1 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X_1),hAPP(nat,int,power_power(int,X_1),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_137_power2__ge__self) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 767 (all X_a (number(X_a) -> (all W_1 all X_2 (number_number_of(X_a,W_1) = ti(X_a,X_2) <-> ti(X_a,X_2) = number_number_of(X_a,W_1))))) # label(fact_83_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 768 (all X_a (semiring_div(X_a) -> (all B all A_1 hAPP(X_a,X_a,div_mod(X_a,A_1),B) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,B),A_1)),B)))) # label(fact_928_mod__add__self1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 769 (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]. 2.34/2.78 770 (all N_1 all J_1 all K_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J_1),K_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,minus_minus(nat,J_1),N_1)),K_1)))) # label(fact_575_less__imp__diff__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 771 (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),M) = hAPP(nat,nat,times_times(nat,K_1),N_1) -> N_1 = M))) # label(fact_881_mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 772 (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_572_nat__neq__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 773 (all K_1 all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,times_times(nat,K_1),I_1)),hAPP(nat,nat,times_times(nat,K_1),J_1)))))) # label(fact_712_mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 774 (all B_2 all A_3 (hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),d22set(A_3))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),B_2)))) # label(fact_859_d22set__g__1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 775 (all B_2 all A_3 all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,bit1(bit0(bit1(pls))))),P_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,minus_minus(int,P_3),one_one(int)))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),B_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_2),hAPP(int,int,minus_minus(int,P_3),one_one(int)))) -> (hBOOL(hAPP(fun(int,bool),bool,member(int,inv(P_3,B_2)),wset(A_3,P_3))) -> hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),wset(A_3,P_3)))))))))) # label(fact_842_wset__inv__mem__mem) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 776 (all X_a (zero(X_a) -> zero_zero(X_a) = ti(X_a,zero_zero(X_a)))) # label(tsy_c_Groups_Ozero__class_Ozero_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 777 (all X_a (linorder(X_a) & dvd(X_a) & plus(X_a) -> (all D all Sa exists Z all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),X)) -> (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),hAPP(X_a,X_a,plus_plus(X_a,X),Sa))) <-> -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),hAPP(X_a,X_a,plus_plus(X_a,X),Sa)))))))) # label(fact_864_pinf_I10_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 778 (all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> hBOOL(hAPP(int,bool,zcong(zfact(hAPP(int,int,minus_minus(int,P_1),one_one(int))),number_number_of(int,min)),P_1)))) # label(fact_753_Wilson__Russ) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 779 (all X_a (linordered_semidom(X_a) -> -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),zero_zero(X_a))))) # label(fact_507_not__one__le__zero) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 780 (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,I_2),U)),Ma) = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,J_2),U)),N) <-> N = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,hAPP(nat,nat,minus_minus(nat,I_2),J_2)),U)),Ma)))) # label(fact_734_nat__eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 781 (all X_a (ordered_ring(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,times_times(X_a,A_1),B)))))))) # label(fact_480_mult__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 782 (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]. 2.34/2.78 783 (all X_1 hAPP(nat,real,power_power(real,hAPP(real,real,times_times(real,number_number_of(real,bit0(bit1(pls)))),X_1)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(real,real,times_times(real,number_number_of(real,bit0(bit0(bit1(pls))))),hAPP(nat,real,power_power(real,X_1),number_number_of(nat,bit0(bit1(pls)))))) # label(fact_175_four__x__squared) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 784 (all B_1_1 all X_a (ab_semigroup_mult(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_0_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 785 (all X_a (linord626643107strict(X_a) -> (all V all U_1 all Y_1 all X_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_1),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),U_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),V)) -> (one_one(X_a) = hAPP(X_a,X_a,plus_plus(X_a,U_1),V) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,U_1),X_1)),hAPP(X_a,X_a,times_times(X_a,V),Y_1))),A_1)))))))))) # label(fact_422_convex__bound__lt) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 786 (all X_a (linordered_semidom(X_a) -> (all N_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N_1)),hAPP(nat,X_a,power_power(X_a,B),N_1))))))))) # label(fact_197_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 787 (all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,minus_minus(int,P_1),one_one(int)))) -> one_one(int) != inv(P_1,A_1))))) # label(fact_822_inv__not__1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.78 788 (all A_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,div_mod(int,A_1),B))))) # label(fact_955_pos__mod__sign) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 789 (all N_1 all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M)) -> M = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,minus_minus(nat,M),N_1)),N_1))) # label(fact_691_le__add__diff__inverse2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 790 (all N all P_2 (-hBOOL(hAPP(nat,bool,P_2,zero_zero(nat))) -> (hBOOL(hAPP(nat,bool,P_2,N)) -> (exists K_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_2),N)) & (all I (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),K_2)) -> -hBOOL(hAPP(nat,bool,P_2,I)))) & hBOOL(hAPP(nat,bool,P_2,hAPP(nat,nat,plus_plus(nat,K_2),one_one(nat))))))))) # label(fact_723_ex__least__nat__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 791 (all A_3 all B_2 all N (N != zero_zero(nat) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_3),B_2)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,A_3),N)),hAPP(nat,int,power_power(int,B_2),N)))))) # label(fact_866_pow__divides__eq__int) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 792 (all X_a (linordered_semidom(X_a) -> (all N_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(nat,X_a,power_power(X_a,A_1),N_1)))))))) # label(fact_321_one__less__power) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 793 (all X_a (number_semiring(X_a) -> one_one(X_a) = number_number_of(X_a,bit1(pls)))) # label(fact_157_semiring__numeral__1__eq__1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 794 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(K)),pls)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),pls)))) # label(fact_55_rel__simps_I29_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 795 (all X_a (linord20386208strict(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),C)))))))) # label(fact_492_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 796 (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_251_zero__power2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 797 (all X_a (power(X_a) -> (all A_1 hAPP(nat,X_a,power_power(X_a,A_1),zero_zero(nat)) = one_one(X_a)))) # label(fact_314_power__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 798 (all X_a (ordered_semiring(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,C),A_1)),hAPP(X_a,X_a,times_times(X_a,C),B)))))))) # label(fact_478_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 799 (all Ma all N all K (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),N)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,times_times(nat,K),Ma)),hAPP(nat,nat,times_times(nat,K),N)))))) # label(fact_728_nat__mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 800 (all X_a (number_ring(X_a) -> zero_zero(X_a) = number_number_of(X_a,pls))) # label(fact_221_semiring__norm_I112_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 801 (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),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),dvd_dvd(nat),Ma),N))))) # label(fact_730_nat__mult__dvd__cancel1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 802 (all X_a all X_c all X_b all B_1_1 all B_2_1 ti(fun(X_a,X_c),combc(X_a,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_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 803 (all A_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),hAPP(int,int,div_mod(int,A_1),B))))) # label(fact_906_neg__mod__bound) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 804 (all X_a (linord20386208strict(X_a) -> (all C all D_2 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),D_2)))))))))) # label(fact_530_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 805 (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]. 2.34/2.79 806 (all I_1 all K_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J_1)) -> hAPP(nat,nat,plus_plus(nat,I_1),hAPP(nat,nat,minus_minus(nat,J_1),K_1)) = hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,plus_plus(nat,I_1),J_1)),K_1))) # label(fact_689_add__diff__assoc) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 807 (all B_1_1 all X_a (semiring_div(X_a) -> div_mod(X_a,B_1_1) = div_mod(X_a,ti(X_a,B_1_1)))) # label(tsy_c_Divides_Odiv__class_Omod_arg1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 808 (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_155_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 809 (all B_2 all A_3 all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,minus_minus(int,P_3),one_one(int)))) -> (hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),wset(A_3,P_3))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_2),hAPP(int,int,minus_minus(int,P_3),one_one(int)))))))) # label(fact_845_wset__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 810 (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]. 2.34/2.79 811 (all A_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,div_mod(int,A_1),B)),zero_zero(int))))) # label(fact_958_neg__mod__sign) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 812 (all X_a (order(X_a) -> ti(fun(X_a,fun(X_a,bool)),ord_less(X_a)) = ord_less(X_a))) # label(tsy_c_Orderings_Oord__class_Oless_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 813 (all W_1 all Z_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),Z_2)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W_1),hAPP(int,int,minus_minus(int,Z_2),one_one(int)))))) # label(fact_386_zle__diff1__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 814 (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]. 2.34/2.79 815 (all Z_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z_2)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z_2)))) # label(fact_243_int__one__le__iff__zero__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 816 (all X_a (semiring_div(X_a) -> (all A_1 all C all B hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),hAPP(X_a,X_a,times_times(X_a,C),B))),B) = hAPP(X_a,X_a,div_mod(X_a,A_1),B)))) # label(fact_902_mod__mult__self1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 817 (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_148_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 818 (all X_a (real_normed_algebra(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_885_mult_Ozero__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 819 (all Ma all K all 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))) <-> (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))))) # label(fact_721_mult__le__cancel2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 820 (all X_1 all Y_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)))) # label(fact_585_dvd_Oless__imp__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 821 (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_560_real__le__linear) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 822 (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),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,A_3),C_1)),hAPP(X_a,X_a,plus_plus(X_a,B_2),C_1))))))) # label(fact_771_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 823 (all N_1 (N_1 != zero_zero(nat) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)))) # label(fact_632_gr0I) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 824 (all K_1 all M all N_1 hAPP(nat,nat,minus_minus(nat,M),N_1) = hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,plus_plus(nat,K_1),M)),hAPP(nat,nat,plus_plus(nat,K_1),N_1))) # label(fact_614_Nat_Odiff__cancel) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 825 (all Ma all N (ti(int,N) = number_number_of(int,min) & ti(int,Ma) = number_number_of(int,min) | ti(int,N) = one_one(int) & ti(int,Ma) = one_one(int) <-> one_one(int) = hAPP(int,int,times_times(int,Ma),N))) # label(fact_320_zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 826 (all X_2 all Y_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),Y_2)) <-> X_2 = Y_2 | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_2),Y_2)))) # label(fact_429_less__eq__real__def) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 827 (all X_a (ordere236663937imp_le(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,C),A_1)),hAPP(X_a,X_a,plus_plus(X_a,C),B))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)))))) # label(fact_777_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 828 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),A_1)) -> ti(int,A_1) = hAPP(int,int,div_mod(int,A_1),B)))) # label(fact_960_mod__neg__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 829 (all N_1 all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),M),N_1)) -> -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N_1),M))))) # label(fact_190_zdvd__not__zless) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 830 (all X_a (comm_semiring_1(X_a) -> (all C all D_2 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),C),D_2)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),D_2)))))))) # label(fact_454_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 831 (all X_a all X_c all B_1_1 all B_2_1 hAPP(X_a,X_c,B_1_1,ti(X_a,B_2_1)) = hAPP(X_a,X_c,B_1_1,B_2_1)) # label(tsy_c_hAPP_arg2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 832 (all X_a (linorder(X_a) & number(X_a) -> (all V_2 all W_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,V_2)),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_2))))))) # label(fact_29_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 833 (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_77_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 834 (all B_1_1 all X_a (dvd(X_a) -> times_times(X_a,B_1_1) = ti(fun(X_a,X_a),times_times(X_a,B_1_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_2_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 835 (all L_1 pls != bit1(L_1)) # label(fact_117_rel__simps_I39_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 836 (all C_1 all X_2 all Ta all A_3 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_3),D)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_3),hAPP(int,int,plus_plus(int,X_2),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_2),hAPP(int,int,times_times(int,C_1),D))),Ta)))))) # label(fact_213_zdvd__period) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 837 (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_144_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 838 (all Y_2 all X_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_2),X_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_2),Y_2)) <-> X_2 = Y_2))) # label(fact_597_dvd_Oantisym__conv) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 839 (all Z_3 all Z_1 all W_2 all W (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_2),W)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_3),Z_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,W_2),Z_3)),hAPP(int,int,plus_plus(int,W),Z_1)))))) # label(fact_32_zadd__zless__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 840 (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]. 2.34/2.79 841 (all K_1 all M all N_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,times_times(int,K_1),M)),hAPP(int,int,times_times(int,K_1),N_1))) -> (ti(int,K_1) != zero_zero(int) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M),N_1))))) # label(fact_192_zdvd__mult__cancel) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 842 (all X_1 all Y_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) | Y_1 = X_1)) # label(fact_596_dvd_Ole__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 843 (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)) & A_1 = hAPP(nat,real,power_power(real,R_2),N_1)))))) # label(fact_551_realpow__pos__nth) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 844 (all X_2 all Y_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),Y_2)) & Y_2 != X_2 <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_2),Y_2)))) # label(fact_430_real__less__def) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 845 (all M hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,times_times(nat,M),hAPP(nat,nat,times_times(nat,M),M))))) # label(fact_702_le__cube) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 846 (all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,minus_minus(int,P_1),one_one(int)))) -> inv(P_1,A_1) != hAPP(int,int,minus_minus(int,P_1),one_one(int)))))) # label(fact_823_inv__not__p__minus__1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 847 (all A_3 (A_3 != zero_zero(nat) <-> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A_3)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_3),zero_zero(nat))))) # label(fact_869_gcd__lcm__complete__lattice__nat_Oless__top) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 848 (all X_a (real_normed_algebra(X_a) -> (all Y_1 zero_zero(X_a) = hAPP(X_a,X_a,times_times(X_a,zero_zero(X_a)),Y_1)))) # label(fact_886_mult__left_Ozero) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 849 (all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> hAPP(int,int,div_mod(int,number_number_of(int,min)),B) = hAPP(int,int,minus_minus(int,B),one_one(int)))) # label(fact_970_zmod__minus1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 850 (all K_1 all I_1 all J_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),J_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,I_1),K_1)),hAPP(int,int,plus_plus(int,J_1),K_1))))) # label(fact_50_zadd__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 851 (all X_a (ordered_semiring(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),C)))))))) # label(fact_479_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 852 (all N_1 all K_1 all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),M)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),N_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),hAPP(nat,nat,minus_minus(nat,M),N_1)))))) # label(fact_605_dvd__diff__nat) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 853 (all X_a (linordered_semidom(X_a) -> (all M all N_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),M)),hAPP(nat,X_a,power_power(X_a,A_1),N_1))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1))))))) # label(fact_274_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 854 (all X_a (semiring_0(X_a) & power(X_a) -> (all N_1 ((N_1 = zero_zero(nat) -> hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),N_1) = one_one(X_a)) & (zero_zero(nat) != N_1 -> hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),N_1) = zero_zero(X_a)))))) # label(fact_195_power__0__left) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 855 (all Z_1 all X_1 all Y_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),Z_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Z_1))))) # label(fact_590_dvd_Oorder__trans) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 856 (all X_a (linordered_idom(X_a) -> (all A_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,bit0(bit1(pls))))))))) # label(fact_253_zero__le__power2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 857 (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 (J_1 = hAPP(nat,nat,plus_plus(nat,I_1),K_2) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_2)))))) # label(fact_861_less__imp__add__positive) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 858 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K)),min)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),min)))) # label(fact_300_rel__simps_I13_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 859 (all Ma all N (hAPP(nat,nat,power_power(nat,Ma),N) = zero_zero(nat) <-> N != zero_zero(nat) & zero_zero(nat) = Ma)) # label(fact_469_nat__power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 860 (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,Ma),N)),Ma))))) # label(fact_554_dvd__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 861 (all Ma all N all K (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K)) -> (hAPP(nat,nat,times_times(nat,K),N) = hAPP(nat,nat,times_times(nat,K),Ma) <-> N = Ma))) # label(fact_727_nat__mult__eq__cancel1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 862 (all M all X_1 hAPP(int,int,div_mod(int,X_1),M) = standardRes(M,X_1)) # label(fact_981_StandardRes__def) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 863 (all N_1 all X_1 all Y_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,X_1),N_1)),hAPP(nat,nat,power_power(nat,Y_1),N_1))))) # label(fact_470_divides__exp) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 864 (all B all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (-hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_1)) & -hBOOL(hAPP(int,bool,zcong(B,zero_zero(int)),P_1)) -> -hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),B),zero_zero(int)),P_1)))))) # label(fact_415_zcong__zprime__prod__zero__contra) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 865 (all M all Y_1 all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),X_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Y_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M)) -> (hBOOL(hAPP(int,bool,zcong(X_1,Y_1),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y_1),M)) -> ti(int,X_1) = ti(int,Y_1)))))))) # label(fact_404_zcong__less__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 866 (all I_1 all J_1 -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,I_1),J_1)),I_1))) # label(fact_653_not__add__less1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 867 (all X_a (power(X_a) -> (all M all N_1 ((zero_zero(nat) = N_1 -> hAPP(nat,X_a,power_power(X_a,M),N_1) = one_one(X_a)) & (zero_zero(nat) != N_1 -> hAPP(nat,X_a,power_power(X_a,M),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))))))))) # label(fact_722_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 868 (all X_a (linordered_semidom(X_a) -> (all X_1 all Y_1 (hAPP(nat,X_a,power_power(X_a,Y_1),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,X_a,power_power(X_a,X_1),number_number_of(nat,bit0(bit1(pls)))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y_1)) -> ti(X_a,X_1) = ti(X_a,Y_1))))))) # label(fact_255_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 869 (all X_1 all Y_1 hAPP(real,real,plus_plus(real,hAPP(real,real,plus_plus(real,hAPP(nat,real,power_power(real,X_1),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,real,power_power(real,Y_1),number_number_of(nat,bit0(bit1(pls)))))),hAPP(real,real,times_times(real,hAPP(real,real,times_times(real,number_number_of(real,bit0(bit1(pls)))),X_1)),Y_1)) = hAPP(nat,real,power_power(real,hAPP(real,real,plus_plus(real,X_1),Y_1)),number_number_of(nat,bit0(bit1(pls))))) # label(fact_174_real__sum__squared__expand) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 870 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(K)),bit1(L))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)))) # label(fact_41_rel__simps_I34_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 871 (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_940_zmod__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 872 (all K_1 hAPP(int,int,plus_plus(int,K_1),pls) = ti(int,K_1)) # label(fact_125_add__Pls__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 873 (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)) -> Z_1 = W))) # label(fact_562_real__le__antisym) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 874 (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,times_times(X_a,A_1),B)),B) = zero_zero(X_a)))) # label(fact_900_mod__mult__self2__is__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 875 (all K all Ma all 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))) <-> zero_zero(nat) = K | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),N)))) # label(fact_729_nat__mult__dvd__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 876 (all N_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N_1))) # label(fact_645_less__eq__nat_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 877 (all V all W hAPP(int,int,plus_plus(int,number_number_of(int,V)),number_number_of(int,W)) = number_number_of(int,hAPP(int,int,plus_plus(int,V),W))) # label(fact_134_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 878 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,times_times(X_a,B),A_1)))))) # label(fact_457_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 879 (all X_a (comm_semiring_1(X_a) -> (all X_1 all Y_1 all Z_1 hAPP(X_a,X_a,times_times(X_a,X_1),hAPP(X_a,X_a,plus_plus(X_a,Y_1),Z_1)) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,X_1),Y_1)),hAPP(X_a,X_a,times_times(X_a,X_1),Z_1))))) # label(fact_112_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 880 (all A_3 all B_2 all Ma ((exists K_2 ti(int,B_2) = hAPP(int,int,plus_plus(int,A_3),hAPP(int,int,times_times(int,Ma),K_2))) <-> hBOOL(hAPP(int,bool,zcong(A_3,B_2),Ma)))) # label(fact_349_zcong__iff__lin) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 881 (all X_a (linordered_idom(X_a) -> (all A_1 -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,bit0(bit1(pls))))),zero_zero(X_a)))))) # label(fact_256_power2__less__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 882 (all X_1 all Y_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) -> -(-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1))))) # label(fact_586_dvd_Oless__not__sym) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 883 (all X_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_1)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,multInv(P_1,X_1)),X_1),one_one(int)),P_1)))))) # label(fact_839_MultInv__prop2a) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 884 (all N_1 -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),N_1))) # label(fact_573_less__not__refl) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 885 (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_71_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 886 (all K all L (ti(int,K) = ti(int,L) <-> bit0(L) = bit0(K))) # label(fact_85_rel__simps_I48_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 887 (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),number_number_of(int,K)),number_number_of(int,L))))) # label(fact_49_less__eq__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 888 (all B_2 all A_3 all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,bit1(bit0(bit1(pls))))),P_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,minus_minus(int,P_3),one_one(int)))) -> (hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),wset(A_3,P_3))) -> hBOOL(hAPP(fun(int,bool),bool,member(int,inv(P_3,B_2)),wset(A_3,P_3)))))))) # label(fact_841_wset__mem__inv__mem) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 889 (all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,minus_minus(int,P_1),one_one(int)))) -> inv(P_1,A_1) != zero_zero(int))))) # label(fact_827_inv__not__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 890 (all K_1 all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,I_1),K_1)),hAPP(nat,nat,plus_plus(nat,J_1),K_1))))) # label(fact_680_add__le__mono1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 891 (all J_1 all K_1 all A_1 all P_1 (hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,J_1),K_1),A_1),P_1)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,multInv(P_1,J_1)),J_1)),K_1),hAPP(int,int,times_times(int,multInv(P_1,J_1)),A_1)),P_1)))) # label(fact_819_aux______3) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 892 (all B_1_1 all B_2_1 ti(fun(int,bool),wset(B_1_1,B_2_1)) = wset(B_1_1,B_2_1)) # label(tsy_c_WilsonRuss_Owset_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 893 (all K all Ma all 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))) <-> (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))))) # label(fact_720_mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 894 (all X_a (linord20386208strict(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),B)),zero_zero(X_a)))))))) # label(fact_494_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 895 (all Y_1 all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,times_times(int,X_1),Y_1)))))) # label(fact_345_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 896 (all X_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_1)) -> -hBOOL(hAPP(int,bool,zcong(multInv(P_1,X_1),zero_zero(int)),P_1)))))) # label(fact_831_MultInv__prop3) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 897 (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_925_mod__mod__cancel) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 898 (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_140_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 899 (all X_a (zero_neq_one(X_a) -> one_one(X_a) != zero_zero(X_a))) # label(fact_446_one__neq__zero) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 900 (all K_1 all A_1 all J_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,zcong(J_1,zero_zero(int)),P_1)) -> (hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,multInv(P_1,J_1)),J_1)),K_1),hAPP(int,int,times_times(int,multInv(P_1,J_1)),A_1)),P_1)) -> hBOOL(hAPP(int,bool,zcong(K_1,hAPP(int,int,times_times(int,A_1),multInv(P_1,J_1))),P_1))))))) # label(fact_833_aux______4) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 901 (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,Ma) & one_one(int) = ti(int,N) <-> hAPP(int,int,times_times(int,Ma),N) = one_one(int)))) # label(fact_244_pos__zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 902 (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)) -> number_number_of(X_a,hAPP(int,int,times_times(int,V),V_1)) = hAPP(X_a,X_a,times_times(X_a,number_number_of(X_a,V)),number_number_of(X_a,V_1))))))) # label(fact_135_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 903 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),zero_zero(X_a)))))))) # label(fact_813_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 904 (all X_a (real_normed_algebra(X_a) -> (all A_1 all B all B_3 hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),B)),hAPP(X_a,X_a,times_times(X_a,A_1),B_3)) = hAPP(X_a,X_a,times_times(X_a,A_1),hAPP(X_a,X_a,plus_plus(X_a,B),B_3))))) # label(fact_891_mult_Oadd__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.79 905 (all X_a (linordered_ring(X_a) -> (all X_1 all Y_1 -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,X_1),X_1)),hAPP(X_a,X_a,times_times(X_a,Y_1),Y_1))),zero_zero(X_a)))))) # label(fact_238_not__sum__squares__lt__zero) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 906 (all Ma all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),N)) <-> Ma = N | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),N)))) # label(fact_665_le__eq__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 907 (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) <-> zero_zero(X_a) = ti(X_a,A_3))))) # label(fact_768_double__zero__sym) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 908 (all X_a (linordered_idom(X_a) -> (all X_2 all Y_2 (zero_zero(X_a) = ti(X_a,Y_2) & ti(X_a,X_2) = 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,hAPP(nat,X_a,power_power(X_a,X_2),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,X_a,power_power(X_a,Y_2),number_number_of(nat,bit0(bit1(pls)))))),zero_zero(X_a))))))) # label(fact_266_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 909 (all M all K_1 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,M),K_1)),N_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)))) # label(fact_683_add__leD1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 910 (all X_a (comm_semiring_1(X_a) -> (all A_1 all M all B hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),M)),hAPP(X_a,X_a,times_times(X_a,B),M)) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),M)))) # label(fact_109_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 911 (all B all D_2 all A_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_2),A_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_2),hAPP(nat,nat,plus_plus(nat,A_1),B))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_2),B))))) # label(fact_464_divides__add__revr) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 912 (all X_a (ab_semigroup_mult(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,times_times(X_a,A_1),hAPP(X_a,X_a,times_times(X_a,B),C)) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),B)),C)))) # label(fact_757_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 913 (all X_a (linordered_semidom(X_a) -> (all N_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N_1)),hAPP(X_a,X_a,times_times(X_a,A_1),hAPP(nat,X_a,power_power(X_a,A_1),N_1)))))))) # label(fact_176_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 914 (all X_a (group_add(X_a) -> (all A_3 all B_2 (ti(X_a,B_2) = ti(X_a,A_3) <-> hAPP(X_a,X_a,minus_minus(X_a,A_3),B_2) = zero_zero(X_a))))) # label(fact_785_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 915 (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]. 2.34/2.80 916 (all J_1 all K_1 all A_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_1)) -> (-hBOOL(hAPP(int,bool,zcong(K_1,zero_zero(int)),P_1)) -> (-hBOOL(hAPP(int,bool,zcong(J_1,zero_zero(int)),P_1)) -> (hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),multInv(P_1,J_1)),hAPP(int,int,times_times(int,A_1),multInv(P_1,K_1))),P_1)) -> hBOOL(hAPP(int,bool,zcong(J_1,K_1),P_1))))))))) # label(fact_836_MultInv__zcong__prop3) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 917 (all Z_2 all W_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_2),W_1)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_2),W_1)) & ti(int,W_1) != ti(int,Z_2))) # label(fact_22_zless__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 918 (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))) -> (zero_zero(nat) != N_1 -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_1),B))))) # label(fact_879_pow__divides__pow__int) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 919 (all B_1_1 all B_2_1 ti(int,standardRes(B_1_1,B_2_1)) = standardRes(B_1_1,B_2_1)) # label(tsy_c_Residues_OStandardRes_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 920 (all A_1 all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M)) -> (exists X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),M)) & hBOOL(hAPP(int,bool,zcong(A_1,X),M)) & (all Y (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y)) & hBOOL(hAPP(int,bool,zcong(A_1,Y),M)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y),M)) -> ti(int,Y) = ti(int,X))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)))))) # label(fact_751_zcong__zless__unique) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 921 (all X_2 all Y_2 all Z_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Z_2)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_2),Y_2)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,X_2),Z_2)),hAPP(real,real,times_times(real,Y_2),Z_2)))))) # label(fact_519_real__mult__less__iff1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 922 (all X_1 all Y_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) -> Y_1 != X_1)) # label(fact_583_dvd_Oless__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 923 (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)) -> hAPP(X_a,X_a,div_mod(X_a,B),A_1) = zero_zero(X_a))))) # label(fact_905_dvd__imp__mod__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 924 (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,C),hAPP(X_a,X_a,plus_plus(X_a,A_1),D_2)) = hAPP(X_a,X_a,plus_plus(X_a,A_1),hAPP(X_a,X_a,plus_plus(X_a,C),D_2))))) # label(fact_80_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 925 (all X_a (linordered_semidom(X_a) -> (all A_1 all N_1 all N_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),N_3)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),one_one(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N_3)),hAPP(nat,X_a,power_power(X_a,A_1),N_1))))))))) # label(fact_216_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 926 (all K_1 all L_1 hAPP(int,int,plus_plus(int,bit1(K_1)),bit0(L_1)) = bit1(hAPP(int,int,plus_plus(int,K_1),L_1))) # label(fact_151_add__Bit1__Bit0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 927 (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),C) = hAPP(X_a,X_a,plus_plus(X_a,A_1),B) -> ti(X_a,C) = ti(X_a,B))))) # label(fact_761_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 928 (all X_a (ring_11004092258visors(X_a) -> (all N_1 all A_1 (ti(X_a,A_1) != zero_zero(X_a) -> hAPP(nat,X_a,power_power(X_a,A_1),N_1) != zero_zero(X_a))))) # label(fact_194_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 929 (all N_1 all M hAPP(nat,nat,minus_minus(nat,N_1),hAPP(nat,nat,plus_plus(nat,N_1),M)) = zero_zero(nat)) # label(fact_643_diff__add__0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 930 (all B_2 all A_3 (hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),d22set(A_3))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),A_3)))) # label(fact_857_d22set__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 931 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),t)) -> (exists X exists Y hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,X),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,Y),number_number_of(nat,bit0(bit1(pls))))) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int))) # label(fact_2__0961_A_060_At_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 932 (all X_a (comm_semiring(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),C) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),C))))) # label(fact_448_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 933 (all X_a (linordered_idom(X_a) -> (all Y_1 all X_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_1),one_one(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,X_1),Y_1)),X_1)))))))) # label(fact_541_mult__right__le__one__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 934 (all B all M all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),A_1)) -> -hBOOL(hAPP(int,bool,zcong(A_1,B),M))))))) # label(fact_347_zcong__not) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 935 (all X_a (linord581940658strict(X_a) -> (all A_3 all B_2 all C_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),A_3)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,C_1),A_3)),hAPP(X_a,X_a,times_times(X_a,C_1),B_2)))))))) # label(fact_529_mult__le__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 936 (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,A_3),D)),Ma)) <-> hBOOL(hAPP(int,bool,zcong(C_1,hAPP(int,int,times_times(int,B_2),D)),Ma))))) # label(fact_397_zcong__zmult__prop1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 937 (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_372_zdiff__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 938 (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_984_mod__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 939 (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_14_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 940 (all B_2 all P_3 all A_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_3)) -> (-hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),wset(hAPP(int,int,minus_minus(int,A_3),one_one(int)),P_3))) -> (hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),wset(A_3,P_3))) -> ti(int,B_2) = inv(P_3,A_3) | ti(int,B_2) = ti(int,A_3))))) # label(fact_847_wset__mem__imp__or) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 941 (all X_a (real_normed_algebra(X_a) -> (all X_1 hAPP(X_a,X_a,times_times(X_a,X_1),zero_zero(X_a)) = zero_zero(X_a)))) # label(fact_884_mult__right_Ozero) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 942 (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),B),zero_zero(X_a))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a))) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),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),zero_zero(X_a)),hAPP(X_a,X_a,times_times(X_a,A_1),B))))))) # label(fact_472_split__mult__pos__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 943 (all A_1 all P_1 ((hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_1)) -> zero_zero(int) = legendre(A_1,P_1)) & (-hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_1)) -> (-hBOOL(hAPP(int,bool,quadRes(P_1),A_1)) -> number_number_of(int,min) = legendre(A_1,P_1)) & (hBOOL(hAPP(int,bool,quadRes(P_1),A_1)) -> one_one(int) = legendre(A_1,P_1))))) # label(fact_418_Legendre__def) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 944 (all A_3 all B_2 (hBOOL(hAPP(int,bool,zcong(A_3,B_2),zero_zero(int))) <-> ti(int,B_2) = ti(int,A_3))) # label(fact_335_IntPrimes_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 945 (all Z1 all Z2 all W hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,Z1),Z2)),W) = hAPP(int,int,minus_minus(int,hAPP(int,int,times_times(int,Z1),W)),hAPP(int,int,times_times(int,Z2),W))) # label(fact_373_zdiff__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 946 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,A_1),B)))))))) # label(fact_755_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 947 (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_447_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 948 (all X_a (comm_semiring_1(X_a) -> (all A_1 all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,power_power(X_a,A_1),M)),hAPP(nat,X_a,power_power(X_a,A_1),N_1))))))) # label(fact_186_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 949 (all B_1_1 zfact(B_1_1) = zfact(ti(int,B_1_1))) # label(tsy_c_IntFact_Ozfact_arg1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 950 (all X_a (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_791_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 951 (all X_a (no_zero_divisors(X_a) -> (all B all A_1 (ti(X_a,A_1) != zero_zero(X_a) -> (zero_zero(X_a) != ti(X_a,B) -> zero_zero(X_a) != hAPP(X_a,X_a,times_times(X_a,A_1),B)))))) # label(fact_441_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 952 (all M all N_1 all K_1 hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,plus_plus(nat,M),N_1)),K_1) = hAPP(nat,nat,plus_plus(nat,M),hAPP(nat,nat,plus_plus(nat,N_1),K_1))) # label(fact_608_nat__add__assoc) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 953 (all K all Ma all N (hAPP(nat,nat,plus_plus(nat,K),Ma) = hAPP(nat,nat,plus_plus(nat,K),N) <-> Ma = N)) # label(fact_609_nat__add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 954 (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),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),A_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,A_1),B)),zero_zero(X_a))))))) # label(fact_471_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 955 (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_381_less__bin__lemma) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 956 (all Y_1 all X_1 all P_1 (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_1)) -> (hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,Y_1),number_number_of(nat,bit0(bit1(pls)))),X_1),P_1)) -> -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),Y_1))))) # label(fact_277_Euler_Oaux____1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 957 (all K_1 all M hBOOL(hAPP(int,bool,zcong(K_1,K_1),M))) # label(fact_327_zcong__refl) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 958 (all L_1 min != bit0(L_1)) # label(fact_296_rel__simps_I42_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 959 (all X_a (ordere779506340up_add(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)),hAPP(X_a,X_a,plus_plus(X_a,B),C))))))) # label(fact_773_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 960 (all X_2 (hAPP(int,int,div_mod(int,X_2),number_number_of(int,bit0(bit1(pls)))) = one_one(int) <-> hAPP(int,int,div_mod(int,X_2),number_number_of(int,bit0(bit1(pls)))) != zero_zero(int))) # label(fact_969_neq__one__mod__two) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 961 (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_322_dvd__power) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 962 (all Ma all Ta all K (zero_zero(int) != ti(int,K) -> (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]. 2.34/2.80 963 (all X_a (comm_semiring_1(X_a) -> (all X_1 hAPP(X_a,X_a,times_times(X_a,X_1),X_1) = hAPP(nat,X_a,power_power(X_a,X_1),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_13_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 964 (all J_1 all A_1 all K_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,zcong(K_1,zero_zero(int)),P_1)) -> (hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,J_1),K_1),hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,A_1),multInv(P_1,K_1))),K_1)),P_1)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,J_1),K_1),A_1),P_1))))))) # label(fact_834_aux______2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 965 (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,D_2) != ti(X_a,C) & ti(X_a,B) = ti(X_a,A_1) -> 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_219_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 966 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K1)),bit0(K2))))) # label(fact_46_less__eq__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 967 (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)),hAPP(X_a,X_a,div_mod(X_a,B),C))),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,minus_minus(X_a,A_1),B)),C)))) # label(fact_948_mod__diff__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 968 (all X_a (linordered_semidom(X_a) -> (all X_1 all Y_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,X_1),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,X_a,power_power(X_a,Y_1),number_number_of(nat,bit0(bit1(pls)))))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))))))) # label(fact_254_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 969 (all K all P_2 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D)) -> ((all X (hBOOL(hAPP(int,bool,P_2,X)) -> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,plus_plus(int,X),D))))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) -> (all X (hBOOL(hAPP(int,bool,P_2,X)) -> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,plus_plus(int,X),hAPP(int,int,times_times(int,K),D)))))))))) # label(fact_749_incr__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 970 (all B_2 all A_3 all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,minus_minus(int,P_3),one_one(int)))) -> (hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),wset(A_3,P_3))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),B_2)))))) # label(fact_846_wset__g__1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 971 (all A_1 all B all P_1 all Q_1 twoSqu1929807760sum2sq(product_Pair(int,int,hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A_1),P_1)),hAPP(int,int,times_times(int,B),Q_1)),hAPP(int,int,minus_minus(int,hAPP(int,int,times_times(int,A_1),Q_1)),hAPP(int,int,times_times(int,B),P_1)))) = hAPP(int,int,times_times(int,twoSqu1929807760sum2sq(product_Pair(int,int,A_1,B))),twoSqu1929807760sum2sq(product_Pair(int,int,P_1,Q_1)))) # label(fact_392_mult__sum2sq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 972 (all X_a (semiring_div(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),hAPP(X_a,X_a,div_mod(X_a,B),C))),C)))) # label(fact_929_mod__add__right__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 973 (all B_1_1 all B_2_1 inv(B_1_1,ti(int,B_2_1)) = inv(B_1_1,B_2_1)) # label(tsy_c_WilsonRuss_Oinv_arg2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 974 (all B all A_1 all C (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),C)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),A_1)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),B))))) # label(fact_334_Euler_Oaux2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 975 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(K)),min)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),min)))) # label(fact_306_rel__simps_I30_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 976 (all X_a (cancel_semigroup_add(X_a) -> (all B_2 all A_3 all C_1 (hAPP(X_a,X_a,plus_plus(X_a,C_1),A_3) = hAPP(X_a,X_a,plus_plus(X_a,B_2),A_3) <-> ti(X_a,C_1) = ti(X_a,B_2))))) # label(fact_760_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 977 (all C_1 all D all A_3 all B_2 all Ma (hBOOL(hAPP(int,bool,zcong(A_3,B_2),Ma)) -> (hBOOL(hAPP(int,bool,zcong(C_1,hAPP(int,int,times_times(int,D),B_2)),Ma)) <-> hBOOL(hAPP(int,bool,zcong(C_1,hAPP(int,int,times_times(int,D),A_3)),Ma))))) # label(fact_398_zcong__zmult__prop2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 978 (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]. 2.34/2.80 979 (all X_a all X_c all X_b all B_1_1 all B_2_1 combs(X_a,X_b,X_c,B_1_1,B_2_1) = ti(fun(X_a,X_c),combs(X_a,X_b,X_c,B_1_1,B_2_1))) # label(tsy_c_COMBS_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 980 (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]. 2.34/2.80 981 (all W_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(W_1)),zero_zero(int))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),zero_zero(int))))) # label(fact_226_bin__less__0__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 982 (all X_a (ordered_ab_group_add(X_a) -> (all A_3 all B_2 all C_1 all D (hAPP(X_a,X_a,minus_minus(X_a,C_1),D) = hAPP(X_a,X_a,minus_minus(X_a,A_3),B_2) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D))))))) # label(fact_790_diff__eq__diff__less) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 983 (all X_2 all Y_2 (zero_zero(real) = Y_2 & X_2 = zero_zero(real) <-> zero_zero(real) = hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,X_2),X_2)),hAPP(real,real,times_times(real,Y_2),Y_2)))) # label(fact_524_real__two__squares__add__zero__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 984 (all X_a (ordere236663937imp_le(X_a) -> (all C_1 all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,C_1),A_3)),hAPP(X_a,X_a,plus_plus(X_a,C_1),B_2))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2)))))) # label(fact_772_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 985 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),N_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N_1),M)) -> M = N_1))) # label(fact_592_dvd__antisym) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 986 (all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),M)) -> -hBOOL(hAPP(int,bool,zcong(one_one(int),number_number_of(int,min)),M)))) # label(fact_365_one__not__neg__one__mod__m) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 987 (all Ma all K all N (zero_zero(nat) = K | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),N)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,times_times(nat,Ma),K)),hAPP(nat,nat,times_times(nat,N),K))))) # label(fact_518_nat__mult__dvd__cancel__disj_H) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 988 -(all T_4 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) != 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_19__096_B_Bthesis_O_A_I_B_Bt_O_As_A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_L_A1_) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 989 (all X_a (linordered_ring(X_a) -> (all A_1 -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),A_1)),zero_zero(X_a)))))) # label(fact_504_not__square__less__zero) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 990 (all X_a all X_c all X_b all B_1_1 all B_2_1 ti(fun(X_a,X_c),combb(X_b,X_c,X_a,B_1_1,B_2_1)) = combb(X_b,X_c,X_a,B_1_1,B_2_1)) # label(tsy_c_COMBB_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 991 (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_110_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 992 (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),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),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_485_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 993 (all X_a (linordered_idom(X_a) -> (all X_1 all Y_1 -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X_1),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,X_a,power_power(X_a,Y_1),number_number_of(nat,bit0(bit1(pls)))))),zero_zero(X_a)))))) # label(fact_267_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 994 (all N (zero_zero(nat) = N <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),zero_zero(nat))))) # label(fact_644_le__0__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 995 (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_98_less__eq__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 996 (all B_1_1 minus_minus(nat,B_1_1) = ti(fun(nat,nat),minus_minus(nat,B_1_1))) # label(tsy_c_Groups_Ominus__class_Ominus_1_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 997 (all X_a (dvd(X_a) & semiring_0(X_a) -> (all P_2 all L ((exists X hBOOL(hAPP(X_a,bool,P_2,hAPP(X_a,X_a,times_times(X_a,L),X)))) <-> (exists X (hBOOL(hAPP(X_a,bool,P_2,X)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),L),hAPP(X_a,X_a,plus_plus(X_a,X),zero_zero(X_a)))))))))) # label(fact_744_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 998 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all Y_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),number_number_of(X_a,Y_2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(pls)),Y_2)))))) # label(fact_105_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 999 (all X_a (ordered_semiring(X_a) -> (all C all D_2 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),D_2)))))))))) # label(fact_473_mult__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 1000 (all I_1 all J_1 all K_1 hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,minus_minus(nat,I_1),K_1)),J_1) = hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,minus_minus(nat,I_1),J_1)),K_1)) # label(fact_558_diff__commute) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 1001 (all B_1_1 all B_2_1 standardRes(B_1_1,ti(int,B_2_1)) = standardRes(B_1_1,B_2_1)) # label(tsy_c_Residues_OStandardRes_arg2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 1002 (all X_a (comm_semiring_1(X_a) -> (all M all A_1 all N_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N_1)),B)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,power_power(X_a,A_1),M)),B))))))) # label(fact_188_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 1003 (all K_1 all L_1 bit1(hAPP(int,int,plus_plus(int,K_1),L_1)) = hAPP(int,int,plus_plus(int,bit0(K_1)),bit1(L_1))) # label(fact_152_add__Bit0__Bit1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.80 1004 (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),A_1),B)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A_1))))) # label(fact_599_dvd_Oneq__le__trans) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1005 (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),B),A_1)) -> (C = B -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),C),A_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C))))) # label(fact_580_dvd_Oord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1006 (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,hAPP(nat,nat,minus_minus(nat,J_2),I_2)),U)),N) = Ma <-> 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_737_nat__eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1007 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx all Ry hAPP(X_a,X_a,times_times(X_a,Rx),hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,Lx),Ly)),Ry)) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,Lx),Ly)),hAPP(X_a,X_a,times_times(X_a,Rx),Ry))))) # label(fact_67_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1008 (all N_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N_1))) # label(fact_555_le0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1009 (all X_a (ordere1490568538miring(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,C),A_1)),hAPP(X_a,X_a,times_times(X_a,C),B)))))))) # label(fact_477_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1010 (all X_1 all Y_1 all M hAPP(int,int,div_mod(int,hAPP(int,int,minus_minus(int,X_1),hAPP(int,int,div_mod(int,Y_1),M))),M) = hAPP(int,int,div_mod(int,hAPP(int,int,minus_minus(int,X_1),Y_1)),M)) # label(fact_945_zdiff__zmod__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1011 (all P_3 all A_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_3)) -> hBOOL(hAPP(fun(int,bool),bool,member(int,A_3),wset(A_3,P_3))))) # label(fact_843_wset__mem__mem) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1012 (all X_a (linordered_semidom(X_a) -> (all N_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),hAPP(nat,X_a,power_power(X_a,A_1),N_1))))))) # label(fact_270_one__le__power) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1013 (all Ma all X_2 (hBOOL(hAPP(int,bool,quadRes(Ma),X_2)) <-> (exists Y hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,Y),number_number_of(nat,bit0(bit1(pls)))),X_2),Ma))))) # label(fact_420_QuadRes__def) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1014 (all Q_1 all R_1 all B all C (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),R_1),B)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,B),hAPP(nat,nat,div_mod(nat,Q_1),C))),R_1)),hAPP(nat,nat,times_times(nat,B),C)))))) # label(fact_992_mod__lemma) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1015 (all I_1 all K_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J_1)) -> hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,plus_plus(nat,J_1),I_1)),K_1) = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,minus_minus(nat,J_1),K_1)),I_1))) # label(fact_694_add__diff__assoc2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1016 (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_66_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1017 (all P_2 all N all K (hBOOL(hAPP(nat,bool,P_2,hAPP(nat,nat,div_mod(nat,N),K))) <-> (zero_zero(nat) = K -> hBOOL(hAPP(nat,bool,P_2,N))) & (K != zero_zero(nat) -> (all I all J (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J),K)) -> (hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,K),I)),J) = N -> hBOOL(hAPP(nat,bool,P_2,J)))))))) # label(fact_991_split__mod) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1018 (all K_1 ti(int,K_1) = hAPP(int,int,minus_minus(int,K_1),pls)) # label(fact_370_diff__bin__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1019 (all A_1 all B all C hAPP(int,int,div_mod(int,hAPP(int,int,times_times(int,A_1),B)),C) = hAPP(int,int,div_mod(int,hAPP(int,int,times_times(int,A_1),hAPP(int,int,div_mod(int,B),C))),C)) # label(fact_919_zmod__zmult1__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1020 (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))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),hAPP(int,int,div_mod(int,A_1),B))))) # label(fact_959_neg__mod__conj) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1021 (all B_1_1 sr(B_1_1) = sr(ti(int,B_1_1))) # label(tsy_c_Residues_OSR_arg1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1022 (all X_2 all P_3 (-hBOOL(hAPP(int,bool,zcong(X_2,zero_zero(int)),P_3)) <-> zero_zero(int) != standardRes(P_3,X_2))) # label(fact_987_StandardRes__prop3) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1023 (all N_1 (number_number_of(int,min) = hAPP(nat,int,power_power(int,number_number_of(int,min)),N_1) | one_one(int) = hAPP(nat,int,power_power(int,number_number_of(int,min)),N_1))) # label(fact_412_neg__one__power) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1024 (all L_1 bit1(hAPP(int,int,minus_minus(int,min),L_1)) = hAPP(int,int,minus_minus(int,min),bit0(L_1))) # label(fact_388_diff__bin__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1025 (all X_2 all Y_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,minus_minus(real,X_2),Y_2)),zero_zero(real))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),Y_2)))) # label(fact_424_real__le__eq__diff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1026 (all X_a (comm_semiring_1(X_a) -> (all X_1 one_one(X_a) = hAPP(nat,X_a,power_power(X_a,X_1),zero_zero(nat))))) # label(fact_313_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1027 (all W hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),W),W))) # label(fact_559_real__le__refl) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1028 (all X_a (number_ring(X_a) -> (all V all W number_number_of(X_a,hAPP(int,int,times_times(int,V),W)) = hAPP(X_a,X_a,times_times(X_a,number_number_of(X_a,V)),number_number_of(X_a,W))))) # label(fact_146_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1029 (all Ma all K all F_1 ((all M_2 all N_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),N_2)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,F_1,M_2)),hAPP(nat,nat,F_1,N_2))))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,F_1,Ma)),K)),hAPP(nat,nat,F_1,hAPP(nat,nat,plus_plus(nat,Ma),K)))))) # label(fact_854_mono__nat__linear__lb) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1030 (all X_a (ordered_ab_group_add(X_a) -> (all A_3 all B_2 all C_1 all D (hAPP(X_a,X_a,minus_minus(X_a,C_1),D) = hAPP(X_a,X_a,minus_minus(X_a,A_3),B_2) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_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_789_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1031 (all X_2 all Y_2 all Z_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Z_2)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),Y_2)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,X_2),Z_2)),hAPP(real,real,times_times(real,Y_2),Z_2)))))) # label(fact_520_real__mult__le__cancel__iff1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1032 (all N_1 all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,power_power(int,X_1),N_1))))) # label(fact_348_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1033 (all N all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Ma)) -> (N = one_one(nat) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,times_times(nat,N),Ma)),Ma))))) # label(fact_553_dvd__mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1034 (all B_1_1 ti(fun(int,bool),sr(B_1_1)) = sr(B_1_1)) # label(tsy_c_Residues_OSR_res) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1035 (all X_a (number_ring(X_a) -> (all X_1 all Y_1 hAPP(X_a,X_a,minus_minus(X_a,hAPP(X_a,X_a,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X_1),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,X_a,power_power(X_a,Y_1),number_number_of(nat,bit0(bit1(pls)))))),hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,number_number_of(X_a,bit0(bit1(pls)))),X_1)),Y_1)) = hAPP(nat,X_a,power_power(X_a,hAPP(X_a,X_a,minus_minus(X_a,X_1),Y_1)),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_401_power2__diff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1036 (all X_a (comm_monoid_mult(X_a) -> (all A_1 hAPP(X_a,X_a,times_times(X_a,A_1),one_one(X_a)) = ti(X_a,A_1)))) # label(fact_794_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1037 (all B_1_1 all B_2_1 wset(B_1_1,B_2_1) = wset(ti(int,B_1_1),B_2_1)) # label(tsy_c_WilsonRuss_Owset_arg1) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1038 (all W all Z1 all Z2 hAPP(int,int,times_times(int,W),hAPP(int,int,plus_plus(int,Z1),Z2)) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,W),Z1)),hAPP(int,int,times_times(int,W),Z2))) # label(fact_133_zadd__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1039 (all X_a (linord581940658strict(X_a) -> (all X_2 all Y_2 (zero_zero(X_a) = ti(X_a,X_2) & zero_zero(X_a) = ti(X_a,Y_2) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,X_2),X_2)),hAPP(X_a,X_a,times_times(X_a,Y_2),Y_2))),zero_zero(X_a))))))) # label(fact_236_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1040 (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_242_power__one__right) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1041 (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,zero_zero(X_a)),number_number_of(X_a,W))),number_number_of(X_a,W)) = number_number_of(X_a,bit0(W))))) # label(fact_241_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1042 (all X_a (linord20386208strict(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,C),A_1)),hAPP(X_a,X_a,times_times(X_a,C),B))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))))))) # label(fact_539_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 2.34/2.81 1043 -(exists X exists Y hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,X),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,Y),number_number_of(nat,bit0(bit1(pls))))) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int))) # label(conj_0) # label(negated_conjecture) # label(non_clause). [assumption]. 2.56/2.83 2.56/2.83 ============================== end of process non-clausal formulas === 2.56/2.83 2.56/2.83 ============================== PROCESS INITIAL CLAUSES =============== 2.56/2.83 2.56/2.83 ============================== PREDICATE ELIMINATION ================= 2.56/2.83 1044 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,D),B)),hAPP(A,A,times_times(A,D),C))) # label(fact_491_mult__strict__left__mono) # label(axiom). [clausify(7)]. 2.56/2.83 1045 linord20386208strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 2.56/2.83 1046 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,times_times(A,B),C))) # label(fact_500_mult__pos__pos) # label(axiom). [clausify(10)]. 2.56/2.83 1047 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),E))) # label(fact_532_mult__less__le__imp__less) # label(axiom). [clausify(16)]. 2.56/2.83 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(1047,a,1045,a)]. 2.56/2.83 1048 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),zero_zero(A))) # label(fact_499_mult__pos__neg) # label(axiom). [clausify(94)]. 2.56/2.83 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(1048,a,1045,a)]. 2.56/2.83 1049 linord20386208strict(int) # label(arity_Int_Oint___Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 2.56/2.83 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(1049,a,1046,a)]. 2.56/2.83 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(1049,a,1047,a)]. 2.56/2.83 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(1049,a,1048,a)]. 2.56/2.83 1050 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),D)) # label(fact_535_mult__less__imp__less__right) # label(axiom). [clausify(326)]. 2.56/2.86 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(1050,a,1045,a)]. 2.56/2.86 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(1050,a,1049,a)]. 2.56/2.86 1051 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,times_times(A,B),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) # label(fact_496_zero__less__mult__pos2) # label(axiom). [clausify(366)]. 2.56/2.86 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(1051,a,1045,a)]. 2.56/2.86 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(1051,a,1049,a)]. 2.56/2.86 1052 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,times_times(A,B),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) # label(fact_497_zero__less__mult__pos) # label(axiom). [clausify(413)]. 2.56/2.86 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(1052,a,1045,a)]. 2.56/2.86 1053 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),E))) # label(fact_531_mult__strict__mono_H) # label(axiom). [clausify(424)]. 2.56/2.86 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(1053,a,1045,a)]. 2.56/2.86 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(1053,a,1049,a)]. 2.56/2.86 1054 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,C),B)),zero_zero(A))) # label(fact_498_mult__pos__neg2) # label(axiom). [clausify(471)]. 2.56/2.89 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(1054,a,1045,a)]. 2.56/2.89 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(1054,a,1049,a)]. 2.56/2.89 1055 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) # label(fact_537_mult__less__imp__less__left) # label(axiom). [clausify(489)]. 2.56/2.89 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,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(1055,a,1045,a)]. 2.56/2.89 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(1055,a,1049,a)]. 2.56/2.89 1056 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),D)) # label(fact_538_mult__right__le__imp__le) # label(axiom). [clausify(521)]. 2.56/2.89 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(1056,a,1049,a)]. 2.56/2.89 1057 linord20386208strict(nat) # label(arity_Nat_Onat___Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 2.56/2.89 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(1057,a,1046,a)]. 2.56/2.89 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(1057,a,1047,a)]. 2.56/2.89 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(1057,a,1050,a)]. 2.56/2.89 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(1057,a,1051,a)]. 2.56/2.89 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(1057,a,1052,a)]. 2.64/2.91 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(1057,a,1053,a)]. 2.64/2.91 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(1057,a,1055,a)]. 2.64/2.91 1058 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),E))) # label(fact_533_mult__le__less__imp__less) # label(axiom). [clausify(672)]. 2.64/2.91 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(1058,a,1045,a)]. 2.64/2.91 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(1058,a,1049,a)]. 2.64/2.91 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(1058,a,1057,a)]. 2.64/2.91 1059 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),D))) # label(fact_492_mult__strict__right__mono) # label(axiom). [clausify(795)]. 2.64/2.91 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),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(1059,a,1049,a)]. 2.64/2.91 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),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(1059,a,1057,a)]. 2.64/2.91 1060 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),E))) # label(fact_530_mult__strict__mono) # label(axiom). [clausify(804)]. 2.75/3.09 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(1060,a,1045,a)]. 2.75/3.09 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(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(1060,a,1049,a)]. 2.75/3.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),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(1060,a,1057,a)]. 2.75/3.09 1061 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),zero_zero(A))) # label(fact_494_mult__neg__pos) # label(axiom). [clausify(894)]. 2.75/3.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),zero_zero(nat))) | -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),hAPP(nat,nat,times_times(nat,A),B)),zero_zero(nat))). [resolve(1061,a,1057,a)]. 2.75/3.09 1062 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) # label(fact_539_mult__left__le__imp__le) # label(axiom). [clausify(1042)]. 2.75/3.09 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(1062,a,1045,a)]. 2.75/3.09 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(1062,a,1049,a)]. 2.75/3.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,times_times(nat,A),B)),hAPP(nat,nat,times_times(nat,A),C))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)). [resolve(1062,a,1057,a)]. 2.75/3.09 1063 linordered_idom(int) # label(arity_Int_Oint___Rings_Olinordered__idom) # label(axiom). [assumption]. 2.75/3.09 1064 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,B)),number_number_of(A,C))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)) # label(fact_31_le__number__of) # label(axiom). [clausify(3)]. 2.75/3.09 1065 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,B)),number_number_of(A,C))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)) # label(fact_31_le__number__of) # label(axiom). [clausify(3)]. 2.75/3.09 1066 -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_247_less__special_I3_J) # label(axiom). [clausify(31)]. 2.75/3.09 1067 -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_247_less__special_I3_J) # label(axiom). [clausify(31)]. 2.75/3.09 1068 -linordered_idom(A) | zero_zero(A) = ti(A,B) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(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))))))) # label(fact_268_sum__power2__gt__zero__iff) # label(axiom). [clausify(44)]. 2.75/3.09 1069 -linordered_idom(A) | zero_zero(A) = ti(A,B) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,plus_plus(A,hAPP(nat,A,power_power(A,C),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_268_sum__power2__gt__zero__iff) # label(axiom). [clausify(44)]. 2.75/3.09 1070 -linordered_idom(A) | zero_zero(A) != ti(A,B) | zero_zero(A) != ti(A,C) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(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))))))) # label(fact_268_sum__power2__gt__zero__iff) # label(axiom). [clausify(44)]. 2.75/3.09 1071 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),pls)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,B)),zero_zero(A))) # label(fact_249_le__special_I3_J) # label(axiom). [clausify(154)]. 2.75/3.09 1072 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),pls)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,B)),zero_zero(A))) # label(fact_249_le__special_I3_J) # label(axiom). [clausify(154)]. 2.75/3.09 1073 -linordered_idom(A) | zero_zero(A) != ti(A,B) | zero_zero(A) != ti(A,C) | zero_zero(A) = hAPP(A,A,plus_plus(A,hAPP(nat,A,power_power(A,C),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls))))) # label(fact_258_sum__power2__eq__zero__iff) # label(axiom). [clausify(160)]. 2.75/3.09 1074 -linordered_idom(A) | zero_zero(A) = ti(A,B) | zero_zero(A) != hAPP(A,A,plus_plus(A,hAPP(nat,A,power_power(A,C),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls))))) # label(fact_258_sum__power2__eq__zero__iff) # label(axiom). [clausify(160)]. 2.75/3.09 1075 -linordered_idom(A) | zero_zero(A) = ti(A,B) | zero_zero(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))))) # label(fact_258_sum__power2__eq__zero__iff) # label(axiom). [clausify(160)]. 2.75/3.09 1076 -linordered_idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))))) | zero_zero(A) != ti(A,B) # label(fact_257_zero__less__power2) # label(axiom). [clausify(165)]. 2.75/3.09 1077 -linordered_idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))))) | zero_zero(A) = ti(A,B) # label(fact_257_zero__less__power2) # label(axiom). [clausify(165)]. 2.75/3.09 1078 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),bit1(pls))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,B)),one_one(A))) # label(fact_106_le__special_I4_J) # label(axiom). [clausify(177)]. 2.75/3.09 1079 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),bit1(pls))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,B)),one_one(A))) # label(fact_106_le__special_I4_J) # label(axiom). [clausify(177)]. 2.75/3.09 1080 -number_ring(A) | -linordered_idom(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,B)),number_number_of(A,C))) # label(fact_30_less__number__of) # label(axiom). [clausify(315)]. 2.75/3.09 1081 -number_ring(A) | -linordered_idom(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,B)),number_number_of(A,C))) # label(fact_30_less__number__of) # label(axiom). [clausify(315)]. 2.75/3.09 1082 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),number_number_of(A,B))) # label(fact_246_less__special_I1_J) # label(axiom). [clausify(323)]. 2.75/3.09 1083 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),number_number_of(A,B))) # label(fact_246_less__special_I1_J) # label(axiom). [clausify(323)]. 2.75/3.09 1084 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),number_number_of(A,B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) # label(fact_248_le__special_I1_J) # label(axiom). [clausify(369)]. 2.75/3.09 1085 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),number_number_of(A,B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) # label(fact_248_le__special_I1_J) # label(axiom). [clausify(369)]. 2.75/3.09 1086 -linordered_idom(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),C),one_one(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,C),B)),B)) # label(fact_540_mult__left__le__one__le) # label(axiom). [clausify(383)]. 2.75/3.09 1087 -linordered_idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(nat,A,power_power(A,B),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),C)))) # label(fact_269_zero__le__even__power_H) # label(axiom). [clausify(453)]. 2.75/3.09 1088 -linordered_idom(A) | ti(A,B) = ti(A,C) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) # label(fact_425_linorder__neqE__linordered__idom) # label(axiom). [clausify(522)]. 2.75/3.09 1089 -linordered_idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),B)),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) # label(fact_217_even__less__0__iff) # label(axiom). [clausify(621)]. 2.75/3.09 1090 -linordered_idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),B)),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) # label(fact_217_even__less__0__iff) # label(axiom). [clausify(621)]. 2.75/3.09 1091 -linordered_idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(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))))))) # label(fact_265_sum__power2__ge__zero) # label(axiom). [clausify(627)]. 2.75/3.09 1092 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),number_number_of(A,B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),B)) # label(fact_107_le__special_I2_J) # label(axiom). [clausify(638)]. 2.75/3.09 1093 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),number_number_of(A,B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),B)) # label(fact_107_le__special_I2_J) # label(axiom). [clausify(638)]. 2.75/3.09 1094 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),bit1(pls))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,B)),one_one(A))) # label(fact_104_less__special_I4_J) # label(axiom). [clausify(704)]. 2.75/3.09 1095 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),bit1(pls))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,B)),one_one(A))) # label(fact_104_less__special_I4_J) # label(axiom). [clausify(704)]. 2.75/3.09 1096 -linordered_idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(nat,A,power_power(A,B),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),C))),zero_zero(A))) | zero_zero(A) = ti(A,B) # label(fact_279_even__power__le__0__imp__0) # label(axiom). [clausify(722)]. 2.75/3.09 1097 -linordered_idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))))) # label(fact_253_zero__le__power2) # label(axiom). [clausify(856)]. 2.75/3.09 Derived: -number_ring(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(1063,a,1066,a)]. 2.75/3.09 Derived: -number_ring(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(1063,a,1067,a)]. 2.75/3.09 Derived: zero_zero(int) = ti(int,A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(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))))))). [resolve(1063,a,1068,a)]. 2.75/3.09 Derived: zero_zero(int) = ti(int,A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,B),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls))))))). [resolve(1063,a,1069,a)]. 2.75/3.09 Derived: zero_zero(int) != ti(int,A) | zero_zero(int) != ti(int,B) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(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))))))). [resolve(1063,a,1070,a)]. 2.75/3.09 Derived: -number_ring(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),pls)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,A)),zero_zero(int))). [resolve(1063,a,1071,a)]. 2.75/3.09 Derived: -number_ring(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),pls)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,A)),zero_zero(int))). [resolve(1063,a,1072,a)]. 2.75/3.09 Derived: zero_zero(int) != ti(int,A) | zero_zero(int) != ti(int,B) | zero_zero(int) = hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,B),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls))))). [resolve(1063,a,1073,a)]. 2.75/3.09 Derived: zero_zero(int) = ti(int,A) | zero_zero(int) != hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,B),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls))))). [resolve(1063,a,1074,a)]. 2.75/3.09 Derived: zero_zero(int) = ti(int,A) | zero_zero(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))))). [resolve(1063,a,1075,a)]. 2.75/3.09 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))))) | zero_zero(int) != ti(int,A). [resolve(1063,a,1076,a)]. 2.75/3.09 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))))) | zero_zero(int) = ti(int,A). [resolve(1063,a,1077,a)]. 2.75/3.09 Derived: -number_ring(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),bit1(pls))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,A)),one_one(int))). [resolve(1063,a,1078,a)]. 2.75/3.09 Derived: -number_ring(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),bit1(pls))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,A)),one_one(int))). [resolve(1063,a,1079,a)]. 2.75/3.09 Derived: -number_ring(int) | -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),number_number_of(int,A)),number_number_of(int,B))). [resolve(1063,a,1080,b)]. 2.75/3.09 Derived: -number_ring(int) | 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),number_number_of(int,A)),number_number_of(int,B))). [resolve(1063,a,1081,b)]. 2.75/3.09 Derived: -number_ring(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(1063,a,1082,a)]. 2.75/3.09 Derived: -number_ring(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(1063,a,1083,a)]. 2.75/3.09 Derived: -number_ring(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)). [resolve(1063,a,1084,a)]. 2.75/3.09 Derived: -number_ring(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)). [resolve(1063,a,1085,a)]. 2.75/3.09 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),B),one_one(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,times_times(int,B),A)),A)). [resolve(1063,a,1086,a)]. 2.75/3.09 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,power_power(int,A),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),B)))). [resolve(1063,a,1087,a)]. 2.75/3.09 Derived: ti(int,A) = ti(int,B) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)). [resolve(1063,a,1088,a)]. 2.75/3.09 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,A),A)),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))). [resolve(1063,a,1089,a)]. 2.75/3.09 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,A),A)),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))). [resolve(1063,a,1090,a)]. 2.75/3.09 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(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))))))). [resolve(1063,a,1091,a)]. 2.75/3.09 Derived: -number_ring(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),number_number_of(int,A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),A)). [resolve(1063,a,1092,a)]. 2.75/3.09 Derived: -number_ring(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),number_number_of(int,A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),A)). [resolve(1063,a,1093,a)]. 2.85/3.10 Derived: -number_ring(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),bit1(pls))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,A)),one_one(int))). [resolve(1063,a,1094,a)]. 2.85/3.10 Derived: -number_ring(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),bit1(pls))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,A)),one_one(int))). [resolve(1063,a,1095,a)]. 2.85/3.10 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,power_power(int,A),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),B))),zero_zero(int))) | zero_zero(int) = ti(int,A). [resolve(1063,a,1096,a)]. 2.85/3.10 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))))). [resolve(1063,a,1097,a)]. 2.85/3.10 1098 -linordered_idom(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))))),zero_zero(A))) # label(fact_256_power2__less__0) # label(axiom). [clausify(881)]. 2.85/3.10 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls))))),zero_zero(int))). [resolve(1098,a,1063,a)]. 2.85/3.10 1099 -linordered_idom(A) | zero_zero(A) != ti(A,B) | zero_zero(A) != ti(A,C) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,hAPP(nat,A,power_power(A,C),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))))),zero_zero(A))) # label(fact_266_sum__power2__le__zero__iff) # label(axiom). [clausify(908)]. 2.85/3.10 Derived: zero_zero(int) != ti(int,A) | zero_zero(int) != ti(int,B) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,B),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))))),zero_zero(int))). [resolve(1099,a,1063,a)]. 2.85/3.10 1100 -linordered_idom(A) | zero_zero(A) = ti(A,B) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,hAPP(nat,A,power_power(A,C),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))))),zero_zero(A))) # label(fact_266_sum__power2__le__zero__iff) # label(axiom). [clausify(908)]. 2.85/3.10 Derived: zero_zero(int) = ti(int,A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,B),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))))),zero_zero(int))). [resolve(1100,a,1063,a)]. 2.85/3.10 1101 -linordered_idom(A) | zero_zero(A) = ti(A,B) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,hAPP(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)))))),zero_zero(A))) # label(fact_266_sum__power2__le__zero__iff) # label(axiom). [clausify(908)]. 2.85/3.10 Derived: zero_zero(int) = ti(int,A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(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)))))),zero_zero(int))). [resolve(1101,a,1063,a)]. 2.85/3.10 1102 linordered_idom(real) # label(arity_RealDef_Oreal___Rings_Olinordered__idom) # label(axiom). [assumption]. 2.85/3.10 Derived: -number_ring(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),number_number_of(real,B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(1102,a,1064,a)]. 2.85/3.10 Derived: -number_ring(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),number_number_of(real,B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(1102,a,1065,a)]. 2.85/3.11 Derived: -number_ring(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(1102,a,1066,a)]. 2.85/3.11 Derived: -number_ring(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(1102,a,1067,a)]. 2.85/3.11 Derived: zero_zero(real) = ti(real,A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(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))))))). [resolve(1102,a,1068,a)]. 2.85/3.11 Derived: zero_zero(real) = ti(real,A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,plus_plus(real,hAPP(nat,real,power_power(real,B),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls))))))). [resolve(1102,a,1069,a)]. 2.85/3.11 Derived: zero_zero(real) != ti(real,A) | zero_zero(real) != ti(real,B) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(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))))))). [resolve(1102,a,1070,a)]. 2.85/3.11 Derived: -number_ring(real) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),pls)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),zero_zero(real))). [resolve(1102,a,1071,a)]. 2.85/3.11 Derived: -number_ring(real) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),pls)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),zero_zero(real))). [resolve(1102,a,1072,a)]. 2.85/3.11 Derived: zero_zero(real) != ti(real,A) | zero_zero(real) != ti(real,B) | zero_zero(real) = hAPP(real,real,plus_plus(real,hAPP(nat,real,power_power(real,B),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls))))). [resolve(1102,a,1073,a)]. 2.85/3.11 Derived: zero_zero(real) = ti(real,A) | zero_zero(real) != hAPP(real,real,plus_plus(real,hAPP(nat,real,power_power(real,B),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls))))). [resolve(1102,a,1074,a)]. 2.85/3.11 Derived: zero_zero(real) = ti(real,A) | zero_zero(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))))). [resolve(1102,a,1075,a)]. 2.85/3.11 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))))) | zero_zero(real) != ti(real,A). [resolve(1102,a,1076,a)]. 2.85/3.11 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))))) | zero_zero(real) = ti(real,A). [resolve(1102,a,1077,a)]. 2.85/3.11 Derived: -number_ring(real) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),bit1(pls))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),one_one(real))). [resolve(1102,a,1078,a)]. 2.85/3.11 Derived: -number_ring(real) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),bit1(pls))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),one_one(real))). [resolve(1102,a,1079,a)]. 2.85/3.11 Derived: -number_ring(real) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,A)),number_number_of(real,B))). [resolve(1102,a,1080,b)]. 2.85/3.11 Derived: -number_ring(real) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,A)),number_number_of(real,B))). [resolve(1102,a,1081,b)]. 2.85/3.11 Derived: -number_ring(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(1102,a,1082,a)]. 2.85/3.11 Derived: -number_ring(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(1102,a,1083,a)]. 2.85/3.11 Derived: -number_ring(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),number_number_of(real,A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)). [resolve(1102,a,1084,a)]. 2.85/3.11 Derived: -number_ring(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),number_number_of(real,A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)). [resolve(1102,a,1085,a)]. 2.85/3.11 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)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),one_one(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,B),A)),A)). [resolve(1102,a,1086,a)]. 2.85/3.11 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,power_power(real,A),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),B)))). [resolve(1102,a,1087,a)]. 2.85/3.11 Derived: ti(real,A) = ti(real,B) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)). [resolve(1102,a,1088,a)]. 2.85/3.11 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,A),A)),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))). [resolve(1102,a,1089,a)]. 2.85/3.11 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,A),A)),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))). [resolve(1102,a,1090,a)]. 2.85/3.11 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(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))))))). [resolve(1102,a,1091,a)]. 2.85/3.11 Derived: -number_ring(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),number_number_of(real,A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),A)). [resolve(1102,a,1092,a)]. 2.85/3.11 Derived: -number_ring(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),number_number_of(real,A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),A)). [resolve(1102,a,1093,a)]. 2.85/3.11 Derived: -number_ring(real) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),bit1(pls))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,A)),one_one(real))). [resolve(1102,a,1094,a)]. 2.85/3.11 Derived: -number_ring(real) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),bit1(pls))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,A)),one_one(real))). [resolve(1102,a,1095,a)]. 2.85/3.11 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,power_power(real,A),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),B))),zero_zero(real))) | zero_zero(real) = ti(real,A). [resolve(1102,a,1096,a)]. 2.85/3.11 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))))). [resolve(1102,a,1097,a)]. 2.85/3.11 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls))))),zero_zero(real))). [resolve(1102,a,1098,a)]. 2.85/3.13 Derived: zero_zero(real) != ti(real,A) | zero_zero(real) != ti(real,B) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,hAPP(nat,real,power_power(real,B),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))))),zero_zero(real))). [resolve(1102,a,1099,a)]. 2.85/3.13 Derived: zero_zero(real) = ti(real,A) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,hAPP(nat,real,power_power(real,B),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))))),zero_zero(real))). [resolve(1102,a,1100,a)]. 2.85/3.13 Derived: zero_zero(real) = ti(real,A) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,hAPP(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)))))),zero_zero(real))). [resolve(1102,a,1101,a)]. 2.85/3.13 1103 -linordered_idom(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),C),one_one(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),C)),B)) # label(fact_541_mult__right__le__one__le) # label(axiom). [clausify(933)]. 2.85/3.13 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_eq(int),zero_zero(int)),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),one_one(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,times_times(int,A),B)),A)). [resolve(1103,a,1063,a)]. 2.85/3.13 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_eq(real),zero_zero(real)),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),one_one(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,A),B)),A)). [resolve(1103,a,1102,a)]. 2.85/3.13 1104 -linordered_idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(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)))))),zero_zero(A))) # label(fact_267_not__sum__power2__lt__zero) # label(axiom). [clausify(993)]. 2.85/3.13 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(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)))))),zero_zero(int))). [resolve(1104,a,1063,a)]. 2.85/3.13 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(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)))))),zero_zero(real))). [resolve(1104,a,1102,a)]. 2.85/3.13 1105 -number_ring(A) | -linordered_idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),number_number_of(A,B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(pls)),B)) # label(fact_105_less__special_I2_J) # label(axiom). [clausify(998)]. 2.85/3.13 Derived: -number_ring(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(1105,b,1063,a)]. 2.85/3.13 Derived: -number_ring(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(1105,b,1102,a)]. 2.85/3.13 1106 -number_ring(A) | -linordered_idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),number_number_of(A,B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(pls)),B)) # label(fact_105_less__special_I2_J) # label(axiom). [clausify(998)]. 2.95/3.20 Derived: -number_ring(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(1106,b,1063,a)]. 2.95/3.20 Derived: -number_ring(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(1106,b,1102,a)]. 2.95/3.20 1107 -number(A) | -monoid_mult(A) | hAPP(A,A,times_times(A,number_number_of(A,B)),number_number_of(A,B)) = hAPP(nat,A,power_power(A,number_number_of(A,B)),number_number_of(nat,bit0(bit1(pls)))) # label(fact_10_power2__eq__square__number__of) # label(axiom). [clausify(45)]. 2.95/3.20 1108 monoid_mult(real) # label(arity_RealDef_Oreal___Groups_Omonoid__mult) # label(axiom). [assumption]. 2.95/3.20 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(1107,b,1108,a)]. 2.95/3.20 1109 -monoid_mult(A) | hAPP(A,A,times_times(A,hAPP(nat,A,power_power(A,B),C)),hAPP(nat,A,power_power(A,B),D)) = hAPP(nat,A,power_power(A,B),hAPP(nat,nat,plus_plus(nat,C),D)) # label(fact_261_power__add) # label(axiom). [clausify(173)]. 2.95/3.20 Derived: hAPP(real,real,times_times(real,hAPP(nat,real,power_power(real,A),B)),hAPP(nat,real,power_power(real,A),C)) = hAPP(nat,real,power_power(real,A),hAPP(nat,nat,plus_plus(nat,B),C)). [resolve(1109,a,1108,a)]. 2.95/3.20 1110 -monoid_mult(A) | hAPP(nat,A,power_power(A,hAPP(nat,A,power_power(A,B),C)),D) = hAPP(nat,A,power_power(A,B),hAPP(nat,nat,times_times(nat,C),D)) # label(fact_263_power__mult) # label(axiom). [clausify(196)]. 2.95/3.20 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(1110,a,1108,a)]. 2.95/3.20 1111 monoid_mult(nat) # label(arity_Nat_Onat___Groups_Omonoid__mult) # label(axiom). [assumption]. 2.95/3.20 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(1111,a,1107,b)]. 2.95/3.20 Derived: hAPP(nat,nat,times_times(nat,hAPP(nat,nat,power_power(nat,A),B)),hAPP(nat,nat,power_power(nat,A),C)) = hAPP(nat,nat,power_power(nat,A),hAPP(nat,nat,plus_plus(nat,B),C)). [resolve(1111,a,1109,a)]. 2.95/3.20 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(1111,a,1110,a)]. 2.95/3.20 1112 -monoid_mult(A) | hAPP(nat,A,power_power(A,hAPP(nat,A,power_power(A,B),C)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,A,power_power(A,B),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),C)) # label(fact_103_power__even__eq) # label(axiom). [clausify(219)]. 2.95/3.20 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(1112,a,1108,a)]. 2.95/3.20 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(1112,a,1111,a)]. 2.95/3.20 1113 -monoid_mult(A) | hAPP(A,A,times_times(A,hAPP(A,A,times_times(A,B),B)),B) = hAPP(nat,A,power_power(A,B),number_number_of(nat,bit1(bit1(pls)))) # label(fact_163_power3__eq__cube) # label(axiom). [clausify(331)]. 2.95/3.20 Derived: hAPP(real,real,times_times(real,hAPP(real,real,times_times(real,A),A)),A) = hAPP(nat,real,power_power(real,A),number_number_of(nat,bit1(bit1(pls)))). [resolve(1113,a,1108,a)]. 2.95/3.20 Derived: hAPP(nat,nat,times_times(nat,hAPP(nat,nat,times_times(nat,A),A)),A) = hAPP(nat,nat,power_power(nat,A),number_number_of(nat,bit1(bit1(pls)))). [resolve(1113,a,1111,a)]. 3.05/3.30 1114 -monoid_mult(A) | ti(A,B) = hAPP(A,A,times_times(A,B),one_one(A)) # label(fact_793_mult__1__right) # label(axiom). [clausify(423)]. 3.05/3.30 Derived: ti(real,A) = hAPP(real,real,times_times(real,A),one_one(real)). [resolve(1114,a,1108,a)]. 3.05/3.30 Derived: ti(nat,A) = hAPP(nat,nat,times_times(nat,A),one_one(nat)). [resolve(1114,a,1111,a)]. 3.05/3.30 1115 monoid_mult(int) # label(arity_Int_Oint___Groups_Omonoid__mult) # label(axiom). [assumption]. 3.05/3.30 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(1115,a,1107,b)]. 3.05/3.30 Derived: hAPP(int,int,times_times(int,hAPP(nat,int,power_power(int,A),B)),hAPP(nat,int,power_power(int,A),C)) = hAPP(nat,int,power_power(int,A),hAPP(nat,nat,plus_plus(nat,B),C)). [resolve(1115,a,1109,a)]. 3.05/3.30 Derived: hAPP(nat,int,power_power(int,hAPP(nat,int,power_power(int,A),B)),C) = hAPP(nat,int,power_power(int,A),hAPP(nat,nat,times_times(nat,B),C)). [resolve(1115,a,1110,a)]. 3.05/3.30 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(1115,a,1112,a)]. 3.05/3.30 Derived: hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,A),A)),A) = hAPP(nat,int,power_power(int,A),number_number_of(nat,bit1(bit1(pls)))). [resolve(1115,a,1113,a)]. 3.05/3.30 Derived: ti(int,A) = hAPP(int,int,times_times(int,A),one_one(int)). [resolve(1115,a,1114,a)]. 3.05/3.30 1116 -monoid_mult(A) | one_one(A) = hAPP(nat,A,power_power(A,one_one(A)),B) # label(fact_262_power__one) # label(axiom). [clausify(650)]. 3.05/3.30 Derived: one_one(real) = hAPP(nat,real,power_power(real,one_one(real)),A). [resolve(1116,a,1108,a)]. 3.05/3.30 Derived: one_one(nat) = hAPP(nat,nat,power_power(nat,one_one(nat)),A). [resolve(1116,a,1111,a)]. 3.05/3.30 Derived: one_one(int) = hAPP(nat,int,power_power(int,one_one(int)),A). [resolve(1116,a,1115,a)]. 3.05/3.30 1117 -monoid_mult(A) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),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) = hAPP(nat,A,power_power(A,C),B) # label(fact_436_realpow__minus__mult) # label(axiom). [clausify(707)]. 3.05/3.30 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),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) = hAPP(nat,real,power_power(real,B),A). [resolve(1117,a,1108,a)]. 3.05/3.30 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),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) = hAPP(nat,nat,power_power(nat,B),A). [resolve(1117,a,1111,a)]. 3.05/3.30 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),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) = hAPP(nat,int,power_power(int,B),A). [resolve(1117,a,1115,a)]. 3.05/3.30 1118 -monoid_mult(A) | hAPP(A,A,times_times(A,hAPP(nat,A,power_power(A,B),C)),B) = hAPP(A,A,times_times(A,B),hAPP(nat,A,power_power(A,B),C)) # label(fact_259_power__commutes) # label(axiom). [clausify(750)]. 3.05/3.30 Derived: hAPP(real,real,times_times(real,hAPP(nat,real,power_power(real,A),B)),A) = hAPP(real,real,times_times(real,A),hAPP(nat,real,power_power(real,A),B)). [resolve(1118,a,1108,a)]. 3.05/3.30 Derived: hAPP(nat,nat,times_times(nat,hAPP(nat,nat,power_power(nat,A),B)),A) = hAPP(nat,nat,times_times(nat,A),hAPP(nat,nat,power_power(nat,A),B)). [resolve(1118,a,1111,a)]. 3.05/3.30 Derived: hAPP(int,int,times_times(int,hAPP(nat,int,power_power(int,A),B)),A) = hAPP(int,int,times_times(int,A),hAPP(nat,int,power_power(int,A),B)). [resolve(1118,a,1115,a)]. 3.05/3.30 1119 -monoid_mult(A) | hAPP(A,A,times_times(A,B),B) = hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))) # label(fact_14_power2__eq__square) # label(axiom). [clausify(939)]. 3.17/3.42 Derived: hAPP(real,real,times_times(real,A),A) = hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))). [resolve(1119,a,1108,a)]. 3.17/3.42 Derived: hAPP(nat,nat,times_times(nat,A),A) = hAPP(nat,nat,power_power(nat,A),number_number_of(nat,bit0(bit1(pls)))). [resolve(1119,a,1111,a)]. 3.17/3.42 Derived: hAPP(int,int,times_times(int,A),A) = hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))). [resolve(1119,a,1115,a)]. 3.17/3.42 1120 -monoid_mult(A) | ti(A,B) = hAPP(A,A,times_times(A,one_one(A)),B) # label(fact_791_mult__1__left) # label(axiom). [clausify(950)]. 3.17/3.42 Derived: ti(real,A) = hAPP(real,real,times_times(real,one_one(real)),A). [resolve(1120,a,1108,a)]. 3.17/3.42 Derived: ti(nat,A) = hAPP(nat,nat,times_times(nat,one_one(nat)),A). [resolve(1120,a,1111,a)]. 3.17/3.42 Derived: ti(int,A) = hAPP(int,int,times_times(int,one_one(int)),A). [resolve(1120,a,1115,a)]. 3.17/3.42 1121 -monoid_mult(A) | ti(A,B) = hAPP(nat,A,power_power(A,B),one_one(nat)) # label(fact_242_power__one__right) # label(axiom). [clausify(1040)]. 3.17/3.42 Derived: ti(real,A) = hAPP(nat,real,power_power(real,A),one_one(nat)). [resolve(1121,a,1108,a)]. 3.17/3.42 Derived: ti(nat,A) = hAPP(nat,nat,power_power(nat,A),one_one(nat)). [resolve(1121,a,1111,a)]. 3.17/3.42 Derived: ti(int,A) = hAPP(nat,int,power_power(int,A),one_one(nat)). [resolve(1121,a,1115,a)]. 3.17/3.42 1122 -no_zero_divisors(A) | ti(fun(A,A),times_times(A,B)) = times_times(A,B) # label(tsy_c_Groups_Otimes__class_Otimes_5_res) # label(axiom). [clausify(256)]. 3.17/3.42 1123 no_zero_divisors(real) # label(arity_RealDef_Oreal___Rings_Ono__zero__divisors) # label(axiom). [assumption]. 3.17/3.42 1124 no_zero_divisors(int) # label(arity_Int_Oint___Rings_Ono__zero__divisors) # label(axiom). [assumption]. 3.17/3.42 Derived: ti(fun(real,real),times_times(real,A)) = times_times(real,A). [resolve(1122,a,1123,a)]. 3.17/3.42 Derived: ti(fun(int,int),times_times(int,A)) = times_times(int,A). [resolve(1122,a,1124,a)]. 3.17/3.42 1125 -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(482)]. 3.17/3.42 Derived: times_times(real,ti(real,A)) = times_times(real,A). [resolve(1125,a,1123,a)]. 3.17/3.42 Derived: times_times(int,ti(int,A)) = times_times(int,A). [resolve(1125,a,1124,a)]. 3.17/3.42 1126 no_zero_divisors(nat) # label(arity_Nat_Onat___Rings_Ono__zero__divisors) # label(axiom). [assumption]. 3.17/3.42 Derived: ti(fun(nat,nat),times_times(nat,A)) = times_times(nat,A). [resolve(1126,a,1122,a)]. 3.17/3.42 Derived: times_times(nat,ti(nat,A)) = times_times(nat,A). [resolve(1126,a,1125,a)]. 3.17/3.42 1127 -power(A) | -zero_neq_one(A) | -no_zero_divisors(A) | -mult_zero(A) | zero_zero(A) != hAPP(nat,A,power_power(A,B),number_number_of(nat,C)) | zero_zero(A) = ti(A,B) # label(fact_343_power__eq__0__iff__number__of) # label(axiom). [clausify(602)]. 3.17/3.42 Derived: -power(real) | -zero_neq_one(real) | -mult_zero(real) | zero_zero(real) != hAPP(nat,real,power_power(real,A),number_number_of(nat,B)) | zero_zero(real) = ti(real,A). [resolve(1127,c,1123,a)]. 3.17/3.42 Derived: -power(int) | -zero_neq_one(int) | -mult_zero(int) | zero_zero(int) != hAPP(nat,int,power_power(int,A),number_number_of(nat,B)) | zero_zero(int) = ti(int,A). [resolve(1127,c,1124,a)]. 3.17/3.42 Derived: -power(nat) | -zero_neq_one(nat) | -mult_zero(nat) | zero_zero(nat) != hAPP(nat,nat,power_power(nat,A),number_number_of(nat,B)) | zero_zero(nat) = ti(nat,A). [resolve(1127,c,1126,a)]. 3.17/3.42 1128 -power(A) | -zero_neq_one(A) | -no_zero_divisors(A) | -mult_zero(A) | zero_zero(A) != hAPP(nat,A,power_power(A,B),number_number_of(nat,C)) | zero_zero(nat) != number_number_of(nat,C) # label(fact_343_power__eq__0__iff__number__of) # label(axiom). [clausify(602)]. 3.17/3.42 Derived: -power(real) | -zero_neq_one(real) | -mult_zero(real) | zero_zero(real) != hAPP(nat,real,power_power(real,A),number_number_of(nat,B)) | zero_zero(nat) != number_number_of(nat,B). [resolve(1128,c,1123,a)]. 3.17/3.42 Derived: -power(int) | -zero_neq_one(int) | -mult_zero(int) | zero_zero(int) != hAPP(nat,int,power_power(int,A),number_number_of(nat,B)) | zero_zero(nat) != number_number_of(nat,B). [resolve(1128,c,1124,a)]. 3.17/3.42 1129 -power(A) | -zero_neq_one(A) | -no_zero_divisors(A) | -mult_zero(A) | zero_zero(A) = hAPP(nat,A,power_power(A,B),number_number_of(nat,C)) | zero_zero(A) != ti(A,B) | zero_zero(nat) = number_number_of(nat,C) # label(fact_343_power__eq__0__iff__number__of) # label(axiom). [clausify(602)]. 3.35/3.61 Derived: -power(real) | -zero_neq_one(real) | -mult_zero(real) | zero_zero(real) = hAPP(nat,real,power_power(real,A),number_number_of(nat,B)) | zero_zero(real) != ti(real,A) | zero_zero(nat) = number_number_of(nat,B). [resolve(1129,c,1123,a)]. 3.35/3.61 Derived: -power(int) | -zero_neq_one(int) | -mult_zero(int) | zero_zero(int) = hAPP(nat,int,power_power(int,A),number_number_of(nat,B)) | zero_zero(int) != ti(int,A) | zero_zero(nat) = number_number_of(nat,B). [resolve(1129,c,1124,a)]. 3.35/3.61 Derived: -power(nat) | -zero_neq_one(nat) | -mult_zero(nat) | zero_zero(nat) = hAPP(nat,nat,power_power(nat,A),number_number_of(nat,B)) | zero_zero(nat) != ti(nat,A) | zero_zero(nat) = number_number_of(nat,B). [resolve(1129,c,1126,a)]. 3.35/3.61 1130 -no_zero_divisors(A) | zero_zero(A) != hAPP(A,A,times_times(A,B),C) | zero_zero(A) = ti(A,C) | zero_zero(A) = ti(A,B) # label(fact_440_divisors__zero) # label(axiom). [clausify(735)]. 3.35/3.61 Derived: zero_zero(real) != hAPP(real,real,times_times(real,A),B) | zero_zero(real) = ti(real,B) | zero_zero(real) = ti(real,A). [resolve(1130,a,1123,a)]. 3.35/3.61 Derived: zero_zero(int) != hAPP(int,int,times_times(int,A),B) | zero_zero(int) = ti(int,B) | zero_zero(int) = ti(int,A). [resolve(1130,a,1124,a)]. 3.35/3.61 Derived: zero_zero(nat) != hAPP(nat,nat,times_times(nat,A),B) | zero_zero(nat) = ti(nat,B) | zero_zero(nat) = ti(nat,A). [resolve(1130,a,1126,a)]. 3.35/3.61 1131 -mult_zero(A) | -zero_neq_one(A) | -no_zero_divisors(A) | -power(A) | zero_zero(A) != hAPP(nat,A,power_power(A,B),C) | zero_zero(A) = ti(A,B) # label(fact_185_power__eq__0__iff) # label(axiom). [clausify(737)]. 3.35/3.61 Derived: -mult_zero(real) | -zero_neq_one(real) | -power(real) | zero_zero(real) != hAPP(nat,real,power_power(real,A),B) | zero_zero(real) = ti(real,A). [resolve(1131,c,1123,a)]. 3.35/3.61 Derived: -mult_zero(int) | -zero_neq_one(int) | -power(int) | zero_zero(int) != hAPP(nat,int,power_power(int,A),B) | zero_zero(int) = ti(int,A). [resolve(1131,c,1124,a)]. 3.35/3.61 Derived: -mult_zero(nat) | -zero_neq_one(nat) | -power(nat) | zero_zero(nat) != hAPP(nat,nat,power_power(nat,A),B) | zero_zero(nat) = ti(nat,A). [resolve(1131,c,1126,a)]. 3.35/3.61 1132 -mult_zero(A) | -zero_neq_one(A) | -no_zero_divisors(A) | -power(A) | zero_zero(A) != hAPP(nat,A,power_power(A,B),C) | zero_zero(nat) != C # label(fact_185_power__eq__0__iff) # label(axiom). [clausify(737)]. 3.35/3.61 Derived: -mult_zero(real) | -zero_neq_one(real) | -power(real) | zero_zero(real) != hAPP(nat,real,power_power(real,A),B) | zero_zero(nat) != B. [resolve(1132,c,1123,a)]. 3.35/3.61 Derived: -mult_zero(int) | -zero_neq_one(int) | -power(int) | zero_zero(int) != hAPP(nat,int,power_power(int,A),B) | zero_zero(nat) != B. [resolve(1132,c,1124,a)]. 3.35/3.61 1133 -mult_zero(A) | -zero_neq_one(A) | -no_zero_divisors(A) | -power(A) | zero_zero(A) = hAPP(nat,A,power_power(A,B),C) | zero_zero(A) != ti(A,B) | zero_zero(nat) = C # label(fact_185_power__eq__0__iff) # label(axiom). [clausify(737)]. 3.35/3.61 Derived: -mult_zero(real) | -zero_neq_one(real) | -power(real) | zero_zero(real) = hAPP(nat,real,power_power(real,A),B) | zero_zero(real) != ti(real,A) | zero_zero(nat) = B. [resolve(1133,c,1123,a)]. 3.35/3.61 Derived: -mult_zero(int) | -zero_neq_one(int) | -power(int) | zero_zero(int) = hAPP(nat,int,power_power(int,A),B) | zero_zero(int) != ti(int,A) | zero_zero(nat) = B. [resolve(1133,c,1124,a)]. 3.35/3.61 Derived: -mult_zero(nat) | -zero_neq_one(nat) | -power(nat) | zero_zero(nat) = hAPP(nat,nat,power_power(nat,A),B) | zero_zero(nat) != ti(nat,A) | zero_zero(nat) = B. [resolve(1133,c,1126,a)]. 3.35/3.61 1134 -no_zero_divisors(A) | zero_zero(A) = ti(A,B) | zero_zero(A) = ti(A,C) | zero_zero(A) != hAPP(A,A,times_times(A,B),C) # label(fact_441_no__zero__divisors) # label(axiom). [clausify(951)]. 3.35/3.61 1135 ordered_ring(real) # label(arity_RealDef_Oreal___Rings_Oordered__ring) # label(axiom). [assumption]. 3.35/3.61 1136 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),D)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,E),C)),F))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,B),E)),C)),D)),F)) # label(fact_545_less__add__iff1) # label(axiom). [clausify(15)]. 3.35/3.61 1137 -ordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),D)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,E),C)),F))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,B),E)),C)),D)),F)) # label(fact_545_less__add__iff1) # label(axiom). [clausify(15)]. 3.35/3.61 1138 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,D),C)),hAPP(A,A,times_times(A,D),B))) # label(fact_475_mult__left__mono__neg) # label(axiom). [clausify(36)]. 3.35/3.61 1139 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,C),D)),E)),F))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,D),E)),B)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),E)),F))) # label(fact_544_le__add__iff2) # label(axiom). [clausify(356)]. 3.35/3.61 1140 -ordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,C),D)),E)),F))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,D),E)),B)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),E)),F))) # label(fact_544_le__add__iff2) # label(axiom). [clausify(356)]. 3.35/3.61 1141 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,C),D)),hAPP(A,A,times_times(A,B),D))) # label(fact_476_mult__right__mono__neg) # label(axiom). [clausify(480)]. 3.35/3.61 1142 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),D)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,E),C)),F))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,B),E)),C)),D)),F)) # label(fact_543_le__add__iff1) # label(axiom). [clausify(565)]. 3.35/3.61 1143 -ordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),D)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,E),C)),F))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,B),E)),C)),D)),F)) # label(fact_543_le__add__iff1) # label(axiom). [clausify(565)]. 3.35/3.61 1144 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,C),D)),E)),F))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,D),E)),B)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),E)),F))) # label(fact_546_less__add__iff2) # label(axiom). [clausify(725)]. 3.35/3.61 1145 -ordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,C),D)),E)),F))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,D),E)),B)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),E)),F))) # label(fact_546_less__add__iff2) # label(axiom). [clausify(725)]. 3.35/3.61 1146 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,times_times(A,B),C))) # label(fact_480_mult__nonpos__nonpos) # label(axiom). [clausify(781)]. 3.35/3.61 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),B)),C)),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,D),B)),E))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,A),D)),B)),C)),E)). [resolve(1135,a,1136,a)]. 3.35/3.61 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),B)),C)),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,D),B)),E))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,A),D)),B)),C)),E)). [resolve(1135,a,1137,a)]. 3.35/3.61 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(1135,a,1138,a)]. 3.35/3.61 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(1135,a,1139,a)]. 3.35/3.61 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(1135,a,1140,a)]. 3.35/3.61 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(1135,a,1141,a)]. 3.35/3.61 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),B)),C)),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,D),B)),E))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,A),D)),B)),C)),E)). [resolve(1135,a,1142,a)]. 3.35/3.61 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),B)),C)),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,D),B)),E))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,A),D)),B)),C)),E)). [resolve(1135,a,1143,a)]. 3.35/3.61 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(1135,a,1144,a)]. 3.35/3.61 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(1135,a,1145,a)]. 3.35/3.63 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(1135,a,1146,a)]. 3.35/3.63 1147 ordered_ring(int) # label(arity_Int_Oint___Rings_Oordered__ring) # label(axiom). [assumption]. 3.35/3.63 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),B)),C)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,D),B)),E))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,A),D)),B)),C)),E)). [resolve(1147,a,1136,a)]. 3.35/3.63 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),B)),C)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,D),B)),E))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,A),D)),B)),C)),E)). [resolve(1147,a,1137,a)]. 3.35/3.63 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(1147,a,1138,a)]. 3.35/3.63 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(1147,a,1139,a)]. 3.35/3.63 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(1147,a,1140,a)]. 3.35/3.63 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(1147,a,1141,a)]. 3.35/3.63 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),B)),C)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,D),B)),E))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,A),D)),B)),C)),E)). [resolve(1147,a,1142,a)]. 3.35/3.63 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),B)),C)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,D),B)),E))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,A),D)),B)),C)),E)). [resolve(1147,a,1143,a)]. 3.35/3.63 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(1147,a,1144,a)]. 3.35/3.63 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(1147,a,1145,a)]. 3.47/3.73 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(1147,a,1146,a)]. 3.47/3.73 1148 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,times_times(A,C),B))) # label(fact_472_split__mult__pos__le) # label(axiom). [clausify(942)]. 3.47/3.73 1149 -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,C),B))) # label(fact_472_split__mult__pos__le) # label(axiom). [clausify(942)]. 3.47/3.73 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,B),A))). [resolve(1149,a,1135,a)]. 3.47/3.73 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,B),A))). [resolve(1149,a,1147,a)]. 3.47/3.73 1150 linordered_semidom(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semidom) # label(axiom). [assumption]. 3.47/3.73 1151 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),one_one(A))) # label(fact_508_zero__less__one) # label(axiom). [clausify(17)]. 3.47/3.73 1152 -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_272_power__inject__exp) # label(axiom). [clausify(101)]. 3.47/3.73 1153 -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_272_power__inject__exp) # label(axiom). [clausify(101)]. 3.47/3.73 1154 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),one_one(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),hAPP(nat,A,power_power(A,B),C))),hAPP(nat,A,power_power(A,B),C))) # label(fact_232_power__Suc__less) # label(axiom). [clausify(172)]. 3.47/3.73 1155 -linordered_semidom(A) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),one_one(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(nat,A,power_power(A,D),C)),hAPP(nat,A,power_power(A,D),B))) # label(fact_215_power__decreasing) # label(axiom). [clausify(212)]. 3.47/3.73 1156 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(nat,A,power_power(A,B),C))) # label(fact_208_zero__le__power) # label(axiom). [clausify(215)]. 3.47/3.73 1157 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(nat,A,power_power(A,B),C)),hAPP(nat,A,power_power(A,B),D))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) # label(fact_178_power__le__imp__le__exp) # label(axiom). [clausify(226)]. 3.47/3.73 1158 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),hAPP(A,A,times_times(A,B),C))) # label(fact_510_less__1__mult) # label(axiom). [clausify(228)]. 3.47/3.73 1159 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),hAPP(A,A,plus_plus(A,B),D))) # label(fact_505_pos__add__strict) # label(axiom). [clausify(287)]. 3.47/3.73 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),one_one(real))). [resolve(1150,a,1151,a)]. 3.47/3.73 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(1150,a,1152,a)]. 3.47/3.73 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(1150,a,1153,a)]. 3.47/3.73 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(1150,a,1154,a)]. 3.47/3.73 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(1150,a,1155,a)]. 3.47/3.73 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(1150,a,1156,a)]. 3.47/3.73 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,power_power(real,A),B)),hAPP(nat,real,power_power(real,A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)). [resolve(1150,a,1157,a)]. 3.47/3.73 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(1150,a,1158,a)]. 3.47/3.73 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(1150,a,1159,a)]. 3.47/3.73 1160 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),one_one(A))) # label(fact_506_zero__le__one) # label(axiom). [clausify(311)]. 3.47/3.73 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),one_one(real))). [resolve(1160,a,1150,a)]. 3.47/3.73 1161 -linordered_semidom(A) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(nat,A,power_power(A,D),B)),hAPP(nat,A,power_power(A,D),C))) # label(fact_271_power__increasing) # label(axiom). [clausify(339)]. 3.47/3.73 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(1161,a,1150,a)]. 3.47/3.75 1162 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(nat,A,power_power(A,B),C))) # label(fact_210_zero__less__power) # label(axiom). [clausify(364)]. 3.47/3.75 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(1162,a,1150,a)]. 3.47/3.75 1163 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,power_power(A,B),C)),hAPP(nat,A,power_power(A,D),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),D)) # label(fact_214_power__less__imp__less__base) # label(axiom). [clausify(418)]. 3.47/3.75 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(1163,a,1150,a)]. 3.47/3.75 1164 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),hAPP(A,A,times_times(A,B),hAPP(nat,A,power_power(A,B),C)))) # label(fact_177_power__gt1__lemma) # label(axiom). [clausify(464)]. 3.47/3.75 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(1164,a,1150,a)]. 3.47/3.75 1165 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,A,power_power(A,C),number_number_of(nat,bit0(bit1(pls)))))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) # label(fact_264_power2__less__imp__less) # label(axiom). [clausify(485)]. 3.47/3.75 1166 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(A,A,plus_plus(A,B),one_one(A)))) # label(fact_511_less__add__one) # label(axiom). [clausify(487)]. 3.47/3.75 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(1166,a,1150,a)]. 3.47/3.75 1167 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(nat,A,power_power(A,B),D)),hAPP(nat,A,power_power(A,C),D))) # label(fact_209_power__mono) # label(axiom). [clausify(509)]. 3.47/3.75 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(1167,a,1150,a)]. 3.47/3.75 1168 -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_179_power__increasing__iff) # label(axiom). [clausify(528)]. 3.47/3.75 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(1168,a,1150,a)]. 3.47/3.75 1169 -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_179_power__increasing__iff) # label(axiom). [clausify(528)]. 3.47/3.79 1170 -linordered_semidom(A) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,power_power(A,D),B)),hAPP(nat,A,power_power(A,D),C))) # label(fact_275_power__strict__increasing) # label(axiom). [clausify(540)]. 3.47/3.79 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,power_power(real,C),A)),hAPP(nat,real,power_power(real,C),B))). [resolve(1170,a,1150,a)]. 3.47/3.79 1171 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),zero_zero(A))) # label(fact_509_not__one__less__zero) # label(axiom). [clausify(562)]. 3.47/3.79 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),zero_zero(real))). [resolve(1171,a,1150,a)]. 3.47/3.79 1172 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,power_power(A,B),C)),hAPP(nat,A,power_power(A,B),D))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),C),D)) # label(fact_273_power__strict__increasing__iff) # label(axiom). [clausify(589)]. 3.47/3.79 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(1172,a,1150,a)]. 3.47/3.79 1173 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,power_power(A,B),C)),hAPP(nat,A,power_power(A,B),D))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),C),D)) # label(fact_273_power__strict__increasing__iff) # label(axiom). [clausify(589)]. 3.47/3.79 1174 -linordered_semidom(A) | hAPP(nat,A,power_power(A,B),C) != hAPP(nat,A,power_power(A,D),C) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C)) | ti(A,B) = ti(A,D) # label(fact_189_power__eq__imp__eq__base) # label(axiom). [clausify(620)]. 3.47/3.79 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(1174,a,1150,a)]. 3.47/3.79 1175 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,plus_plus(A,one_one(A)),one_one(A)))) # label(fact_542_zero__less__two) # label(axiom). [clausify(708)]. 3.47/3.79 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(1175,a,1150,a)]. 3.47/3.79 1176 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),zero_zero(A))) # label(fact_507_not__one__le__zero) # label(axiom). [clausify(779)]. 3.47/3.79 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),zero_zero(real))). [resolve(1176,a,1150,a)]. 3.47/3.79 1177 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,power_power(A,B),D)),hAPP(nat,A,power_power(A,C),D))) # label(fact_197_power__strict__mono) # label(axiom). [clausify(786)]. 3.47/3.80 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_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(1177,a,1150,a)]. 3.47/3.80 1178 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),hAPP(nat,A,power_power(A,B),C))) # label(fact_321_one__less__power) # label(axiom). [clausify(792)]. 3.47/3.80 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(1178,a,1150,a)]. 3.47/3.80 1179 linordered_semidom(int) # label(arity_Int_Oint___Rings_Olinordered__semidom) # label(axiom). [assumption]. 3.47/3.80 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),one_one(int))). [resolve(1179,a,1151,a)]. 3.47/3.80 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(1179,a,1152,a)]. 3.47/3.80 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(1179,a,1153,a)]. 3.47/3.80 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(1179,a,1154,a)]. 3.47/3.80 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(1179,a,1155,a)]. 3.47/3.80 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,power_power(int,A),B)),hAPP(nat,int,power_power(int,A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)). [resolve(1179,a,1157,a)]. 3.47/3.80 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(1179,a,1158,a)]. 3.47/3.80 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(1179,a,1159,a)]. 3.47/3.80 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),one_one(int))). [resolve(1179,a,1160,a)]. 3.47/3.80 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(1179,a,1161,a)]. 3.47/3.80 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(1179,a,1162,a)]. 3.56/3.81 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(1179,a,1163,a)]. 3.56/3.81 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(1179,a,1164,a)]. 3.56/3.81 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(1179,a,1166,a)]. 3.56/3.81 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(1179,a,1167,a)]. 3.56/3.81 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(1179,a,1168,a)]. 3.56/3.81 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,power_power(int,C),A)),hAPP(nat,int,power_power(int,C),B))). [resolve(1179,a,1170,a)]. 3.56/3.81 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),zero_zero(int))). [resolve(1179,a,1171,a)]. 3.56/3.81 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(1179,a,1172,a)]. 3.56/3.81 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(1179,a,1174,a)]. 3.56/3.81 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(1179,a,1175,a)]. 3.56/3.81 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),zero_zero(int))). [resolve(1179,a,1176,a)]. 3.56/3.81 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(1179,a,1177,a)]. 3.56/3.81 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(1179,a,1178,a)]. 3.56/3.81 1180 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,power_power(A,B),C)),hAPP(nat,A,power_power(A,B),D))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),C),D)) # label(fact_274_power__less__imp__less__exp) # label(axiom). [clausify(853)]. 3.56/3.81 1181 -linordered_semidom(A) | hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))) != hAPP(nat,A,power_power(A,C),number_number_of(nat,bit0(bit1(pls)))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | ti(A,B) = ti(A,C) # label(fact_255_power2__eq__imp__eq) # label(axiom). [clausify(868)]. 3.56/3.83 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(1181,a,1150,a)]. 3.56/3.83 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(1181,a,1179,a)]. 3.56/3.83 1182 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,power_power(A,B),C)),hAPP(A,A,times_times(A,B),hAPP(nat,A,power_power(A,B),C)))) # label(fact_176_power__less__power__Suc) # label(axiom). [clausify(913)]. 3.56/3.83 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(1182,a,1150,a)]. 3.56/3.83 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(1182,a,1179,a)]. 3.56/3.83 1183 -linordered_semidom(A) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),one_one(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,power_power(A,D),C)),hAPP(nat,A,power_power(A,D),B))) # label(fact_216_power__strict__decreasing) # label(axiom). [clausify(925)]. 3.56/3.83 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(1183,a,1150,a)]. 3.56/3.83 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(1183,a,1179,a)]. 3.56/3.83 1184 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,A,power_power(A,C),number_number_of(nat,bit0(bit1(pls)))))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_254_power2__le__imp__le) # label(axiom). [clausify(968)]. 3.56/3.83 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(1184,a,1150,a)]. 3.56/3.83 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(1184,a,1179,a)]. 3.56/3.83 1185 linordered_semidom(nat) # label(arity_Nat_Onat___Rings_Olinordered__semidom) # label(axiom). [assumption]. 3.56/3.83 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),one_one(nat))). [resolve(1185,a,1151,a)]. 3.56/3.83 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(1185,a,1152,a)]. 3.56/3.83 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(1185,a,1153,a)]. 3.56/3.83 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(1185,a,1154,a)]. 3.56/3.83 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(1185,a,1155,a)]. 3.56/3.83 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,power_power(nat,A),B)),hAPP(nat,nat,power_power(nat,A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)). [resolve(1185,a,1157,a)]. 3.56/3.83 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(1185,a,1158,a)]. 3.56/3.83 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),hAPP(nat,nat,plus_plus(nat,A),C))). [resolve(1185,a,1159,a)]. 3.56/3.83 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(1185,a,1161,a)]. 3.56/3.83 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,power_power(nat,A),B))). [resolve(1185,a,1162,a)]. 3.56/3.83 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(1185,a,1163,a)]. 3.56/3.83 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(1185,a,1164,a)]. 3.56/3.83 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(1185,a,1166,a)]. 3.56/3.83 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(1185,a,1167,a)]. 3.56/3.83 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(1185,a,1168,a)]. 3.56/3.84 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,power_power(nat,C),A)),hAPP(nat,nat,power_power(nat,C),B))). [resolve(1185,a,1170,a)]. 3.56/3.84 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),zero_zero(nat))). [resolve(1185,a,1171,a)]. 3.56/3.84 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(1185,a,1172,a)]. 3.56/3.84 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(1185,a,1174,a)]. 3.56/3.84 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(1185,a,1175,a)]. 3.56/3.84 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),zero_zero(nat))). [resolve(1185,a,1176,a)]. 3.56/3.84 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(1185,a,1177,a)]. 3.56/3.84 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(1185,a,1178,a)]. 3.56/3.84 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(1185,a,1181,a)]. 3.56/3.84 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(1185,a,1182,a)]. 3.56/3.84 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(1185,a,1183,a)]. 3.56/3.84 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(1185,a,1184,a)]. 3.56/3.84 1186 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),hAPP(nat,A,power_power(A,B),C))) # label(fact_270_one__le__power) # label(axiom). [clausify(1012)]. 3.65/3.93 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(1186,a,1150,a)]. 3.65/3.93 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(1186,a,1179,a)]. 3.65/3.93 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(1186,a,1185,a)]. 3.65/3.93 1187 semiring_div(int) # label(arity_Int_Oint___Divides_Osemiring__div) # label(axiom). [assumption]. 3.65/3.93 1188 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,B),C)),C) = hAPP(A,A,div_mod(A,B),C) # label(fact_927_mod__add__self2) # label(axiom). [clausify(25)]. 3.65/3.93 1189 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | zero_zero(A) = hAPP(A,A,div_mod(A,C),B) # label(fact_904_dvd__eq__mod__eq__0) # label(axiom). [clausify(39)]. 3.65/3.93 1190 -semiring_div(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | zero_zero(A) != hAPP(A,A,div_mod(A,C),B) # label(fact_904_dvd__eq__mod__eq__0) # label(axiom). [clausify(39)]. 3.65/3.93 1191 -semiring_div(A) | ti(A,B) = hAPP(A,A,div_mod(A,B),zero_zero(A)) # label(fact_943_mod__by__0) # label(axiom). [clausify(51)]. 3.65/3.93 1192 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,hAPP(A,A,div_mod(A,B),C)),D)),C) = hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,B),D)),C) # label(fact_933_zmod__simps_I1_J) # label(axiom). [clausify(188)]. 3.65/3.93 1193 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D)) = hAPP(A,A,times_times(A,B),hAPP(A,A,div_mod(A,C),D)) # label(fact_938_mod__mult__mult1) # label(axiom). [clausify(207)]. 3.65/3.93 1194 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,B),hAPP(A,A,div_mod(A,C),D))),D) = hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,B),C)),D) # label(fact_935_mod__mult__right__eq) # label(axiom). [clausify(211)]. 3.65/3.93 1195 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C)) = hAPP(A,A,times_times(A,hAPP(A,A,div_mod(A,B),D)),C) # label(fact_939_mod__mult__mult2) # label(axiom). [clausify(218)]. 3.65/3.93 1196 -semiring_div(A) | -number(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),number_number_of(A,B)),number_number_of(A,C))) | zero_zero(A) = hAPP(A,A,div_mod(A,number_number_of(A,C)),number_number_of(A,B)) # label(fact_951_dvd__eq__mod__eq__0__number__of) # label(axiom). [clausify(242)]. 3.65/3.93 1197 -semiring_div(A) | -number(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),number_number_of(A,B)),number_number_of(A,C))) | zero_zero(A) != hAPP(A,A,div_mod(A,number_number_of(A,C)),number_number_of(A,B)) # label(fact_951_dvd__eq__mod__eq__0__number__of) # label(axiom). [clausify(242)]. 3.65/3.93 1198 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,B),hAPP(A,A,times_times(A,C),D))),C) = hAPP(A,A,div_mod(A,B),C) # label(fact_903_mod__mult__self2) # label(axiom). [clausify(245)]. 3.65/3.93 1199 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,div_mod(A,C),D))) # label(fact_924_dvd__mod) # label(axiom). [clausify(261)]. 3.65/3.93 1200 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,div_mod(A,C),D))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) # label(fact_923_dvd__mod__imp__dvd) # label(axiom). [clausify(262)]. 3.65/3.93 1201 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,div_mod(A,B),C)),C) = hAPP(A,A,div_mod(A,B),C) # label(fact_918_mod__mod__trivial) # label(axiom). [clausify(270)]. 3.65/3.93 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(1187,a,1188,a)]. 3.65/3.93 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | zero_zero(int) = hAPP(int,int,div_mod(int,B),A). [resolve(1187,a,1189,a)]. 3.65/3.96 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | zero_zero(int) != hAPP(int,int,div_mod(int,B),A). [resolve(1187,a,1190,a)]. 3.65/3.96 Derived: ti(int,A) = hAPP(int,int,div_mod(int,A),zero_zero(int)). [resolve(1187,a,1191,a)]. 3.65/3.96 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(1187,a,1192,a)]. 3.65/3.96 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(1187,a,1193,a)]. 3.65/3.96 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(1187,a,1195,a)]. 3.65/3.96 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(1187,a,1198,a)]. 3.65/3.96 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(1187,a,1201,a)]. 3.65/3.96 1202 -semiring_div(A) | hAPP(A,A,div_mod(A,B),C) != hAPP(A,A,div_mod(A,D),C) | hAPP(A,A,div_mod(A,E),C) != hAPP(A,A,div_mod(A,F),C) | hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,B),E)),C) = hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,D),F)),C) # label(fact_934_mod__add__cong) # label(axiom). [clausify(309)]. 3.65/3.96 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(1202,a,1187,a)]. 3.65/3.96 1203 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,hAPP(A,A,div_mod(A,B),C)),D)),C) = hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,B),D)),C) # label(fact_930_mod__add__left__eq) # label(axiom). [clausify(386)]. 3.65/3.96 1204 -semiring_div(A) | zero_zero(A) = hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,B),C)),B) # label(fact_899_mod__mult__self1__is__0) # label(axiom). [clausify(406)]. 3.65/3.96 Derived: zero_zero(int) = hAPP(int,int,div_mod(int,hAPP(int,int,times_times(int,A),B)),A). [resolve(1204,a,1187,a)]. 3.65/3.96 1205 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,B),hAPP(A,A,div_mod(A,C),D))),D) = hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,B),C)),D) # label(fact_932_zmod__simps_I2_J) # label(axiom). [clausify(449)]. 3.65/3.96 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(1205,a,1187,a)]. 3.65/3.96 1206 -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(466)]. 3.65/3.96 Derived: div_mod(int,A) = ti(fun(int,int),div_mod(int,A)). [resolve(1206,a,1187,a)]. 3.65/3.96 1207 -semiring_div(A) | zero_zero(A) = hAPP(A,A,div_mod(A,B),one_one(A)) # label(fact_901_mod__by__1) # label(axiom). [clausify(498)]. 3.65/3.96 Derived: zero_zero(int) = hAPP(int,int,div_mod(int,A),one_one(int)). [resolve(1207,a,1187,a)]. 3.65/3.96 1208 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,hAPP(A,A,div_mod(A,B),C)),hAPP(A,A,div_mod(A,D),C))),C) = hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,B),D)),C) # label(fact_937_mod__mult__eq) # label(axiom). [clausify(555)]. 3.65/3.96 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(1208,a,1187,a)]. 3.65/3.96 1209 -semiring_div(A) | zero_zero(A) = hAPP(A,A,div_mod(A,zero_zero(A)),B) # label(fact_944_mod__0) # label(axiom). [clausify(564)]. 3.65/3.96 Derived: zero_zero(int) = hAPP(int,int,div_mod(int,zero_zero(int)),A). [resolve(1209,a,1187,a)]. 3.65/3.96 1210 semiring_div(nat) # label(arity_Nat_Onat___Divides_Osemiring__div) # label(axiom). [assumption]. 3.65/3.96 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(1210,a,1188,a)]. 3.65/3.97 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | zero_zero(nat) = hAPP(nat,nat,div_mod(nat,B),A). [resolve(1210,a,1189,a)]. 3.65/3.97 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | zero_zero(nat) != hAPP(nat,nat,div_mod(nat,B),A). [resolve(1210,a,1190,a)]. 3.65/3.97 Derived: ti(nat,A) = hAPP(nat,nat,div_mod(nat,A),zero_zero(nat)). [resolve(1210,a,1191,a)]. 3.65/3.97 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(1210,a,1192,a)]. 3.65/3.97 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(1210,a,1194,a)]. 3.65/3.97 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(1210,a,1198,a)]. 3.65/3.97 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,div_mod(nat,B),C))). [resolve(1210,a,1199,a)]. 3.65/3.97 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(1210,a,1200,a)]. 3.65/3.97 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(1210,a,1201,a)]. 3.65/3.97 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(1210,a,1202,a)]. 3.65/3.97 Derived: zero_zero(nat) = hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,times_times(nat,A),B)),A). [resolve(1210,a,1204,a)]. 3.65/3.97 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(1210,a,1205,a)]. 3.65/3.97 Derived: div_mod(nat,A) = ti(fun(nat,nat),div_mod(nat,A)). [resolve(1210,a,1206,a)]. 3.65/3.97 Derived: zero_zero(nat) = hAPP(nat,nat,div_mod(nat,A),one_one(nat)). [resolve(1210,a,1207,a)]. 3.65/3.97 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(1210,a,1208,a)]. 3.65/3.97 Derived: zero_zero(nat) = hAPP(nat,nat,div_mod(nat,zero_zero(nat)),A). [resolve(1210,a,1209,a)]. 3.65/3.97 1211 -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,D),C))) # label(fact_926_dvd__mod__iff) # label(axiom). [clausify(581)]. 3.65/3.97 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,div_mod(int,C),B))). [resolve(1211,a,1187,a)]. 3.65/3.97 1212 -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,D),C))) # label(fact_926_dvd__mod__iff) # label(axiom). [clausify(581)]. 3.65/3.97 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,div_mod(int,C),B))). [resolve(1212,a,1187,a)]. 3.65/3.97 1213 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,hAPP(A,A,div_mod(A,B),C)),hAPP(A,A,div_mod(A,D),C))),C) = hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,B),D)),C) # label(fact_931_mod__add__eq) # label(axiom). [clausify(591)]. 3.65/3.97 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(1213,a,1187,a)]. 3.77/4.02 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(1213,a,1210,a)]. 3.77/4.02 1214 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,hAPP(A,A,div_mod(A,B),C)),D)),C) = hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,B),D)),C) # label(fact_936_mod__mult__left__eq) # label(axiom). [clausify(609)]. 3.77/4.02 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(1214,a,1187,a)]. 3.77/4.02 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(1214,a,1210,a)]. 3.77/4.02 1215 -semiring_div(A) | zero_zero(A) = hAPP(A,A,div_mod(A,B),B) # label(fact_942_mod__self) # label(axiom). [clausify(683)]. 3.77/4.02 Derived: zero_zero(int) = hAPP(int,int,div_mod(int,A),A). [resolve(1215,a,1187,a)]. 3.77/4.02 Derived: zero_zero(nat) = hAPP(nat,nat,div_mod(nat,A),A). [resolve(1215,a,1210,a)]. 3.77/4.02 1216 -semiring_div(A) | hAPP(A,A,div_mod(A,B),C) != hAPP(A,A,div_mod(A,D),C) | hAPP(A,A,div_mod(A,E),C) != hAPP(A,A,div_mod(A,F),C) | hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,B),E)),C) = hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,D),F)),C) # label(fact_941_mod__mult__cong) # label(axiom). [clausify(689)]. 3.77/4.02 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(1216,a,1187,a)]. 3.77/4.02 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(1216,a,1210,a)]. 3.77/4.02 1217 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,B),C)),B) = hAPP(A,A,div_mod(A,C),B) # label(fact_928_mod__add__self1) # label(axiom). [clausify(768)]. 3.77/4.02 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(1217,a,1187,a)]. 3.77/4.02 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(1217,a,1210,a)]. 3.77/4.02 1218 -semiring_div(A) | div_mod(A,ti(A,B)) = div_mod(A,B) # label(tsy_c_Divides_Odiv__class_Omod_arg1) # label(axiom). [clausify(807)]. 3.77/4.02 Derived: div_mod(int,ti(int,A)) = div_mod(int,A). [resolve(1218,a,1187,a)]. 3.77/4.02 Derived: div_mod(nat,ti(nat,A)) = div_mod(nat,A). [resolve(1218,a,1210,a)]. 3.77/4.02 1219 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,B),hAPP(A,A,times_times(A,C),D))),D) = hAPP(A,A,div_mod(A,B),D) # label(fact_902_mod__mult__self1) # label(axiom). [clausify(816)]. 3.77/4.02 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(1219,a,1187,a)]. 3.77/4.02 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(1219,a,1210,a)]. 3.77/4.02 1220 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,hAPP(A,A,div_mod(A,B),C)),D)),C) = hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,B),D)),C) # label(fact_940_zmod__simps_I4_J) # label(axiom). [clausify(871)]. 3.77/4.02 1221 -semiring_div(A) | zero_zero(A) = hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,B),C)),C) # label(fact_900_mod__mult__self2__is__0) # label(axiom). [clausify(874)]. 3.77/4.02 Derived: zero_zero(int) = hAPP(int,int,div_mod(int,hAPP(int,int,times_times(int,A),B)),B). [resolve(1221,a,1187,a)]. 3.77/4.02 Derived: zero_zero(nat) = hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,times_times(nat,A),B)),B). [resolve(1221,a,1210,a)]. 3.93/4.20 1222 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | hAPP(A,A,div_mod(A,hAPP(A,A,div_mod(A,D),C)),B) = hAPP(A,A,div_mod(A,D),B) # label(fact_925_mod__mod__cancel) # label(axiom). [clausify(897)]. 3.93/4.20 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(1222,a,1187,a)]. 3.93/4.20 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(1222,a,1210,a)]. 3.93/4.20 1223 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | zero_zero(A) = hAPP(A,A,div_mod(A,C),B) # label(fact_905_dvd__imp__mod__0) # label(axiom). [clausify(923)]. 3.93/4.20 1224 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,B),hAPP(A,A,div_mod(A,C),D))),D) = hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,B),C)),D) # label(fact_929_mod__add__right__eq) # label(axiom). [clausify(972)]. 3.93/4.20 1225 -group_add(A) | ti(fun(A,A),minus_minus(A,B)) = minus_minus(A,B) # label(tsy_c_Groups_Ominus__class_Ominus_0_res) # label(axiom). [clausify(33)]. 3.93/4.20 1226 group_add(int) # label(arity_Int_Oint___Groups_Ogroup__add) # label(axiom). [assumption]. 3.93/4.20 Derived: ti(fun(int,int),minus_minus(int,A)) = minus_minus(int,A). [resolve(1225,a,1226,a)]. 3.93/4.20 1227 -group_add(A) | ti(A,B) = hAPP(A,A,minus_minus(A,B),zero_zero(A)) # label(fact_788_diff__0__right) # label(axiom). [clausify(110)]. 3.93/4.20 Derived: ti(int,A) = hAPP(int,int,minus_minus(int,A),zero_zero(int)). [resolve(1227,a,1226,a)]. 3.93/4.20 1228 -group_add(A) | ti(A,B) = hAPP(A,A,plus_plus(A,hAPP(A,A,minus_minus(A,B),C)),C) # label(fact_796_diff__add__cancel) # label(axiom). [clausify(112)]. 3.93/4.20 Derived: ti(int,A) = hAPP(int,int,plus_plus(int,hAPP(int,int,minus_minus(int,A),B)),B). [resolve(1228,a,1226,a)]. 3.93/4.20 1229 group_add(real) # label(arity_RealDef_Oreal___Groups_Ogroup__add) # label(axiom). [assumption]. 3.93/4.20 Derived: ti(fun(real,real),minus_minus(real,A)) = minus_minus(real,A). [resolve(1229,a,1225,a)]. 3.93/4.20 Derived: ti(real,A) = hAPP(real,real,minus_minus(real,A),zero_zero(real)). [resolve(1229,a,1227,a)]. 3.93/4.20 Derived: ti(real,A) = hAPP(real,real,plus_plus(real,hAPP(real,real,minus_minus(real,A),B)),B). [resolve(1229,a,1228,a)]. 3.93/4.20 1230 -group_add(A) | zero_zero(A) = hAPP(A,A,minus_minus(A,B),B) # label(fact_787_diff__self) # label(axiom). [clausify(552)]. 3.93/4.20 Derived: zero_zero(int) = hAPP(int,int,minus_minus(int,A),A). [resolve(1230,a,1226,a)]. 3.93/4.20 Derived: zero_zero(real) = hAPP(real,real,minus_minus(real,A),A). [resolve(1230,a,1229,a)]. 3.93/4.20 1231 -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(695)]. 3.93/4.20 Derived: minus_minus(int,ti(int,A)) = minus_minus(int,A). [resolve(1231,a,1226,a)]. 3.93/4.20 Derived: minus_minus(real,ti(real,A)) = minus_minus(real,A). [resolve(1231,a,1229,a)]. 3.93/4.20 1232 -group_add(A) | ti(A,B) = hAPP(A,A,minus_minus(A,hAPP(A,A,plus_plus(A,B),C)),C) # label(fact_795_add__diff__cancel) # label(axiom). [clausify(755)]. 3.93/4.20 Derived: ti(int,A) = hAPP(int,int,minus_minus(int,hAPP(int,int,plus_plus(int,A),B)),B). [resolve(1232,a,1226,a)]. 3.93/4.20 Derived: ti(real,A) = hAPP(real,real,minus_minus(real,hAPP(real,real,plus_plus(real,A),B)),B). [resolve(1232,a,1229,a)]. 3.93/4.20 1233 -group_add(A) | ti(A,B) != ti(A,C) | zero_zero(A) = hAPP(A,A,minus_minus(A,C),B) # label(fact_785_right__minus__eq) # label(axiom). [clausify(914)]. 3.93/4.20 Derived: ti(int,A) != ti(int,B) | zero_zero(int) = hAPP(int,int,minus_minus(int,B),A). [resolve(1233,a,1226,a)]. 3.93/4.20 Derived: ti(real,A) != ti(real,B) | zero_zero(real) = hAPP(real,real,minus_minus(real,B),A). [resolve(1233,a,1229,a)]. 3.93/4.20 1234 -group_add(A) | ti(A,B) = ti(A,C) | zero_zero(A) != hAPP(A,A,minus_minus(A,C),B) # label(fact_785_right__minus__eq) # label(axiom). [clausify(914)]. 3.93/4.20 Derived: ti(int,A) = ti(int,B) | zero_zero(int) != hAPP(int,int,minus_minus(int,B),A). [resolve(1234,a,1226,a)]. 3.93/4.20 Derived: ti(real,A) = ti(real,B) | zero_zero(real) != hAPP(real,real,minus_minus(real,B),A). [resolve(1234,a,1229,a)]. 4.23/4.48 1235 -semiring(A) | ti(fun(A,A),times_times(A,B)) = times_times(A,B) # label(tsy_c_Groups_Otimes__class_Otimes_3_res) # label(axiom). [clausify(60)]. 4.23/4.48 1236 semiring(real) # label(arity_RealDef_Oreal___Rings_Osemiring) # label(axiom). [assumption]. 4.23/4.48 1237 semiring(int) # label(arity_Int_Oint___Rings_Osemiring) # label(axiom). [assumption]. 4.23/4.48 1238 -number(A) | -semiring(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,number_number_of(A,B)),C)),hAPP(A,A,times_times(A,number_number_of(A,B)),D)) = hAPP(A,A,times_times(A,number_number_of(A,B)),hAPP(A,A,plus_plus(A,C),D)) # label(fact_139_right__distrib__number__of) # label(axiom). [clausify(432)]. 4.23/4.48 Derived: -number(real) | hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,number_number_of(real,A)),B)),hAPP(real,real,times_times(real,number_number_of(real,A)),C)) = hAPP(real,real,times_times(real,number_number_of(real,A)),hAPP(real,real,plus_plus(real,B),C)). [resolve(1238,b,1236,a)]. 4.23/4.48 1239 -semiring(A) | times_times(A,ti(A,B)) = times_times(A,B) # label(tsy_c_Groups_Otimes__class_Otimes_3_arg1) # label(axiom). [clausify(686)]. 4.23/4.48 1240 -semiring(A) | -number(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),number_number_of(A,C))),hAPP(A,A,times_times(A,D),number_number_of(A,C))) = hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),D)),number_number_of(A,C)) # label(fact_138_left__distrib__number__of) # label(axiom). [clausify(764)]. 4.23/4.48 Derived: -number(real) | hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),number_number_of(real,B))),hAPP(real,real,times_times(real,C),number_number_of(real,B))) = hAPP(real,real,times_times(real,hAPP(real,real,plus_plus(real,A),C)),number_number_of(real,B)). [resolve(1240,a,1236,a)]. 4.23/4.48 Derived: -number(int) | hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),number_number_of(int,B))),hAPP(int,int,times_times(int,C),number_number_of(int,B))) = hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,A),C)),number_number_of(int,B)). [resolve(1240,a,1237,a)]. 4.23/4.48 1241 -semiring(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),C)),D)),E) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),D)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),D)),E)) # label(fact_447_combine__common__factor) # label(axiom). [clausify(947)]. 4.23/4.48 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(1241,a,1236,a)]. 4.23/4.48 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(1241,a,1237,a)]. 4.23/4.48 1242 semiring(nat) # label(arity_Nat_Onat___Rings_Osemiring) # label(axiom). [assumption]. 4.23/4.48 Derived: -number(nat) | hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,number_number_of(nat,A)),B)),hAPP(nat,nat,times_times(nat,number_number_of(nat,A)),C)) = hAPP(nat,nat,times_times(nat,number_number_of(nat,A)),hAPP(nat,nat,plus_plus(nat,B),C)). [resolve(1242,a,1238,b)]. 4.23/4.48 Derived: -number(nat) | hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,A),number_number_of(nat,B))),hAPP(nat,nat,times_times(nat,C),number_number_of(nat,B))) = hAPP(nat,nat,times_times(nat,hAPP(nat,nat,plus_plus(nat,A),C)),number_number_of(nat,B)). [resolve(1242,a,1240,a)]. 4.23/4.48 Derived: hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,hAPP(nat,nat,plus_plus(nat,A),B)),C)),D) = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,A),C)),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,B),C)),D)). [resolve(1242,a,1241,a)]. 4.23/4.48 1243 -ordere453448008miring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),C)),zero_zero(A))) # label(fact_481_mult__nonpos__nonneg) # label(axiom). [clausify(269)]. 4.23/4.56 1244 ordere453448008miring(nat) # label(arity_Nat_Onat___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 4.23/4.56 1245 ordere453448008miring(int) # label(arity_Int_Oint___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 4.23/4.56 1246 ordere453448008miring(real) # label(arity_RealDef_Oreal___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 4.23/4.56 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(1243,a,1244,a)]. 4.23/4.56 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),zero_zero(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,times_times(int,A),B)),zero_zero(int))). [resolve(1243,a,1245,a)]. 4.23/4.56 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),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,A),B)),zero_zero(real))). [resolve(1243,a,1246,a)]. 4.23/4.56 1247 -ordere453448008miring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,C),B)),zero_zero(A))) # label(fact_482_mult__nonneg__nonpos2) # label(axiom). [clausify(370)]. 4.23/4.56 1248 -ordere453448008miring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,times_times(A,B),C))) # label(fact_484_mult__nonneg__nonneg) # label(axiom). [clausify(566)]. 4.23/4.56 1249 -ordere453448008miring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),C)),zero_zero(A))) # label(fact_483_mult__nonneg__nonpos) # label(axiom). [clausify(569)]. 4.23/4.56 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(1249,a,1244,a)]. 4.23/4.56 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),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,times_times(int,A),B)),zero_zero(int))). [resolve(1249,a,1245,a)]. 4.23/4.56 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),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,A),B)),zero_zero(real))). [resolve(1249,a,1246,a)]. 4.23/4.56 1250 -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,C),B)),zero_zero(A))) # label(fact_471_split__mult__neg__le) # label(axiom). [clausify(954)]. 4.23/4.56 1251 -ordere453448008miring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,C),B)),zero_zero(A))) # label(fact_471_split__mult__neg__le) # label(axiom). [clausify(954)]. 4.37/4.68 1252 ring(int) # label(arity_Int_Oint___Rings_Oring) # label(axiom). [assumption]. 4.37/4.68 1253 -number(A) | -ring(A) | hAPP(A,A,minus_minus(A,hAPP(A,A,times_times(A,B),number_number_of(A,C))),hAPP(A,A,times_times(A,D),number_number_of(A,C))) = hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,B),D)),number_number_of(A,C)) # label(fact_377_left__diff__distrib__number__of) # label(axiom). [clausify(35)]. 4.37/4.68 1254 -ring(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),D) != hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,E),C)),F) | ti(A,D) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,E),B)),C)),F) # label(fact_514_eq__add__iff1) # label(axiom). [clausify(99)]. 4.37/4.68 1255 -ring(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),D) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,E),C)),F) | ti(A,D) != hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,E),B)),C)),F) # label(fact_514_eq__add__iff1) # label(axiom). [clausify(99)]. 4.37/4.68 Derived: hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),B)),C) != hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,D),B)),E) | ti(int,C) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,D),A)),B)),E). [resolve(1252,a,1254,a)]. 4.37/4.68 Derived: hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),B)),C) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,D),B)),E) | ti(int,C) != hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,D),A)),B)),E). [resolve(1252,a,1255,a)]. 4.37/4.68 1256 ring(real) # label(arity_RealDef_Oreal___Rings_Oring) # label(axiom). [assumption]. 4.37/4.68 Derived: -number(real) | hAPP(real,real,minus_minus(real,hAPP(real,real,times_times(real,A),number_number_of(real,B))),hAPP(real,real,times_times(real,C),number_number_of(real,B))) = hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,A),C)),number_number_of(real,B)). [resolve(1256,a,1253,b)]. 4.37/4.68 Derived: hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),B)),C) != hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,D),B)),E) | ti(real,C) = hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,D),A)),B)),E). [resolve(1256,a,1254,a)]. 4.37/4.68 Derived: hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),B)),C) = hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,D),B)),E) | ti(real,C) != hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,D),A)),B)),E). [resolve(1256,a,1255,a)]. 4.37/4.68 1257 -ring(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),hAPP(A,A,minus_minus(A,C),D))),hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,B),E)),D)) = hAPP(A,A,minus_minus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,E),D)) # label(fact_516_mult__diff__mult) # label(axiom). [clausify(398)]. 4.37/4.68 Derived: hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),hAPP(int,int,minus_minus(int,B),C))),hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,A),D)),C)) = hAPP(int,int,minus_minus(int,hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,D),C)). [resolve(1257,a,1252,a)]. 4.37/4.68 Derived: hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),hAPP(real,real,minus_minus(real,B),C))),hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,A),D)),C)) = hAPP(real,real,minus_minus(real,hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,D),C)). [resolve(1257,a,1256,a)]. 4.37/4.68 1258 -ring(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),D) != hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,E),C)),F) | ti(A,F) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,B),E)),C)),D) # label(fact_515_eq__add__iff2) # label(axiom). [clausify(425)]. 4.37/4.68 Derived: hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),B)),C) != hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,D),B)),E) | ti(int,E) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,A),D)),B)),C). [resolve(1258,a,1252,a)]. 4.37/4.68 Derived: hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),B)),C) != hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,D),B)),E) | ti(real,E) = hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,A),D)),B)),C). [resolve(1258,a,1256,a)]. 4.51/4.86 1259 -ring(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),D) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,E),C)),F) | ti(A,F) != hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,B),E)),C)),D) # label(fact_515_eq__add__iff2) # label(axiom). [clausify(425)]. 4.51/4.86 Derived: hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),B)),C) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,D),B)),E) | ti(int,E) != hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,A),D)),B)),C). [resolve(1259,a,1252,a)]. 4.51/4.86 Derived: hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),B)),C) = hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,D),B)),E) | ti(real,E) != hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,A),D)),B)),C). [resolve(1259,a,1256,a)]. 4.51/4.86 1260 -ring(A) | -number(A) | hAPP(A,A,minus_minus(A,hAPP(A,A,times_times(A,number_number_of(A,B)),C)),hAPP(A,A,times_times(A,number_number_of(A,B)),D)) = hAPP(A,A,times_times(A,number_number_of(A,B)),hAPP(A,A,minus_minus(A,C),D)) # label(fact_376_right__diff__distrib__number__of) # label(axiom). [clausify(543)]. 4.51/4.86 Derived: -number(real) | hAPP(real,real,minus_minus(real,hAPP(real,real,times_times(real,number_number_of(real,A)),B)),hAPP(real,real,times_times(real,number_number_of(real,A)),C)) = hAPP(real,real,times_times(real,number_number_of(real,A)),hAPP(real,real,minus_minus(real,B),C)). [resolve(1260,a,1256,a)]. 4.51/4.86 1261 comm_semiring_1(int) # label(arity_Int_Oint___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 4.51/4.86 1262 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(A,A,times_times(A,B),C)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),D)) # label(fact_451_dvd__mult__right) # label(axiom). [clausify(37)]. 4.51/4.86 1263 -comm_semiring_1(A) | zero_zero(A) = hAPP(A,A,times_times(A,zero_zero(A)),B) # label(fact_198_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom). [clausify(70)]. 4.51/4.86 1264 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) # label(fact_427_dvd__trans) # label(axiom). [clausify(72)]. 4.51/4.86 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(1261,a,1262,a)]. 4.51/4.86 Derived: zero_zero(int) = hAPP(int,int,times_times(int,zero_zero(int)),A). [resolve(1261,a,1263,a)]. 4.51/4.86 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(1261,a,1264,a)]. 4.51/4.86 1265 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,B),B) = hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,one_one(A)),one_one(A))),B) # label(fact_142_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom). [clausify(157)]. 4.51/4.86 Derived: hAPP(int,int,plus_plus(int,A),A) = hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,one_one(int)),one_one(int))),A). [resolve(1265,a,1261,a)]. 4.51/4.86 1266 -comm_semiring_1(A) | zero_zero(A) = hAPP(A,A,times_times(A,B),zero_zero(A)) # label(fact_199_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom). [clausify(171)]. 4.51/4.86 Derived: zero_zero(int) = hAPP(int,int,times_times(int,A),zero_zero(int)). [resolve(1266,a,1261,a)]. 4.51/4.86 1267 -comm_semiring_1(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),zero_zero(A))) # label(fact_423_dvd__0__right) # label(axiom). [clausify(178)]. 4.51/4.86 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),zero_zero(int))). [resolve(1267,a,1261,a)]. 4.51/4.86 1268 -comm_semiring_1(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),one_one(A)),B)) # label(fact_460_one__dvd) # label(axiom). [clausify(208)]. 4.61/4.88 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),one_one(int)),A)). [resolve(1268,a,1261,a)]. 4.61/4.88 1269 -comm_semiring_1(A) | hAPP(A,A,times_times(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),E)) = hAPP(A,A,times_times(A,B),hAPP(A,A,times_times(A,C),hAPP(A,A,times_times(A,D),E))) # label(fact_68_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom). [clausify(254)]. 4.61/4.88 1270 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,plus_plus(A,B),C)),D) = hAPP(A,A,plus_plus(A,B),hAPP(A,A,plus_plus(A,C),D)) # label(fact_78_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom). [clausify(260)]. 4.61/4.88 1271 -comm_semiring_1(A) | hAPP(A,A,times_times(A,B),hAPP(A,A,times_times(A,C),D)) = hAPP(A,A,times_times(A,C),hAPP(A,A,times_times(A,B),D)) # label(fact_72_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom). [clausify(274)]. 4.61/4.88 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(1271,a,1261,a)]. 4.61/4.88 1272 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,B),hAPP(A,A,times_times(A,C),B)) = hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,C),one_one(A))),B) # label(fact_141_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom). [clausify(290)]. 4.61/4.88 Derived: hAPP(int,int,plus_plus(int,A),hAPP(int,int,times_times(int,B),A)) = hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,B),one_one(int))),A). [resolve(1272,a,1261,a)]. 4.61/4.88 1273 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,times_times(A,C),D))) # label(fact_456_dvd__mult2) # label(axiom). [clausify(306)]. 4.61/4.88 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(1273,a,1261,a)]. 4.61/4.88 1274 -comm_semiring_1(A) | hAPP(A,A,times_times(A,B),C) = hAPP(A,A,times_times(A,C),B) # label(fact_73_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom). [clausify(349)]. 4.61/4.88 Derived: hAPP(int,int,times_times(int,A),B) = hAPP(int,int,times_times(int,B),A). [resolve(1274,a,1261,a)]. 4.61/4.88 1275 -comm_semiring_1(A) | hAPP(A,A,times_times(A,hAPP(nat,A,power_power(A,B),C)),hAPP(nat,A,power_power(A,B),C)) = hAPP(nat,A,power_power(A,B),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),C)) # label(fact_15_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom). [clausify(382)]. 4.61/4.88 Derived: hAPP(int,int,times_times(int,hAPP(nat,int,power_power(int,A),B)),hAPP(nat,int,power_power(int,A),B)) = hAPP(nat,int,power_power(int,A),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),B)). [resolve(1275,a,1261,a)]. 4.61/4.88 1276 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,plus_plus(A,B),C)),hAPP(A,A,plus_plus(A,D),E)) = hAPP(A,A,plus_plus(A,hAPP(A,A,plus_plus(A,B),D)),hAPP(A,A,plus_plus(A,C),E)) # label(fact_74_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom). [clausify(384)]. 4.61/4.88 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(1276,a,1261,a)]. 4.61/4.88 1277 -comm_semiring_1(A) | ti(A,B) = hAPP(A,A,times_times(A,one_one(A)),B) # label(fact_114_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom). [clausify(385)]. 4.61/4.88 1278 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,B),C) = hAPP(A,A,plus_plus(A,C),B) # label(fact_81_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom). [clausify(396)]. 4.61/4.88 1279 -comm_semiring_1(A) | hAPP(A,A,times_times(A,hAPP(nat,A,power_power(A,B),C)),hAPP(nat,A,power_power(A,D),C)) = hAPP(nat,A,power_power(A,hAPP(A,A,times_times(A,B),D)),C) # label(fact_115_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom). [clausify(412)]. 4.64/4.90 Derived: hAPP(int,int,times_times(int,hAPP(nat,int,power_power(int,A),B)),hAPP(nat,int,power_power(int,C),B)) = hAPP(nat,int,power_power(int,hAPP(int,int,times_times(int,A),C)),B). [resolve(1279,a,1261,a)]. 4.64/4.90 1280 -comm_semiring_1(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),B)) # label(fact_426_dvd__refl) # label(axiom). [clausify(483)]. 4.64/4.90 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),A)). [resolve(1280,a,1261,a)]. 4.64/4.90 1281 -comm_semiring_1(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,times_times(A,B),C))) # label(fact_458_dvd__triv__left) # label(axiom). [clausify(486)]. 4.64/4.90 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,times_times(int,A),B))). [resolve(1281,a,1261,a)]. 4.64/4.90 1282 -comm_semiring_1(A) | ti(A,B) = hAPP(A,A,times_times(A,B),one_one(A)) # label(fact_113_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom). [clausify(501)]. 4.64/4.90 1283 -comm_semiring_1(A) | hAPP(A,A,times_times(A,hAPP(A,A,times_times(A,B),C)),D) = hAPP(A,A,times_times(A,B),hAPP(A,A,times_times(A,C),D)) # label(fact_70_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom). [clausify(517)]. 4.64/4.90 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(1283,a,1261,a)]. 4.64/4.90 1284 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,times_times(A,D),C))) # label(fact_455_dvd__mult) # label(axiom). [clausify(524)]. 4.64/4.90 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(1284,a,1261,a)]. 4.64/4.90 1285 -comm_semiring_1(A) | ti(A,B) = hAPP(nat,A,power_power(A,B),one_one(nat)) # label(fact_27_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom). [clausify(530)]. 4.64/4.90 1286 -comm_semiring_1(A) | hAPP(A,A,times_times(A,hAPP(A,A,times_times(A,B),C)),D) = hAPP(A,A,times_times(A,hAPP(A,A,times_times(A,B),D)),C) # label(fact_69_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom). [clausify(532)]. 4.64/4.90 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(1286,a,1261,a)]. 4.64/4.90 1287 -comm_semiring_1(A) | hAPP(nat,A,power_power(A,hAPP(nat,A,power_power(A,B),C)),D) = hAPP(nat,A,power_power(A,B),hAPP(nat,nat,times_times(nat,C),D)) # label(fact_26_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom). [clausify(536)]. 4.64/4.90 1288 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(nat,A,power_power(A,B),D)),hAPP(nat,A,power_power(A,C),E))) # label(fact_187_dvd__power__le) # label(axiom). [clausify(571)]. 4.64/4.90 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(1288,a,1261,a)]. 4.64/4.90 1289 -comm_semiring_1(A) | hAPP(A,A,times_times(A,hAPP(nat,A,power_power(A,B),C)),hAPP(nat,A,power_power(A,B),D)) = hAPP(nat,A,power_power(A,B),hAPP(nat,nat,plus_plus(nat,C),D)) # label(fact_33_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom). [clausify(642)]. 4.64/4.90 1290 -comm_semiring_1(A) | ti(A,B) = hAPP(A,A,plus_plus(A,B),zero_zero(A)) # label(fact_201_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom). [clausify(654)]. 4.64/4.90 Derived: ti(int,A) = hAPP(int,int,plus_plus(int,A),zero_zero(int)). [resolve(1290,a,1261,a)]. 4.64/4.90 1291 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,plus_plus(A,B),C)),D) = hAPP(A,A,plus_plus(A,B),hAPP(A,A,plus_plus(A,C),D)) # label(fact_79_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom). [clausify(657)]. 4.64/4.91 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(1291,a,1261,a)]. 4.64/4.91 1292 -comm_semiring_1(A) | ti(A,B) = hAPP(A,A,plus_plus(A,zero_zero(A)),B) # label(fact_200_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom). [clausify(669)]. 4.64/4.91 Derived: ti(int,A) = hAPP(int,int,plus_plus(int,zero_zero(int)),A). [resolve(1292,a,1261,a)]. 4.64/4.91 1293 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(nat,A,power_power(A,B),D)),hAPP(nat,A,power_power(A,C),D))) # label(fact_193_dvd__power__same) # label(axiom). [clausify(688)]. 4.64/4.91 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(1293,a,1261,a)]. 4.64/4.91 1294 comm_semiring_1(real) # label(arity_RealDef_Oreal___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 4.64/4.91 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(1294,a,1262,a)]. 4.64/4.91 Derived: zero_zero(real) = hAPP(real,real,times_times(real,zero_zero(real)),A). [resolve(1294,a,1263,a)]. 4.64/4.91 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(1294,a,1264,a)]. 4.64/4.91 Derived: hAPP(real,real,plus_plus(real,A),A) = hAPP(real,real,times_times(real,hAPP(real,real,plus_plus(real,one_one(real)),one_one(real))),A). [resolve(1294,a,1265,a)]. 4.64/4.91 Derived: zero_zero(real) = hAPP(real,real,times_times(real,A),zero_zero(real)). [resolve(1294,a,1266,a)]. 4.64/4.91 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),zero_zero(real))). [resolve(1294,a,1267,a)]. 4.64/4.91 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),one_one(real)),A)). [resolve(1294,a,1268,a)]. 4.64/4.91 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,A),hAPP(real,real,times_times(real,B),hAPP(real,real,times_times(real,C),D))). [resolve(1294,a,1269,a)]. 4.64/4.91 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(1294,a,1270,a)]. 4.64/4.91 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(1294,a,1271,a)]. 4.64/4.91 Derived: hAPP(real,real,plus_plus(real,A),hAPP(real,real,times_times(real,B),A)) = hAPP(real,real,times_times(real,hAPP(real,real,plus_plus(real,B),one_one(real))),A). [resolve(1294,a,1272,a)]. 4.64/4.91 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(1294,a,1273,a)]. 4.64/4.91 Derived: hAPP(real,real,times_times(real,A),B) = hAPP(real,real,times_times(real,B),A). [resolve(1294,a,1274,a)]. 4.64/4.91 Derived: hAPP(real,real,times_times(real,hAPP(nat,real,power_power(real,A),B)),hAPP(nat,real,power_power(real,A),B)) = hAPP(nat,real,power_power(real,A),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),B)). [resolve(1294,a,1275,a)]. 4.64/4.91 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(1294,a,1276,a)]. 4.64/4.91 Derived: hAPP(real,real,plus_plus(real,A),B) = hAPP(real,real,plus_plus(real,B),A). [resolve(1294,a,1278,a)]. 4.64/4.91 Derived: hAPP(real,real,times_times(real,hAPP(nat,real,power_power(real,A),B)),hAPP(nat,real,power_power(real,C),B)) = hAPP(nat,real,power_power(real,hAPP(real,real,times_times(real,A),C)),B). [resolve(1294,a,1279,a)]. 4.67/4.94 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),A)). [resolve(1294,a,1280,a)]. 4.67/4.94 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(real,real,times_times(real,A),B))). [resolve(1294,a,1281,a)]. 4.67/4.94 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(1294,a,1283,a)]. 4.67/4.94 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(1294,a,1284,a)]. 4.67/4.94 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(1294,a,1286,a)]. 4.67/4.94 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(1294,a,1288,a)]. 4.67/4.94 Derived: ti(real,A) = hAPP(real,real,plus_plus(real,A),zero_zero(real)). [resolve(1294,a,1290,a)]. 4.67/4.94 Derived: ti(real,A) = hAPP(real,real,plus_plus(real,zero_zero(real)),A). [resolve(1294,a,1292,a)]. 4.67/4.94 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(1294,a,1293,a)]. 4.67/4.94 1295 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(A,A,times_times(A,B),C)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) # label(fact_452_dvd__mult__left) # label(axiom). [clausify(690)]. 4.67/4.94 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(1295,a,1261,a)]. 4.67/4.94 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(1295,a,1294,a)]. 4.67/4.94 1296 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,plus_plus(A,C),D))) # label(fact_459_dvd__add) # label(axiom). [clausify(709)]. 4.67/4.94 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(1296,a,1261,a)]. 4.67/4.94 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(1296,a,1294,a)]. 4.67/4.94 1297 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),zero_zero(A)),B)) | zero_zero(A) = ti(A,B) # label(fact_449_dvd__0__left) # label(axiom). [clausify(717)]. 4.67/4.94 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),zero_zero(int)),A)) | zero_zero(int) = ti(int,A). [resolve(1297,a,1261,a)]. 4.67/4.94 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),zero_zero(real)),A)) | zero_zero(real) = ti(real,A). [resolve(1297,a,1294,a)]. 4.67/4.94 1298 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),E))) # label(fact_454_mult__dvd__mono) # label(axiom). [clausify(830)]. 4.67/4.94 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(1298,a,1261,a)]. 4.67/4.95 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(1298,a,1294,a)]. 4.67/4.95 1299 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,plus_plus(A,B),C)),D) = hAPP(A,A,plus_plus(A,hAPP(A,A,plus_plus(A,B),D)),C) # label(fact_77_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom). [clausify(833)]. 4.67/4.95 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(1299,a,1261,a)]. 4.67/4.95 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(1299,a,1294,a)]. 4.67/4.95 1300 -comm_semiring_1(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,times_times(A,C),B))) # label(fact_457_dvd__triv__right) # label(axiom). [clausify(878)]. 4.67/4.95 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,times_times(int,B),A))). [resolve(1300,a,1261,a)]. 4.67/4.95 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(real,real,times_times(real,B),A))). [resolve(1300,a,1294,a)]. 4.67/4.95 1301 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D)) = hAPP(A,A,times_times(A,B),hAPP(A,A,plus_plus(A,C),D)) # label(fact_112_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom). [clausify(879)]. 4.67/4.95 Derived: hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,A),C)) = hAPP(real,real,times_times(real,A),hAPP(real,real,plus_plus(real,B),C)). [resolve(1301,a,1294,a)]. 4.67/4.95 1302 -comm_semiring_1(A) | hAPP(A,A,times_times(A,hAPP(A,A,times_times(A,B),C)),D) = hAPP(A,A,times_times(A,B),hAPP(A,A,times_times(A,C),D)) # label(fact_71_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom). [clausify(885)]. 4.67/4.95 1303 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),C) = hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),one_one(A))),C) # label(fact_140_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom). [clausify(898)]. 4.67/4.95 Derived: hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),B)),B) = hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,A),one_one(int))),B). [resolve(1303,a,1261,a)]. 4.67/4.95 Derived: hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),B)),B) = hAPP(real,real,times_times(real,hAPP(real,real,plus_plus(real,A),one_one(real))),B). [resolve(1303,a,1294,a)]. 4.67/4.95 1304 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C)) = hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),D)),C) # label(fact_109_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom). [clausify(910)]. 4.67/4.95 Derived: hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,C),B)) = hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,A),C)),B). [resolve(1304,a,1261,a)]. 4.67/4.95 Derived: hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,C),B)) = hAPP(real,real,times_times(real,hAPP(real,real,plus_plus(real,A),C)),B). [resolve(1304,a,1294,a)]. 4.67/4.95 1305 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,B),hAPP(A,A,plus_plus(A,C),D)) = hAPP(A,A,plus_plus(A,C),hAPP(A,A,plus_plus(A,B),D)) # label(fact_80_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom). [clausify(924)]. 4.67/4.95 Derived: hAPP(int,int,plus_plus(int,A),hAPP(int,int,plus_plus(int,B),C)) = hAPP(int,int,plus_plus(int,B),hAPP(int,int,plus_plus(int,A),C)). [resolve(1305,a,1261,a)]. 4.67/4.95 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(1305,a,1294,a)]. 4.67/4.97 1306 -comm_semiring_1(A) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(nat,A,power_power(A,D),B)),hAPP(nat,A,power_power(A,D),C))) # label(fact_186_le__imp__power__dvd) # label(axiom). [clausify(948)]. 4.67/4.97 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(1306,a,1261,a)]. 4.67/4.97 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(1306,a,1294,a)]. 4.67/4.97 1307 -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_322_dvd__power) # label(axiom). [clausify(961)]. 4.67/4.97 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(1307,a,1261,a)]. 4.67/4.97 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(1307,a,1294,a)]. 4.67/4.97 1308 -comm_semiring_1(A) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),hAPP(nat,A,power_power(A,C),B))) # label(fact_322_dvd__power) # label(axiom). [clausify(961)]. 4.67/4.97 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(1308,a,1261,a)]. 4.67/4.97 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(1308,a,1294,a)]. 4.67/4.97 1309 -comm_semiring_1(A) | hAPP(A,A,times_times(A,B),B) = hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))) # label(fact_13_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom). [clausify(963)]. 4.67/4.97 1310 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C)) = hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),D)),C) # label(fact_110_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom). [clausify(991)]. 4.67/4.97 1311 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(nat,A,power_power(A,B),C)),D)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),E),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(nat,A,power_power(A,B),E)),D)) # label(fact_188_power__le__dvd) # label(axiom). [clausify(1002)]. 4.67/4.97 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(1311,a,1261,a)]. 4.67/4.97 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(1311,a,1294,a)]. 4.67/4.97 1312 -comm_semiring_1(A) | hAPP(A,A,times_times(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),E)) = hAPP(A,A,times_times(A,D),hAPP(A,A,times_times(A,hAPP(A,A,times_times(A,B),C)),E)) # label(fact_67_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom). [clausify(1007)]. 4.67/4.97 1313 comm_semiring_1(nat) # label(arity_Nat_Onat___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 4.67/4.97 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(1313,a,1262,a)]. 4.67/4.98 Derived: zero_zero(nat) = hAPP(nat,nat,times_times(nat,zero_zero(nat)),A). [resolve(1313,a,1263,a)]. 4.67/4.98 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(1313,a,1264,a)]. 4.67/4.98 Derived: hAPP(nat,nat,plus_plus(nat,A),A) = hAPP(nat,nat,times_times(nat,hAPP(nat,nat,plus_plus(nat,one_one(nat)),one_one(nat))),A). [resolve(1313,a,1265,a)]. 4.67/4.98 Derived: zero_zero(nat) = hAPP(nat,nat,times_times(nat,A),zero_zero(nat)). [resolve(1313,a,1266,a)]. 4.67/4.98 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),zero_zero(nat))). [resolve(1313,a,1267,a)]. 4.67/4.98 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),one_one(nat)),A)). [resolve(1313,a,1268,a)]. 4.67/4.98 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,A),hAPP(nat,nat,times_times(nat,B),hAPP(nat,nat,times_times(nat,C),D))). [resolve(1313,a,1269,a)]. 4.67/4.98 Derived: hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,plus_plus(nat,A),B)),C) = hAPP(nat,nat,plus_plus(nat,A),hAPP(nat,nat,plus_plus(nat,B),C)). [resolve(1313,a,1270,a)]. 4.67/4.98 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(1313,a,1271,a)]. 4.67/4.98 Derived: hAPP(nat,nat,plus_plus(nat,A),hAPP(nat,nat,times_times(nat,B),A)) = hAPP(nat,nat,times_times(nat,hAPP(nat,nat,plus_plus(nat,B),one_one(nat))),A). [resolve(1313,a,1272,a)]. 4.67/4.98 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(1313,a,1273,a)]. 4.67/4.98 Derived: hAPP(nat,nat,times_times(nat,A),B) = hAPP(nat,nat,times_times(nat,B),A). [resolve(1313,a,1274,a)]. 4.67/4.98 Derived: hAPP(nat,nat,times_times(nat,hAPP(nat,nat,power_power(nat,A),B)),hAPP(nat,nat,power_power(nat,A),B)) = hAPP(nat,nat,power_power(nat,A),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),B)). [resolve(1313,a,1275,a)]. 4.67/4.98 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(1313,a,1276,a)]. 4.67/4.98 Derived: hAPP(nat,nat,plus_plus(nat,A),B) = hAPP(nat,nat,plus_plus(nat,B),A). [resolve(1313,a,1278,a)]. 4.67/4.98 Derived: hAPP(nat,nat,times_times(nat,hAPP(nat,nat,power_power(nat,A),B)),hAPP(nat,nat,power_power(nat,C),B)) = hAPP(nat,nat,power_power(nat,hAPP(nat,nat,times_times(nat,A),C)),B). [resolve(1313,a,1279,a)]. 4.67/4.98 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),A)). [resolve(1313,a,1280,a)]. 4.67/4.98 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,times_times(nat,A),B))). [resolve(1313,a,1281,a)]. 4.67/4.98 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(1313,a,1283,a)]. 4.67/4.98 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(1313,a,1284,a)]. 4.67/4.98 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(1313,a,1286,a)]. 4.67/4.98 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(1313,a,1288,a)]. 4.67/4.98 Derived: ti(nat,A) = hAPP(nat,nat,plus_plus(nat,A),zero_zero(nat)). [resolve(1313,a,1290,a)]. 4.67/4.98 Derived: ti(nat,A) = hAPP(nat,nat,plus_plus(nat,zero_zero(nat)),A). [resolve(1313,a,1292,a)]. 4.67/4.98 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,A),C)),hAPP(nat,nat,power_power(nat,B),C))). [resolve(1313,a,1293,a)]. 4.67/4.98 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(1313,a,1295,a)]. 4.67/4.98 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(1313,a,1296,a)]. 4.67/4.98 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A)) | zero_zero(nat) = ti(nat,A). [resolve(1313,a,1297,a)]. 4.67/4.98 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(1313,a,1298,a)]. 4.67/4.98 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(1313,a,1299,a)]. 4.67/4.98 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,times_times(nat,B),A))). [resolve(1313,a,1300,a)]. 4.67/4.98 Derived: hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,A),B)),hAPP(nat,nat,times_times(nat,A),C)) = hAPP(nat,nat,times_times(nat,A),hAPP(nat,nat,plus_plus(nat,B),C)). [resolve(1313,a,1301,a)]. 4.67/4.98 Derived: hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,A),B)),B) = hAPP(nat,nat,times_times(nat,hAPP(nat,nat,plus_plus(nat,A),one_one(nat))),B). [resolve(1313,a,1303,a)]. 4.67/4.98 Derived: hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,A),B)),hAPP(nat,nat,times_times(nat,C),B)) = hAPP(nat,nat,times_times(nat,hAPP(nat,nat,plus_plus(nat,A),C)),B). [resolve(1313,a,1304,a)]. 4.67/4.98 Derived: hAPP(nat,nat,plus_plus(nat,A),hAPP(nat,nat,plus_plus(nat,B),C)) = hAPP(nat,nat,plus_plus(nat,B),hAPP(nat,nat,plus_plus(nat,A),C)). [resolve(1313,a,1305,a)]. 4.67/4.98 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(1313,a,1306,a)]. 4.67/4.98 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(1313,a,1307,a)]. 4.67/4.98 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(1313,a,1308,a)]. 4.67/4.98 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(1313,a,1311,a)]. 4.67/4.98 1314 -comm_semiring_1(A) | hAPP(A,A,times_times(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),E)) = hAPP(A,A,times_times(A,hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),E)) # label(fact_66_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom). [clausify(1016)]. 4.67/4.98 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(1314,a,1261,a)]. 4.67/4.98 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(1314,a,1294,a)]. 4.67/4.98 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(1314,a,1313,a)]. 4.82/5.15 1315 -comm_semiring_1(A) | one_one(A) = hAPP(nat,A,power_power(A,B),zero_zero(nat)) # label(fact_313_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom). [clausify(1026)]. 4.82/5.15 Derived: one_one(int) = hAPP(nat,int,power_power(int,A),zero_zero(nat)). [resolve(1315,a,1261,a)]. 4.82/5.15 Derived: one_one(real) = hAPP(nat,real,power_power(real,A),zero_zero(nat)). [resolve(1315,a,1294,a)]. 4.82/5.15 Derived: one_one(nat) = hAPP(nat,nat,power_power(nat,A),zero_zero(nat)). [resolve(1315,a,1313,a)]. 4.82/5.15 1316 ordered_semiring(real) # label(arity_RealDef_Oreal___Rings_Oordered__semiring) # label(axiom). [assumption]. 4.82/5.15 1317 -ordered_semiring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),E))) # label(fact_474_mult__mono_H) # label(axiom). [clausify(38)]. 4.82/5.15 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(1316,a,1317,a)]. 4.82/5.15 1318 ordered_semiring(nat) # label(arity_Nat_Onat___Rings_Oordered__semiring) # label(axiom). [assumption]. 4.82/5.15 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),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(1318,a,1317,a)]. 4.82/5.15 1319 ordered_semiring(int) # label(arity_Int_Oint___Rings_Oordered__semiring) # label(axiom). [assumption]. 4.82/5.15 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(1319,a,1317,a)]. 4.82/5.15 1320 -ordered_semiring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,D),B)),hAPP(A,A,times_times(A,D),C))) # label(fact_478_mult__left__mono) # label(axiom). [clausify(798)]. 4.82/5.15 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(1320,a,1316,a)]. 4.82/5.15 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(1320,a,1319,a)]. 4.82/5.15 1321 -ordered_semiring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),D))) # label(fact_479_mult__right__mono) # label(axiom). [clausify(851)]. 5.20/5.48 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(1321,a,1316,a)]. 5.20/5.48 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(1321,a,1318,a)]. 5.20/5.48 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(1321,a,1319,a)]. 5.20/5.48 1322 -ordered_semiring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),E))) # label(fact_473_mult__mono) # label(axiom). [clausify(999)]. 5.20/5.48 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(1322,a,1316,a)]. 5.20/5.48 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(1322,a,1318,a)]. 5.20/5.48 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(1322,a,1319,a)]. 5.20/5.48 1323 -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(200)]. 5.20/5.48 1324 semiring_1(nat) # label(arity_Nat_Onat___Rings_Osemiring__1) # label(axiom). [assumption]. 5.20/5.48 1325 semiring_1(int) # label(arity_Int_Oint___Rings_Osemiring__1) # label(axiom). [assumption]. 5.20/5.48 1326 semiring_1(real) # label(arity_RealDef_Oreal___Rings_Osemiring__1) # label(axiom). [assumption]. 5.20/5.48 1327 -semiring_1(A) | zero_zero(A) = hAPP(nat,A,power_power(A,zero_zero(A)),number_number_of(nat,bit0(bit1(pls)))) # label(fact_251_zero__power2) # label(axiom). [clausify(796)]. 5.20/5.48 Derived: zero_zero(nat) = hAPP(nat,nat,power_power(nat,zero_zero(nat)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1327,a,1324,a)]. 5.20/5.48 Derived: zero_zero(int) = hAPP(nat,int,power_power(int,zero_zero(int)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1327,a,1325,a)]. 5.20/5.48 Derived: zero_zero(real) = hAPP(nat,real,power_power(real,zero_zero(real)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1327,a,1326,a)]. 5.20/5.48 1328 dvd(nat) # label(arity_Nat_Onat___Rings_Odvd) # label(axiom). [assumption]. 5.27/5.53 1329 -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(47)]. 5.27/5.53 1330 -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_453_dvdI) # label(axiom). [clausify(48)]. 5.27/5.53 1331 -dvd(A) | times_times(A,ti(A,B)) = times_times(A,B) # label(tsy_c_Groups_Otimes__class_Otimes_2_arg1) # label(axiom). [clausify(190)]. 5.27/5.53 1332 -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(247)]. 5.27/5.53 1333 -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_I3_J) # label(axiom). [clausify(279)]. 5.27/5.53 1334 -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_I3_J) # label(axiom). [clausify(279)]. 5.27/5.53 1335 -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,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_I4_J) # label(axiom). [clausify(348)]. 5.27/5.53 1336 -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,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_I4_J) # label(axiom). [clausify(348)]. 5.27/5.53 Derived: -linorder(nat) | -plus(nat) | ti(fun(nat,nat),plus_plus(nat,A)) = plus_plus(nat,A). [resolve(1328,a,1329,a)]. 5.27/5.53 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(1328,a,1330,a)]. 5.27/5.53 Derived: dvd_dvd(nat) = ti(fun(nat,fun(nat,bool)),dvd_dvd(nat)). [resolve(1328,a,1332,a)]. 5.27/5.53 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,hAPP(nat,nat,minus_minus(nat,C),hAPP(nat,nat,times_times(nat,D),B))),E))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,plus_plus(nat,C),E))). [resolve(1328,a,1333,a)]. 5.27/5.53 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,hAPP(nat,nat,minus_minus(nat,C),hAPP(nat,nat,times_times(nat,D),B))),E))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,plus_plus(nat,C),E))). [resolve(1328,a,1334,a)]. 5.27/5.53 1337 -plus(A) | -dvd(A) | -linorder(A) | plus_plus(A,ti(A,B)) = plus_plus(A,B) # label(tsy_c_Groups_Oplus__class_Oplus_0_arg1) # label(axiom). [clausify(391)]. 5.27/5.53 Derived: -plus(nat) | -linorder(nat) | plus_plus(nat,ti(nat,A)) = plus_plus(nat,A). [resolve(1337,b,1328,a)]. 5.27/5.53 1338 -dvd(A) | ti(fun(A,A),times_times(A,B)) = times_times(A,B) # label(tsy_c_Groups_Otimes__class_Otimes_2_res) # label(axiom). [clausify(834)]. 5.27/5.53 1339 dvd(int) # label(arity_Int_Oint___Rings_Odvd) # label(axiom). [assumption]. 5.27/5.53 Derived: -linorder(int) | -plus(int) | ti(fun(int,int),plus_plus(int,A)) = plus_plus(int,A). [resolve(1339,a,1329,a)]. 5.27/5.53 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(1339,a,1330,a)]. 5.27/5.53 Derived: dvd_dvd(int) = ti(fun(int,fun(int,bool)),dvd_dvd(int)). [resolve(1339,a,1332,a)]. 5.27/5.53 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,hAPP(int,int,minus_minus(int,C),hAPP(int,int,times_times(int,D),B))),E))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,plus_plus(int,C),E))). [resolve(1339,a,1333,a)]. 5.27/5.56 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,hAPP(int,int,minus_minus(int,C),hAPP(int,int,times_times(int,D),B))),E))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,plus_plus(int,C),E))). [resolve(1339,a,1334,a)]. 5.27/5.56 Derived: -plus(int) | -linorder(int) | plus_plus(int,ti(int,A)) = plus_plus(int,A). [resolve(1339,a,1337,b)]. 5.27/5.56 1340 dvd(real) # label(arity_RealDef_Oreal___Rings_Odvd) # label(axiom). [assumption]. 5.27/5.56 Derived: -linorder(real) | -plus(real) | ti(fun(real,real),plus_plus(real,A)) = plus_plus(real,A). [resolve(1340,a,1329,a)]. 5.27/5.56 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(1340,a,1330,a)]. 5.27/5.56 Derived: dvd_dvd(real) = ti(fun(real,fun(real,bool)),dvd_dvd(real)). [resolve(1340,a,1332,a)]. 5.27/5.56 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,hAPP(real,real,minus_minus(real,C),hAPP(real,real,times_times(real,D),B))),E))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(real,real,plus_plus(real,C),E))). [resolve(1340,a,1333,a)]. 5.27/5.56 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,hAPP(real,real,minus_minus(real,C),hAPP(real,real,times_times(real,D),B))),E))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(real,real,plus_plus(real,C),E))). [resolve(1340,a,1334,a)]. 5.27/5.56 Derived: -plus(real) | -linorder(real) | plus_plus(real,ti(real,A)) = plus_plus(real,A). [resolve(1340,a,1337,b)]. 5.27/5.56 1341 -dvd(A) | -semiring_0(A) | -hBOOL(hAPP(A,bool,B,hAPP(A,A,times_times(A,C),D))) | hBOOL(hAPP(A,bool,B,f33(A,B,C))) # label(fact_744_unity__coeff__ex) # label(axiom). [clausify(997)]. 5.27/5.56 Derived: -semiring_0(nat) | -hBOOL(hAPP(nat,bool,A,hAPP(nat,nat,times_times(nat,B),C))) | hBOOL(hAPP(nat,bool,A,f33(nat,A,B))). [resolve(1341,a,1328,a)]. 5.27/5.56 Derived: -semiring_0(int) | -hBOOL(hAPP(int,bool,A,hAPP(int,int,times_times(int,B),C))) | hBOOL(hAPP(int,bool,A,f33(int,A,B))). [resolve(1341,a,1339,a)]. 5.27/5.56 Derived: -semiring_0(real) | -hBOOL(hAPP(real,bool,A,hAPP(real,real,times_times(real,B),C))) | hBOOL(hAPP(real,bool,A,f33(real,A,B))). [resolve(1341,a,1340,a)]. 5.27/5.56 1342 -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,f33(A,B,C)),zero_zero(A)))) # label(fact_744_unity__coeff__ex) # label(axiom). [clausify(997)]. 5.27/5.56 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,f33(nat,A,B)),zero_zero(nat)))). [resolve(1342,a,1328,a)]. 5.27/5.56 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,f33(int,A,B)),zero_zero(int)))). [resolve(1342,a,1339,a)]. 5.27/5.56 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,f33(real,A,B)),zero_zero(real)))). [resolve(1342,a,1340,a)]. 5.27/5.56 1343 -dvd(A) | -semiring_0(A) | hBOOL(hAPP(A,bool,B,hAPP(A,A,times_times(A,C),f34(A,B,C)))) | -hBOOL(hAPP(A,bool,B,D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),hAPP(A,A,plus_plus(A,D),zero_zero(A)))) # label(fact_744_unity__coeff__ex) # label(axiom). [clausify(997)]. 5.35/5.69 Derived: -semiring_0(nat) | hBOOL(hAPP(nat,bool,A,hAPP(nat,nat,times_times(nat,B),f34(nat,A,B)))) | -hBOOL(hAPP(nat,bool,A,C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),hAPP(nat,nat,plus_plus(nat,C),zero_zero(nat)))). [resolve(1343,a,1328,a)]. 5.35/5.69 Derived: -semiring_0(int) | hBOOL(hAPP(int,bool,A,hAPP(int,int,times_times(int,B),f34(int,A,B)))) | -hBOOL(hAPP(int,bool,A,C)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),hAPP(int,int,plus_plus(int,C),zero_zero(int)))). [resolve(1343,a,1339,a)]. 5.35/5.69 Derived: -semiring_0(real) | hBOOL(hAPP(real,bool,A,hAPP(real,real,times_times(real,B),f34(real,A,B)))) | -hBOOL(hAPP(real,bool,A,C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),B),hAPP(real,real,plus_plus(real,C),zero_zero(real)))). [resolve(1343,a,1340,a)]. 5.35/5.69 1344 ordere236663937imp_le(nat) # label(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 5.35/5.69 1345 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),C)),hAPP(A,A,plus_plus(A,B),D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) # label(fact_784_add__less__imp__less__left) # label(axiom). [clausify(49)]. 5.35/5.69 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,A),B)),hAPP(nat,nat,plus_plus(nat,A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)). [resolve(1344,a,1345,a)]. 5.35/5.69 1346 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,B),C)),hAPP(A,A,plus_plus(A,D),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),D)) # label(fact_776_add__le__imp__le__right) # label(axiom). [clausify(263)]. 5.35/5.69 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(1346,a,1344,a)]. 5.35/5.69 1347 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),C)),hAPP(A,A,plus_plus(A,D),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),D)) # label(fact_778_add__less__cancel__right) # label(axiom). [clausify(293)]. 5.35/5.69 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,A),B)),hAPP(nat,nat,plus_plus(nat,C),B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),C)). [resolve(1347,a,1344,a)]. 5.35/5.69 1348 -ordere236663937imp_le(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),C)),hAPP(A,A,plus_plus(A,D),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),D)) # label(fact_778_add__less__cancel__right) # label(axiom). [clausify(293)]. 5.35/5.69 1349 ordere236663937imp_le(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 5.35/5.69 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,A),B)),hAPP(real,real,plus_plus(real,A),C))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),C)). [resolve(1349,a,1345,a)]. 5.35/5.69 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(1349,a,1346,a)]. 5.35/5.69 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,A),B)),hAPP(real,real,plus_plus(real,C),B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),C)). [resolve(1349,a,1347,a)]. 5.35/5.69 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,A),B)),hAPP(real,real,plus_plus(real,C),B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),C)). [resolve(1349,a,1348,a)]. 5.35/5.69 1350 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),C)),hAPP(A,A,plus_plus(A,D),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),D)) # label(fact_783_add__less__imp__less__right) # label(axiom). [clausify(458)]. 5.43/5.73 1351 -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_779_add__less__cancel__left) # label(axiom). [clausify(572)]. 5.43/5.73 1352 -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_779_add__less__cancel__left) # label(axiom). [clausify(572)]. 5.43/5.73 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,A),B)),hAPP(nat,nat,plus_plus(nat,A),C))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)). [resolve(1352,a,1344,a)]. 5.43/5.73 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,A),B)),hAPP(real,real,plus_plus(real,A),C))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),C)). [resolve(1352,a,1349,a)]. 5.43/5.73 1353 ordere236663937imp_le(int) # label(arity_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 5.43/5.73 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,A),B)),hAPP(int,int,plus_plus(int,A),C))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)). [resolve(1353,a,1345,a)]. 5.43/5.73 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(1353,a,1346,a)]. 5.43/5.73 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,A),B)),hAPP(int,int,plus_plus(int,C),B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),C)). [resolve(1353,a,1347,a)]. 5.43/5.73 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,A),B)),hAPP(int,int,plus_plus(int,A),C))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)). [resolve(1353,a,1352,a)]. 5.43/5.73 1354 -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,B),D)),hAPP(A,A,plus_plus(A,C),D))) # label(fact_771_add__le__cancel__right) # label(axiom). [clausify(822)]. 5.43/5.73 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,A),C)),hAPP(real,real,plus_plus(real,B),C))). [resolve(1354,a,1349,a)]. 5.43/5.73 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,A),C)),hAPP(int,int,plus_plus(int,B),C))). [resolve(1354,a,1353,a)]. 5.43/5.73 1355 -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,B),D)),hAPP(A,A,plus_plus(A,C),D))) # label(fact_771_add__le__cancel__right) # label(axiom). [clausify(822)]. 5.43/5.73 1356 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,B),C)),hAPP(A,A,plus_plus(A,B),D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) # label(fact_777_add__le__imp__le__left) # label(axiom). [clausify(827)]. 5.43/5.73 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(1356,a,1344,a)]. 5.43/5.73 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(1356,a,1349,a)]. 5.43/5.73 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(1356,a,1353,a)]. 5.55/5.90 1357 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,B),C)),hAPP(A,A,plus_plus(A,B),D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) # label(fact_772_add__le__cancel__left) # label(axiom). [clausify(984)]. 5.55/5.90 1358 -ordere236663937imp_le(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,B),C)),hAPP(A,A,plus_plus(A,B),D))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) # label(fact_772_add__le__cancel__left) # label(axiom). [clausify(984)]. 5.55/5.90 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(1358,a,1344,a)]. 5.55/5.90 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(1358,a,1349,a)]. 5.55/5.90 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(1358,a,1353,a)]. 5.55/5.90 1359 linord581940658strict(int) # label(arity_Int_Oint___Rings_Olinordered__ring__strict) # label(axiom). [assumption]. 5.55/5.90 1360 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,B),C))) # label(fact_495_mult__less__cancel__left__neg) # label(axiom). [clausify(52)]. 5.55/5.90 1361 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,B),C))) # label(fact_495_mult__less__cancel__left__neg) # label(axiom). [clausify(52)]. 5.55/5.90 1362 -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_528_mult__le__cancel__left__pos) # label(axiom). [clausify(167)]. 5.55/5.90 1363 -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_528_mult__le__cancel__left__pos) # label(axiom). [clausify(167)]. 5.55/5.90 1364 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,C),D)),hAPP(A,A,times_times(A,B),D))) # label(fact_489_mult__strict__right__mono__neg) # label(axiom). [clausify(187)]. 5.55/5.90 1365 -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),zero_zero(A)),B)) # label(fact_502_mult__less__cancel__left__disj) # label(axiom). [clausify(222)]. 5.55/5.90 1366 -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),C),D)) # label(fact_502_mult__less__cancel__left__disj) # label(axiom). [clausify(222)]. 5.55/5.90 1367 -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),D),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) # label(fact_502_mult__less__cancel__left__disj) # label(axiom). [clausify(222)]. 5.55/5.90 1368 -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),D),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) # label(fact_502_mult__less__cancel__left__disj) # label(axiom). [clausify(222)]. 5.55/5.90 1369 -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_502_mult__less__cancel__left__disj) # label(axiom). [clausify(222)]. 5.55/5.90 1370 -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_502_mult__less__cancel__left__disj) # label(axiom). [clausify(222)]. 5.55/5.90 1371 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,times_times(A,C),B))) # label(fact_486_zero__le__mult__iff) # label(axiom). [clausify(355)]. 5.55/5.90 1372 -linord581940658strict(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,C),B))) # label(fact_486_zero__le__mult__iff) # label(axiom). [clausify(355)]. 5.55/5.90 1373 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,times_times(A,C),B))) # label(fact_486_zero__le__mult__iff) # label(axiom). [clausify(355)]. 5.55/5.90 1374 -linord581940658strict(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),zero_zero(A)),hAPP(A,A,times_times(A,C),B))) # label(fact_486_zero__le__mult__iff) # label(axiom). [clausify(355)]. 5.55/5.90 1375 -linord581940658strict(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),zero_zero(A)),hAPP(A,A,times_times(A,B),C))) # label(fact_486_zero__le__mult__iff) # label(axiom). [clausify(355)]. 5.55/5.90 1376 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -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_486_zero__le__mult__iff) # label(axiom). [clausify(355)]. 5.55/5.90 1377 -linord581940658strict(A) | zero_zero(A) = ti(A,B) | 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,C),C)),hAPP(A,A,times_times(A,B),B)))) # label(fact_239_sum__squares__gt__zero__iff) # label(axiom). [clausify(365)]. 5.55/5.90 1378 -linord581940658strict(A) | zero_zero(A) = ti(A,B) | 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)))) # label(fact_239_sum__squares__gt__zero__iff) # label(axiom). [clausify(365)]. 5.55/5.90 1379 -linord581940658strict(A) | zero_zero(A) != ti(A,B) | zero_zero(A) != ti(A,C) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(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_239_sum__squares__gt__zero__iff) # label(axiom). [clausify(365)]. 5.63/5.90 1380 -linord581940658strict(A) | zero_zero(A) != ti(A,B) | zero_zero(A) != ti(A,C) | 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_218_sum__squares__eq__zero__iff) # label(axiom). [clausify(445)]. 5.63/5.90 1381 -linord581940658strict(A) | zero_zero(A) = ti(A,B) | 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_218_sum__squares__eq__zero__iff) # label(axiom). [clausify(445)]. 5.63/5.90 1382 -linord581940658strict(A) | zero_zero(A) = ti(A,B) | 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_218_sum__squares__eq__zero__iff) # label(axiom). [clausify(445)]. 5.63/5.90 1383 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),D)) # label(fact_503_mult__less__cancel__right__disj) # label(axiom). [clausify(467)]. 5.63/5.90 1384 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) # label(fact_503_mult__less__cancel__right__disj) # label(axiom). [clausify(467)]. 5.63/5.90 1385 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),D)) # label(fact_503_mult__less__cancel__right__disj) # label(axiom). [clausify(467)]. 5.63/5.90 1386 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) # label(fact_503_mult__less__cancel__right__disj) # label(axiom). [clausify(467)]. 5.63/5.90 1387 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),zero_zero(A))) # label(fact_503_mult__less__cancel__right__disj) # label(axiom). [clausify(467)]. 5.63/5.90 1388 -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),B),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) # label(fact_503_mult__less__cancel__right__disj) # label(axiom). [clausify(467)]. 5.63/5.90 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(1359,a,1360,a)]. 5.63/5.90 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(1359,a,1361,a)]. 5.63/5.90 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_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(1359,a,1362,a)]. 5.63/5.90 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),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(1359,a,1364,a)]. 5.63/5.91 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),zero_zero(int)),A)). [resolve(1359,a,1365,a)]. 5.63/5.91 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),B),C)). [resolve(1359,a,1366,a)]. 5.63/5.91 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),C),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)). [resolve(1359,a,1367,a)]. 5.63/5.91 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),C),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)). [resolve(1359,a,1368,a)]. 5.63/5.91 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),B),C)). [resolve(1359,a,1370,a)]. 5.63/5.91 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),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,times_times(int,B),A))). [resolve(1359,a,1373,a)]. 5.63/5.91 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),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,B),A))). [resolve(1359,a,1374,a)]. 5.63/5.91 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),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(1359,a,1375,a)]. 5.63/5.91 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),zero_zero(int)),A)) | -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(1359,a,1376,a)]. 5.63/5.91 Derived: zero_zero(int) = ti(int,A) | 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,B),B)),hAPP(int,int,times_times(int,A),A)))). [resolve(1359,a,1377,a)]. 5.63/5.91 Derived: zero_zero(int) = ti(int,A) | 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)))). [resolve(1359,a,1378,a)]. 5.63/5.91 Derived: zero_zero(int) != ti(int,A) | zero_zero(int) != ti(int,B) | -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,B),B)),hAPP(int,int,times_times(int,A),A)))). [resolve(1359,a,1379,a)]. 5.63/5.91 Derived: zero_zero(int) != ti(int,A) | zero_zero(int) != ti(int,B) | 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(1359,a,1380,a)]. 5.63/5.91 Derived: zero_zero(int) = ti(int,A) | 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(1359,a,1381,a)]. 5.63/5.95 Derived: zero_zero(int) = ti(int,A) | 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(1359,a,1382,a)]. 5.63/5.95 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(1359,a,1383,a)]. 5.63/5.95 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(1359,a,1384,a)]. 5.63/5.95 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(1359,a,1385,a)]. 5.63/5.95 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(1359,a,1386,a)]. 5.63/5.95 1389 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,D),C)),hAPP(A,A,times_times(A,D),B))) # label(fact_488_mult__strict__left__mono__neg) # label(axiom). [clausify(699)]. 5.63/5.95 1390 -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),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)) # label(fact_501_mult__less__cancel__left__pos) # label(axiom). [clausify(702)]. 5.63/5.95 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),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)). [resolve(1390,a,1359,a)]. 5.63/5.95 1391 -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),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)) # label(fact_501_mult__less__cancel__left__pos) # label(axiom). [clausify(702)]. 5.63/5.95 1392 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,times_times(A,B),C))) # label(fact_493_mult__neg__neg) # label(axiom). [clausify(763)]. 5.63/5.95 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(1392,a,1359,a)]. 5.63/5.95 1393 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,B),C))) # label(fact_529_mult__le__cancel__left__neg) # label(axiom). [clausify(935)]. 5.63/5.95 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,times_times(int,A),C)),hAPP(int,int,times_times(int,A),B))). [resolve(1393,a,1359,a)]. 5.63/5.97 1394 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,B),C))) # label(fact_529_mult__le__cancel__left__neg) # label(axiom). [clausify(935)]. 5.63/5.97 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,times_times(int,A),C)),hAPP(int,int,times_times(int,A),B))). [resolve(1394,a,1359,a)]. 5.63/5.97 1395 -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_485_mult__le__0__iff) # label(axiom). [clausify(992)]. 5.63/5.97 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(1395,a,1359,a)]. 5.63/5.97 1396 -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_485_mult__le__0__iff) # label(axiom). [clausify(992)]. 5.63/5.97 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(1396,a,1359,a)]. 5.63/5.97 1397 -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_485_mult__le__0__iff) # label(axiom). [clausify(992)]. 5.63/5.97 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(1397,a,1359,a)]. 5.63/5.97 1398 -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_485_mult__le__0__iff) # label(axiom). [clausify(992)]. 5.63/5.97 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(1398,a,1359,a)]. 5.63/5.97 1399 -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)),B)) # label(fact_485_mult__le__0__iff) # label(axiom). [clausify(992)]. 5.63/5.97 1400 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),C)),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) # label(fact_485_mult__le__0__iff) # label(axiom). [clausify(992)]. 5.63/5.97 1401 linord581940658strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__ring__strict) # label(axiom). [assumption]. 5.63/5.97 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(1401,a,1360,a)]. 5.63/5.97 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(1401,a,1361,a)]. 5.63/5.97 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(1401,a,1362,a)]. 5.63/5.97 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(1401,a,1364,a)]. 5.63/5.97 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),zero_zero(real)),A)). [resolve(1401,a,1365,a)]. 5.63/5.97 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),B),C)). [resolve(1401,a,1366,a)]. 5.63/5.97 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),C),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)). [resolve(1401,a,1367,a)]. 5.63/5.97 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),C),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),C)). [resolve(1401,a,1368,a)]. 5.63/5.97 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),B),C)). [resolve(1401,a,1370,a)]. 5.63/5.97 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),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,times_times(real,B),A))). [resolve(1401,a,1373,a)]. 5.63/5.97 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),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,B),A))). [resolve(1401,a,1374,a)]. 5.63/5.97 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),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(1401,a,1375,a)]. 5.63/5.98 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),zero_zero(real)),A)) | -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(1401,a,1376,a)]. 5.63/5.98 Derived: zero_zero(real) = ti(real,A) | 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,B),B)),hAPP(real,real,times_times(real,A),A)))). [resolve(1401,a,1377,a)]. 5.63/5.98 Derived: zero_zero(real) = ti(real,A) | 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)))). [resolve(1401,a,1378,a)]. 5.63/5.98 Derived: zero_zero(real) != ti(real,A) | zero_zero(real) != ti(real,B) | -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,B),B)),hAPP(real,real,times_times(real,A),A)))). [resolve(1401,a,1379,a)]. 5.63/5.98 Derived: zero_zero(real) != ti(real,A) | zero_zero(real) != ti(real,B) | 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(1401,a,1380,a)]. 5.63/5.98 Derived: zero_zero(real) = ti(real,A) | 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(1401,a,1381,a)]. 5.63/5.98 Derived: zero_zero(real) = ti(real,A) | 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(1401,a,1382,a)]. 5.63/5.98 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(1401,a,1383,a)]. 5.63/5.98 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(1401,a,1384,a)]. 5.63/5.98 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(1401,a,1385,a)]. 5.63/5.98 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(1401,a,1386,a)]. 5.63/5.98 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),A),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)). [resolve(1401,a,1388,a)]. 5.63/5.98 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),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)). [resolve(1401,a,1390,a)]. 5.63/5.98 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(1401,a,1392,a)]. 5.72/5.99 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,A),C)),hAPP(real,real,times_times(real,A),B))). [resolve(1401,a,1393,a)]. 5.72/5.99 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,A),C)),hAPP(real,real,times_times(real,A),B))). [resolve(1401,a,1394,a)]. 5.72/5.99 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),zero_zero(real)),B)). [resolve(1401,a,1395,a)]. 5.72/5.99 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(1401,a,1396,a)]. 5.72/5.99 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(1401,a,1397,a)]. 5.72/5.99 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),A),zero_zero(real))). [resolve(1401,a,1398,a)]. 5.72/5.99 1402 -linord581940658strict(A) | zero_zero(A) != ti(A,B) | zero_zero(A) != ti(A,C) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),B)),hAPP(A,A,times_times(A,C),C))),zero_zero(A))) # label(fact_236_sum__squares__le__zero__iff) # label(axiom). [clausify(1039)]. 5.72/5.99 Derived: zero_zero(int) != ti(int,A) | zero_zero(int) != ti(int,B) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),A)),hAPP(int,int,times_times(int,B),B))),zero_zero(int))). [resolve(1402,a,1359,a)]. 5.72/5.99 Derived: zero_zero(real) != ti(real,A) | zero_zero(real) != ti(real,B) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),A)),hAPP(real,real,times_times(real,B),B))),zero_zero(real))). [resolve(1402,a,1401,a)]. 5.72/5.99 1403 -linord581940658strict(A) | zero_zero(A) = ti(A,B) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),B)),hAPP(A,A,times_times(A,C),C))),zero_zero(A))) # label(fact_236_sum__squares__le__zero__iff) # label(axiom). [clausify(1039)]. 5.72/5.99 Derived: zero_zero(int) = ti(int,A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),A)),hAPP(int,int,times_times(int,B),B))),zero_zero(int))). [resolve(1403,a,1359,a)]. 5.72/5.99 Derived: zero_zero(real) = ti(real,A) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),A)),hAPP(real,real,times_times(real,B),B))),zero_zero(real))). [resolve(1403,a,1401,a)]. 5.72/5.99 1404 -linord581940658strict(A) | zero_zero(A) = ti(A,B) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),C)),hAPP(A,A,times_times(A,B),B))),zero_zero(A))) # label(fact_236_sum__squares__le__zero__iff) # label(axiom). [clausify(1039)]. 5.72/5.99 Derived: zero_zero(int) = ti(int,A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),B)),hAPP(int,int,times_times(int,A),A))),zero_zero(int))). [resolve(1404,a,1359,a)]. 5.85/6.15 Derived: zero_zero(real) = ti(real,A) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,B),B)),hAPP(real,real,times_times(real,A),A))),zero_zero(real))). [resolve(1404,a,1401,a)]. 5.85/6.15 1405 -ordere223160158up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),D)),hAPP(A,A,plus_plus(A,C),E))) # label(fact_782_add__strict__mono) # label(axiom). [clausify(380)]. 5.85/6.15 1406 ordere223160158up_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 5.85/6.15 1407 ordere223160158up_add(nat) # label(arity_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 5.85/6.15 1408 ordere223160158up_add(int) # label(arity_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 5.85/6.15 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(1405,a,1406,a)]. 5.85/6.15 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),hAPP(nat,nat,plus_plus(nat,A),C)),hAPP(nat,nat,plus_plus(nat,B),D))). [resolve(1405,a,1407,a)]. 5.85/6.15 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(1405,a,1408,a)]. 5.85/6.15 1409 -ordere223160158up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,D),B)),hAPP(A,A,plus_plus(A,D),C))) # label(fact_781_add__strict__left__mono) # label(axiom). [clausify(647)]. 5.85/6.15 1410 -ordere223160158up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),D)),hAPP(A,A,plus_plus(A,C),E))) # label(fact_809_add__le__less__mono) # label(axiom). [clausify(727)]. 5.85/6.15 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(1410,a,1406,a)]. 5.85/6.15 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),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(1410,a,1407,a)]. 5.85/6.15 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(1410,a,1408,a)]. 5.85/6.15 1411 -ordere223160158up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),D)),hAPP(A,A,plus_plus(A,C),E))) # label(fact_808_add__less__le__mono) # label(axiom). [clausify(731)]. 5.85/6.15 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(1411,a,1406,a)]. 6.03/6.33 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_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(1411,a,1407,a)]. 6.03/6.33 1412 -ordere223160158up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),D)),hAPP(A,A,plus_plus(A,C),D))) # label(fact_780_add__strict__right__mono) # label(axiom). [clausify(745)]. 6.03/6.33 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,A),C)),hAPP(nat,nat,plus_plus(nat,B),C))). [resolve(1412,a,1407,a)]. 6.03/6.33 1413 -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_804_zero__less__double__add__iff__zero__less__single__add) # label(axiom). [clausify(426)]. 6.03/6.33 1414 linord219039673up_add(int) # label(arity_Int_Oint___Groups_Olinordered__ab__group__add) # label(axiom). [assumption]. 6.03/6.33 1415 linord219039673up_add(real) # label(arity_RealDef_Oreal___Groups_Olinordered__ab__group__add) # label(axiom). [assumption]. 6.03/6.33 1416 -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_804_zero__less__double__add__iff__zero__less__single__add) # label(axiom). [clausify(426)]. 6.03/6.33 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(1416,a,1414,a)]. 6.03/6.33 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(real),zero_zero(real)),hAPP(real,real,plus_plus(real,A),A))). [resolve(1416,a,1415,a)]. 6.03/6.33 1417 -linord219039673up_add(A) | zero_zero(A) != ti(A,B) | zero_zero(A) = hAPP(A,A,plus_plus(A,B),B) # label(fact_203_double__eq__0__iff) # label(axiom). [clausify(577)]. 6.03/6.33 Derived: zero_zero(int) != ti(int,A) | zero_zero(int) = hAPP(int,int,plus_plus(int,A),A). [resolve(1417,a,1414,a)]. 6.03/6.33 Derived: zero_zero(real) != ti(real,A) | zero_zero(real) = hAPP(real,real,plus_plus(real,A),A). [resolve(1417,a,1415,a)]. 6.03/6.33 1418 -linord219039673up_add(A) | zero_zero(A) = ti(A,B) | zero_zero(A) != hAPP(A,A,plus_plus(A,B),B) # label(fact_203_double__eq__0__iff) # label(axiom). [clausify(577)]. 6.03/6.33 Derived: zero_zero(int) = ti(int,A) | zero_zero(int) != hAPP(int,int,plus_plus(int,A),A). [resolve(1418,a,1414,a)]. 6.03/6.33 Derived: zero_zero(real) = ti(real,A) | zero_zero(real) != hAPP(real,real,plus_plus(real,A),A). [resolve(1418,a,1415,a)]. 6.03/6.33 1419 -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_805_double__add__less__zero__iff__single__add__less__zero) # label(axiom). [clausify(606)]. 6.03/6.33 1420 -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_805_double__add__less__zero__iff__single__add__less__zero) # label(axiom). [clausify(606)]. 6.03/6.33 1421 -linord219039673up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,plus_plus(A,B),B))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) # label(fact_797_zero__le__double__add__iff__zero__le__single__add) # label(axiom). [clausify(677)]. 6.03/6.33 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,plus_plus(int,A),A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)). [resolve(1421,a,1414,a)]. 6.23/6.51 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,plus_plus(real,A),A))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)). [resolve(1421,a,1415,a)]. 6.23/6.51 1422 -linord219039673up_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,plus_plus(A,B),B))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) # label(fact_797_zero__le__double__add__iff__zero__le__single__add) # label(axiom). [clausify(677)]. 6.23/6.51 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,plus_plus(int,A),A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)). [resolve(1422,a,1414,a)]. 6.23/6.51 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,plus_plus(real,A),A))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)). [resolve(1422,a,1415,a)]. 6.23/6.51 1423 -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_798_double__add__le__zero__iff__single__add__le__zero) # label(axiom). [clausify(720)]. 6.23/6.51 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(1423,a,1414,a)]. 6.23/6.51 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(1423,a,1415,a)]. 6.23/6.51 1424 -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_798_double__add__le__zero__iff__single__add__le__zero) # label(axiom). [clausify(720)]. 6.23/6.51 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(1424,a,1414,a)]. 6.23/6.51 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(1424,a,1415,a)]. 6.23/6.51 1425 -linord219039673up_add(A) | zero_zero(A) != hAPP(A,A,plus_plus(A,B),B) | zero_zero(A) = ti(A,B) # label(fact_768_double__zero__sym) # label(axiom). [clausify(907)]. 6.23/6.51 1426 -linord219039673up_add(A) | zero_zero(A) = hAPP(A,A,plus_plus(A,B),B) | zero_zero(A) != ti(A,B) # label(fact_768_double__zero__sym) # label(axiom). [clausify(907)]. 6.23/6.51 1427 -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(194)]. 6.23/6.51 1428 ab_semigroup_add(real) # label(arity_RealDef_Oreal___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 6.23/6.51 1429 ab_semigroup_add(nat) # label(arity_Nat_Onat___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 6.23/6.51 Derived: ti(fun(real,real),plus_plus(real,A)) = plus_plus(real,A). [resolve(1427,a,1428,a)]. 6.23/6.51 Derived: ti(fun(nat,nat),plus_plus(nat,A)) = plus_plus(nat,A). [resolve(1427,a,1429,a)]. 6.23/6.51 1430 -ab_semigroup_add(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,plus_plus(A,B),C)),D) = hAPP(A,A,plus_plus(A,B),hAPP(A,A,plus_plus(A,C),D)) # label(fact_758_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom). [clausify(316)]. 6.23/6.51 1431 -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(343)]. 6.23/6.51 Derived: plus_plus(real,ti(real,A)) = plus_plus(real,A). [resolve(1431,a,1428,a)]. 6.23/6.51 Derived: plus_plus(nat,ti(nat,A)) = plus_plus(nat,A). [resolve(1431,a,1429,a)]. 6.43/6.70 1432 ab_semigroup_add(int) # label(arity_Int_Oint___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 6.43/6.70 Derived: ti(fun(int,int),plus_plus(int,A)) = plus_plus(int,A). [resolve(1432,a,1427,a)]. 6.43/6.70 Derived: plus_plus(int,ti(int,A)) = plus_plus(int,A). [resolve(1432,a,1431,a)]. 6.43/6.70 1433 ordere216010020id_add(nat) # label(arity_Nat_Onat___Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 6.43/6.70 1434 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | zero_zero(A) != ti(A,C) | zero_zero(A) != ti(A,B) | zero_zero(A) = hAPP(A,A,plus_plus(A,B),C) # label(fact_800_add__nonneg__eq__0__iff) # label(axiom). [clausify(68)]. 6.43/6.70 1435 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | zero_zero(A) = ti(A,C) | zero_zero(A) != hAPP(A,A,plus_plus(A,B),C) # label(fact_800_add__nonneg__eq__0__iff) # label(axiom). [clausify(68)]. 6.43/6.70 1436 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | zero_zero(A) = ti(A,B) | zero_zero(A) != hAPP(A,A,plus_plus(A,B),C) # label(fact_800_add__nonneg__eq__0__iff) # label(axiom). [clausify(68)]. 6.43/6.70 1437 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),hAPP(A,A,plus_plus(A,B),D))) # label(fact_801_add__increasing) # label(axiom). [clausify(239)]. 6.43/6.70 1438 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),C)),zero_zero(A))) # label(fact_807_add__neg__neg) # label(axiom). [clausify(246)]. 6.43/6.70 1439 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),hAPP(A,A,plus_plus(A,D),B))) # label(fact_802_add__increasing2) # label(axiom). [clausify(281)]. 6.43/6.70 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | zero_zero(nat) != ti(nat,B) | zero_zero(nat) != ti(nat,A) | zero_zero(nat) = hAPP(nat,nat,plus_plus(nat,A),B). [resolve(1433,a,1434,a)]. 6.43/6.70 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | zero_zero(nat) = ti(nat,B) | zero_zero(nat) != hAPP(nat,nat,plus_plus(nat,A),B). [resolve(1433,a,1435,a)]. 6.43/6.70 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | zero_zero(nat) = ti(nat,A) | zero_zero(nat) != hAPP(nat,nat,plus_plus(nat,A),B). [resolve(1433,a,1436,a)]. 6.43/6.70 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),hAPP(nat,nat,plus_plus(nat,A),C))). [resolve(1433,a,1437,a)]. 6.43/6.70 1440 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,plus_plus(A,B),C))) # label(fact_806_add__pos__pos) # label(axiom). [clausify(298)]. 6.43/6.70 1441 ordere216010020id_add(int) # label(arity_Int_Oint___Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 6.43/6.70 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | zero_zero(int) != ti(int,B) | zero_zero(int) != ti(int,A) | zero_zero(int) = hAPP(int,int,plus_plus(int,A),B). [resolve(1441,a,1434,a)]. 6.43/6.71 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | zero_zero(int) = ti(int,B) | zero_zero(int) != hAPP(int,int,plus_plus(int,A),B). [resolve(1441,a,1435,a)]. 6.43/6.71 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | zero_zero(int) = ti(int,A) | zero_zero(int) != hAPP(int,int,plus_plus(int,A),B). [resolve(1441,a,1436,a)]. 6.43/6.71 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(1441,a,1437,a)]. 6.43/6.71 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(1441,a,1438,a)]. 6.43/6.71 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(1441,a,1439,a)]. 6.43/6.71 1442 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),C)),zero_zero(A))) # label(fact_812_add__nonpos__neg) # label(axiom). [clausify(344)]. 6.43/6.71 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(1442,a,1441,a)]. 6.43/6.71 1443 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,B),C)),zero_zero(A))) # label(fact_803_add__nonpos__nonpos) # label(axiom). [clausify(346)]. 6.43/6.71 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(1443,a,1433,a)]. 6.43/6.71 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(1443,a,1441,a)]. 6.43/6.71 1444 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,plus_plus(A,B),C))) # label(fact_816_add__nonneg__pos) # label(axiom). [clausify(354)]. 6.43/6.71 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),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(1444,a,1441,a)]. 6.43/6.71 1445 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,plus_plus(A,B),C))) # label(fact_799_add__nonneg__nonneg) # label(axiom). [clausify(367)]. 6.43/6.74 1446 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),hAPP(A,A,plus_plus(A,B),D))) # label(fact_814_add__strict__increasing2) # label(axiom). [clausify(547)]. 6.43/6.74 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(1446,a,1441,a)]. 6.43/6.74 1447 ordere216010020id_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 6.43/6.74 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | zero_zero(real) != ti(real,B) | zero_zero(real) != ti(real,A) | zero_zero(real) = hAPP(real,real,plus_plus(real,A),B). [resolve(1447,a,1434,a)]. 6.43/6.74 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | zero_zero(real) = ti(real,B) | zero_zero(real) != hAPP(real,real,plus_plus(real,A),B). [resolve(1447,a,1435,a)]. 6.43/6.74 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | zero_zero(real) = ti(real,A) | zero_zero(real) != hAPP(real,real,plus_plus(real,A),B). [resolve(1447,a,1436,a)]. 6.43/6.74 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(1447,a,1437,a)]. 6.43/6.74 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(1447,a,1438,a)]. 6.43/6.74 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(1447,a,1439,a)]. 6.43/6.74 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(1447,a,1442,a)]. 6.43/6.74 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(1447,a,1443,a)]. 6.43/6.74 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),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(1447,a,1444,a)]. 6.43/6.74 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(1447,a,1446,a)]. 6.83/7.11 1448 -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(769)]. 6.83/7.11 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(1448,a,1433,a)]. 6.83/7.11 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(1448,a,1441,a)]. 6.83/7.11 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(1448,a,1447,a)]. 6.83/7.11 1449 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),C)),zero_zero(A))) # label(fact_813_add__neg__nonpos) # label(axiom). [clausify(903)]. 6.83/7.11 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(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(nat),hAPP(nat,nat,plus_plus(nat,A),B)),zero_zero(nat))). [resolve(1449,a,1433,a)]. 6.83/7.11 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(1449,a,1441,a)]. 6.83/7.11 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(1449,a,1447,a)]. 6.83/7.11 1450 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,plus_plus(A,B),C))) # label(fact_755_add__pos__nonneg) # label(axiom). [clausify(946)]. 6.83/7.11 1451 -linordered_semiring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),D)) # label(fact_534_mult__right__less__imp__less) # label(axiom). [clausify(195)]. 6.83/7.11 1452 linordered_semiring(int) # label(arity_Int_Oint___Rings_Olinordered__semiring) # label(axiom). [assumption]. 6.83/7.11 1453 linordered_semiring(nat) # label(arity_Nat_Onat___Rings_Olinordered__semiring) # label(axiom). [assumption]. 6.83/7.11 1454 -linordered_semiring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) # label(fact_536_mult__left__less__imp__less) # label(axiom). [clausify(573)]. 6.83/7.11 1455 linordered_semiring(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring) # label(axiom). [assumption]. 6.83/7.11 1456 -monoid_add(A) | ti(A,B) = hAPP(A,A,plus_plus(A,B),zero_zero(A)) # label(fact_769_add__0__right) # label(axiom). [clausify(328)]. 6.83/7.11 1457 monoid_add(real) # label(arity_RealDef_Oreal___Groups_Omonoid__add) # label(axiom). [assumption]. 7.42/7.68 1458 -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(330)]. 7.42/7.68 1459 monoid_add(int) # label(arity_Int_Oint___Groups_Omonoid__add) # label(axiom). [assumption]. 7.42/7.68 1460 -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(359)]. 7.42/7.68 1461 monoid_add(nat) # label(arity_Nat_Onat___Groups_Omonoid__add) # label(axiom). [assumption]. 7.42/7.68 1462 -monoid_add(A) | ti(A,B) = hAPP(A,A,plus_plus(A,zero_zero(A)),B) # label(fact_766_add__0__left) # label(axiom). [clausify(427)]. 7.42/7.68 1463 ab_group_add(real) # label(arity_RealDef_Oreal___Groups_Oab__group__add) # label(axiom). [assumption]. 7.42/7.68 1464 -ab_group_add(A) | hAPP(A,A,minus_minus(A,B),C) != hAPP(A,A,minus_minus(A,D),E) | ti(A,C) != ti(A,B) | ti(A,E) = ti(A,D) # label(fact_765_diff__eq__diff__eq) # label(axiom). [clausify(76)]. 7.42/7.68 1465 -ab_group_add(A) | hAPP(A,A,minus_minus(A,B),C) != hAPP(A,A,minus_minus(A,D),E) | ti(A,C) = ti(A,B) | ti(A,E) != ti(A,D) # label(fact_765_diff__eq__diff__eq) # label(axiom). [clausify(76)]. 7.42/7.68 Derived: hAPP(real,real,minus_minus(real,A),B) != hAPP(real,real,minus_minus(real,C),D) | ti(real,B) != ti(real,A) | ti(real,D) = ti(real,C). [resolve(1463,a,1464,a)]. 7.42/7.68 Derived: hAPP(real,real,minus_minus(real,A),B) != hAPP(real,real,minus_minus(real,C),D) | ti(real,B) = ti(real,A) | ti(real,D) != ti(real,C). [resolve(1463,a,1465,a)]. 7.42/7.68 1466 -ab_group_add(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,minus_minus(A,B),C)),hAPP(A,A,minus_minus(A,D),E)) = hAPP(A,A,minus_minus(A,hAPP(A,A,plus_plus(A,B),D)),hAPP(A,A,plus_plus(A,C),E)) # label(fact_450_add__diff__add) # label(axiom). [clausify(548)]. 7.42/7.68 Derived: hAPP(real,real,plus_plus(real,hAPP(real,real,minus_minus(real,A),B)),hAPP(real,real,minus_minus(real,C),D)) = hAPP(real,real,minus_minus(real,hAPP(real,real,plus_plus(real,A),C)),hAPP(real,real,plus_plus(real,B),D)). [resolve(1466,a,1463,a)]. 7.42/7.68 1467 -ab_group_add(A) | ti(A,B) != ti(A,C) | zero_zero(A) = hAPP(A,A,minus_minus(A,C),B) # label(fact_786_eq__iff__diff__eq__0) # label(axiom). [clausify(576)]. 7.42/7.68 1468 -ab_group_add(A) | ti(A,B) = ti(A,C) | zero_zero(A) != hAPP(A,A,minus_minus(A,C),B) # label(fact_786_eq__iff__diff__eq__0) # label(axiom). [clausify(576)]. 7.42/7.68 1469 ab_group_add(int) # label(arity_Int_Oint___Groups_Oab__group__add) # label(axiom). [assumption]. 7.42/7.68 Derived: hAPP(int,int,minus_minus(int,A),B) != hAPP(int,int,minus_minus(int,C),D) | ti(int,B) != ti(int,A) | ti(int,D) = ti(int,C). [resolve(1469,a,1464,a)]. 7.42/7.68 Derived: hAPP(int,int,minus_minus(int,A),B) != hAPP(int,int,minus_minus(int,C),D) | ti(int,B) = ti(int,A) | ti(int,D) != ti(int,C). [resolve(1469,a,1465,a)]. 7.42/7.68 Derived: hAPP(int,int,plus_plus(int,hAPP(int,int,minus_minus(int,A),B)),hAPP(int,int,minus_minus(int,C),D)) = hAPP(int,int,minus_minus(int,hAPP(int,int,plus_plus(int,A),C)),hAPP(int,int,plus_plus(int,B),D)). [resolve(1469,a,1466,a)]. 7.42/7.68 1470 -comm_monoid_add(A) | ti(A,B) = hAPP(A,A,plus_plus(A,B),zero_zero(A)) # label(fact_770_add_Ocomm__neutral) # label(axiom). [clausify(614)]. 7.42/7.68 1471 comm_monoid_add(real) # label(arity_RealDef_Oreal___Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 7.42/7.68 1472 comm_monoid_add(nat) # label(arity_Nat_Onat___Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 7.42/7.68 1473 comm_monoid_add(int) # label(arity_Int_Oint___Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 7.42/7.68 1474 -comm_monoid_add(A) | ti(A,B) = hAPP(A,A,plus_plus(A,zero_zero(A)),B) # label(fact_767_add__0) # label(axiom). [clausify(618)]. 7.42/7.68 1475 cancel_semigroup_add(int) # label(arity_Int_Oint___Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 7.42/7.68 1476 -cancel_semigroup_add(A) | hAPP(A,A,plus_plus(A,B),C) != hAPP(A,A,plus_plus(A,B),D) | ti(A,C) = ti(A,D) # label(fact_759_add__left__cancel) # label(axiom). [clausify(96)]. 7.42/7.68 1477 -cancel_semigroup_add(A) | hAPP(A,A,plus_plus(A,B),C) = hAPP(A,A,plus_plus(A,B),D) | ti(A,C) != ti(A,D) # label(fact_759_add__left__cancel) # label(axiom). [clausify(96)]. 7.42/7.68 1478 -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(117)]. 7.76/8.06 Derived: hAPP(int,int,plus_plus(int,A),B) != hAPP(int,int,plus_plus(int,A),C) | ti(int,B) = ti(int,C). [resolve(1475,a,1476,a)]. 7.76/8.06 Derived: hAPP(int,int,plus_plus(int,A),B) = hAPP(int,int,plus_plus(int,A),C) | ti(int,B) != ti(int,C). [resolve(1475,a,1477,a)]. 7.76/8.06 1479 cancel_semigroup_add(nat) # label(arity_Nat_Onat___Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 7.76/8.06 Derived: hAPP(nat,nat,plus_plus(nat,A),B) != hAPP(nat,nat,plus_plus(nat,A),C) | ti(nat,B) = ti(nat,C). [resolve(1479,a,1476,a)]. 7.76/8.06 Derived: hAPP(nat,nat,plus_plus(nat,A),B) = hAPP(nat,nat,plus_plus(nat,A),C) | ti(nat,B) != ti(nat,C). [resolve(1479,a,1477,a)]. 7.76/8.06 1480 cancel_semigroup_add(real) # label(arity_RealDef_Oreal___Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 7.76/8.06 Derived: hAPP(real,real,plus_plus(real,A),B) != hAPP(real,real,plus_plus(real,A),C) | ti(real,B) = ti(real,C). [resolve(1480,a,1476,a)]. 7.76/8.06 Derived: hAPP(real,real,plus_plus(real,A),B) = hAPP(real,real,plus_plus(real,A),C) | ti(real,B) != ti(real,C). [resolve(1480,a,1477,a)]. 7.76/8.06 1481 -cancel_semigroup_add(A) | hAPP(A,A,plus_plus(A,B),C) != hAPP(A,A,plus_plus(A,D),C) | ti(A,B) = ti(A,D) # label(fact_763_add__right__imp__eq) # label(axiom). [clausify(637)]. 7.76/8.06 Derived: hAPP(int,int,plus_plus(int,A),B) != hAPP(int,int,plus_plus(int,C),B) | ti(int,A) = ti(int,C). [resolve(1481,a,1475,a)]. 7.76/8.06 Derived: hAPP(nat,nat,plus_plus(nat,A),B) != hAPP(nat,nat,plus_plus(nat,C),B) | ti(nat,A) = ti(nat,C). [resolve(1481,a,1479,a)]. 7.76/8.06 Derived: hAPP(real,real,plus_plus(real,A),B) != hAPP(real,real,plus_plus(real,C),B) | ti(real,A) = ti(real,C). [resolve(1481,a,1480,a)]. 7.76/8.06 1482 -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(915)]. 7.76/8.06 1483 -cancel_semigroup_add(A) | hAPP(A,A,plus_plus(A,B),C) != hAPP(A,A,plus_plus(A,B),D) | ti(A,C) = ti(A,D) # label(fact_761_add__left__imp__eq) # label(axiom). [clausify(927)]. 7.76/8.06 1484 -cancel_semigroup_add(A) | hAPP(A,A,plus_plus(A,B),C) != hAPP(A,A,plus_plus(A,D),C) | ti(A,B) = ti(A,D) # label(fact_760_add__right__cancel) # label(axiom). [clausify(976)]. 7.76/8.06 1485 -cancel_semigroup_add(A) | hAPP(A,A,plus_plus(A,B),C) = hAPP(A,A,plus_plus(A,D),C) | ti(A,B) != ti(A,D) # label(fact_760_add__right__cancel) # label(axiom). [clausify(976)]. 7.76/8.06 Derived: hAPP(int,int,plus_plus(int,A),B) = hAPP(int,int,plus_plus(int,C),B) | ti(int,A) != ti(int,C). [resolve(1485,a,1475,a)]. 7.76/8.06 Derived: hAPP(nat,nat,plus_plus(nat,A),B) = hAPP(nat,nat,plus_plus(nat,C),B) | ti(nat,A) != ti(nat,C). [resolve(1485,a,1479,a)]. 7.76/8.06 Derived: hAPP(real,real,plus_plus(real,A),B) = hAPP(real,real,plus_plus(real,C),B) | ti(real,A) != ti(real,C). [resolve(1485,a,1480,a)]. 7.76/8.06 1486 -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(719)]. 7.76/8.06 1487 ab_semigroup_mult(int) # label(arity_Int_Oint___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 7.76/8.06 1488 ab_semigroup_mult(real) # label(arity_RealDef_Oreal___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 7.76/8.06 1489 ab_semigroup_mult(nat) # label(arity_Nat_Onat___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 7.76/8.06 1490 -ab_semigroup_mult(A) | ti(fun(A,A),times_times(A,B)) = times_times(A,B) # label(tsy_c_Groups_Otimes__class_Otimes_0_res) # label(axiom). [clausify(784)]. 7.76/8.06 1491 -ab_semigroup_mult(A) | hAPP(A,A,times_times(A,hAPP(A,A,times_times(A,B),C)),D) = hAPP(A,A,times_times(A,B),hAPP(A,A,times_times(A,C),D)) # label(fact_757_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom). [clausify(912)]. 7.76/8.06 1492 -ring_n68954251visors(A) | zero_zero(A) != hAPP(A,A,times_times(A,B),C) | zero_zero(A) = ti(A,B) | zero_zero(A) = ti(A,C) # label(fact_442_mult__eq__0__iff) # label(axiom). [clausify(289)]. 7.76/8.06 1493 ring_n68954251visors(real) # label(arity_RealDef_Oreal___Rings_Oring__no__zero__divisors) # label(axiom). [assumption]. 7.76/8.06 1494 -ring_n68954251visors(A) | zero_zero(A) = hAPP(A,A,times_times(A,B),C) | zero_zero(A) != ti(A,B) # label(fact_442_mult__eq__0__iff) # label(axiom). [clausify(289)]. 8.33/8.63 Derived: zero_zero(real) = hAPP(real,real,times_times(real,A),B) | zero_zero(real) != ti(real,A). [resolve(1494,a,1493,a)]. 8.33/8.63 1495 -ring_n68954251visors(A) | zero_zero(A) = hAPP(A,A,times_times(A,B),C) | zero_zero(A) != ti(A,C) # label(fact_442_mult__eq__0__iff) # label(axiom). [clausify(289)]. 8.33/8.63 Derived: zero_zero(real) = hAPP(real,real,times_times(real,A),B) | zero_zero(real) != ti(real,B). [resolve(1495,a,1493,a)]. 8.33/8.63 1496 ring_n68954251visors(int) # label(arity_Int_Oint___Rings_Oring__no__zero__divisors) # label(axiom). [assumption]. 8.33/8.63 Derived: zero_zero(int) = hAPP(int,int,times_times(int,A),B) | zero_zero(int) != ti(int,A). [resolve(1496,a,1494,a)]. 8.33/8.63 Derived: zero_zero(int) = hAPP(int,int,times_times(int,A),B) | zero_zero(int) != ti(int,B). [resolve(1496,a,1495,a)]. 8.33/8.63 1497 ring_char_0(int) # label(arity_Int_Oint___Int_Oring__char__0) # label(axiom). [assumption]. 8.33/8.63 1498 -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_82_eq__number__of) # label(axiom). [clausify(113)]. 8.33/8.63 1499 -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_82_eq__number__of) # label(axiom). [clausify(113)]. 8.33/8.63 Derived: -number_ring(int) | number_number_of(int,A) != number_number_of(int,B) | ti(int,A) = ti(int,B). [resolve(1497,a,1498,b)]. 8.33/8.63 Derived: -number_ring(int) | number_number_of(int,A) = number_number_of(int,B) | ti(int,A) != ti(int,B). [resolve(1497,a,1499,b)]. 8.33/8.63 1500 ring_char_0(real) # label(arity_RealDef_Oreal___Int_Oring__char__0) # label(axiom). [assumption]. 8.33/8.63 Derived: -number_ring(real) | number_number_of(real,A) != number_number_of(real,B) | ti(int,A) = ti(int,B). [resolve(1500,a,1498,b)]. 8.33/8.63 Derived: -number_ring(real) | number_number_of(real,A) = number_number_of(real,B) | ti(int,A) != ti(int,B). [resolve(1500,a,1499,b)]. 8.33/8.63 1501 -linord626643107strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),F)) | one_one(A) != hAPP(A,A,plus_plus(A,E),F) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,E),B)),hAPP(A,A,times_times(A,F),D))),C)) # label(fact_422_convex__bound__lt) # label(axiom). [clausify(785)]. 8.33/8.63 1502 linord626643107strict(int) # label(arity_Int_Oint___Rings_Olinordered__semiring__1__strict) # label(axiom). [assumption]. 8.33/8.63 1503 linord626643107strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring__1__strict) # label(axiom). [assumption]. 8.33/8.63 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(1501,a,1502,a)]. 8.33/8.63 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),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(1501,a,1503,a)]. 8.33/8.63 1504 real_normed_algebra(real) # label(arity_RealDef_Oreal___RealVector_Oreal__normed__algebra) # label(axiom). [assumption]. 8.33/8.63 1505 -real_normed_algebra(A) | zero_zero(A) = hAPP(A,A,times_times(A,zero_zero(A)),B) # label(fact_887_mult_Ozero__left) # label(axiom). [clausify(125)]. 8.55/8.89 1506 -real_normed_algebra(A) | hAPP(A,A,minus_minus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D)) = hAPP(A,A,times_times(A,B),hAPP(A,A,minus_minus(A,C),D)) # label(fact_893_mult__right_Odiff) # label(axiom). [clausify(128)]. 8.55/8.89 1507 -real_normed_algebra(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C)) = hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),D)),C) # label(fact_889_mult_Oadd__left) # label(axiom). [clausify(292)]. 8.55/8.89 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(1504,a,1506,a)]. 8.55/8.89 1508 -real_normed_algebra(A) | hAPP(A,A,minus_minus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C)) = hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,B),D)),C) # label(fact_895_mult__left_Odiff) # label(axiom). [clausify(499)]. 8.55/8.89 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(1508,a,1504,a)]. 8.55/8.89 1509 -real_normed_algebra(A) | hAPP(A,A,minus_minus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C)) = hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,B),D)),C) # label(fact_894_mult_Odiff__left) # label(axiom). [clausify(675)]. 8.55/8.89 1510 -real_normed_algebra(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D)) = hAPP(A,A,times_times(A,B),hAPP(A,A,plus_plus(A,C),D)) # label(fact_890_mult__right_Oadd) # label(axiom). [clausify(692)]. 8.55/8.89 1511 -real_normed_algebra(A) | hAPP(A,A,minus_minus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D)) = hAPP(A,A,times_times(A,B),hAPP(A,A,minus_minus(A,C),D)) # label(fact_892_mult_Odiff__right) # label(axiom). [clausify(728)]. 8.55/8.89 1512 -real_normed_algebra(A) | 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),C)),hAPP(A,A,minus_minus(A,D),E))),hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,B),C)),E))),hAPP(A,A,times_times(A,C),hAPP(A,A,minus_minus(A,D),E))) = hAPP(A,A,minus_minus(A,hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),E)) # label(fact_882_mult_Oprod__diff__prod) # label(axiom). [clausify(743)]. 8.55/8.89 Derived: 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),B)),hAPP(real,real,minus_minus(real,C),D))),hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,A),B)),D))),hAPP(real,real,times_times(real,B),hAPP(real,real,minus_minus(real,C),D))) = hAPP(real,real,minus_minus(real,hAPP(real,real,times_times(real,A),C)),hAPP(real,real,times_times(real,B),D)). [resolve(1512,a,1504,a)]. 8.55/8.89 1513 -real_normed_algebra(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C)) = hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),D)),C) # label(fact_888_mult__left_Oadd) # label(axiom). [clausify(753)]. 8.55/8.89 1514 -real_normed_algebra(A) | zero_zero(A) = hAPP(A,A,times_times(A,B),zero_zero(A)) # label(fact_885_mult_Ozero__right) # label(axiom). [clausify(818)]. 8.55/8.89 1515 -real_normed_algebra(A) | zero_zero(A) = hAPP(A,A,times_times(A,zero_zero(A)),B) # label(fact_886_mult__left_Ozero) # label(axiom). [clausify(848)]. 8.55/8.89 1516 -real_normed_algebra(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D)) = hAPP(A,A,times_times(A,B),hAPP(A,A,plus_plus(A,C),D)) # label(fact_891_mult_Oadd__right) # label(axiom). [clausify(904)]. 8.55/8.89 1517 -real_normed_algebra(A) | zero_zero(A) = hAPP(A,A,times_times(A,B),zero_zero(A)) # label(fact_884_mult__right_Ozero) # label(axiom). [clausify(941)]. 8.55/8.89 1518 ring_div(int) # label(arity_Int_Oint___Divides_Oring__div) # label(axiom). [assumption]. 8.55/8.89 1519 -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(127)]. 8.78/9.08 1520 -ring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,minus_minus(A,hAPP(A,A,div_mod(A,B),C)),D)),C) = hAPP(A,A,div_mod(A,hAPP(A,A,minus_minus(A,B),D)),C) # label(fact_949_mod__diff__left__eq) # label(axiom). [clausify(387)]. 8.78/9.08 1521 -ring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,minus_minus(A,B),hAPP(A,A,div_mod(A,C),D))),D) = hAPP(A,A,div_mod(A,hAPP(A,A,minus_minus(A,B),C)),D) # label(fact_950_mod__diff__right__eq) # label(axiom). [clausify(511)]. 8.78/9.08 1522 -ring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,minus_minus(A,hAPP(A,A,div_mod(A,B),C)),hAPP(A,A,div_mod(A,D),C))),C) = hAPP(A,A,div_mod(A,hAPP(A,A,minus_minus(A,B),D)),C) # label(fact_948_mod__diff__eq) # label(axiom). [clausify(967)]. 8.78/9.08 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(1518,a,1519,a)]. 8.78/9.08 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(1518,a,1520,a)]. 8.78/9.08 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(1518,a,1522,a)]. 8.78/9.08 1523 -linordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,times_times(A,B),B))) # label(fact_487_zero__le__square) # label(axiom). [clausify(636)]. 8.78/9.08 1524 linordered_ring(int) # label(arity_Int_Oint___Rings_Olinordered__ring) # label(axiom). [assumption]. 8.78/9.08 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(1523,a,1524,a)]. 8.78/9.08 1525 -linordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),B)),hAPP(A,A,times_times(A,C),C)))) # label(fact_235_sum__squares__ge__zero) # label(axiom). [clausify(664)]. 8.78/9.08 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(1525,a,1524,a)]. 8.78/9.08 1526 linordered_ring(real) # label(arity_RealDef_Oreal___Rings_Olinordered__ring) # label(axiom). [assumption]. 8.78/9.08 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(1526,a,1523,a)]. 8.78/9.08 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(1526,a,1525,a)]. 8.78/9.08 1527 -linordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),B)),hAPP(A,A,times_times(A,C),C))),zero_zero(A))) # label(fact_238_not__sum__squares__lt__zero) # label(axiom). [clausify(905)]. 8.78/9.08 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(1527,a,1524,a)]. 8.78/9.08 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(1527,a,1526,a)]. 8.78/9.08 1528 -linordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),B)),zero_zero(A))) # label(fact_504_not__square__less__zero) # label(axiom). [clausify(989)]. 8.78/9.08 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(1528,a,1524,a)]. 8.78/9.08 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(1528,a,1526,a)]. 8.93/9.23 1529 -idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),D)) | zero_zero(A) = ti(A,B) # label(fact_512_dvd__mult__cancel__left) # label(axiom). [clausify(310)]. 8.93/9.23 1530 idom(int) # label(arity_Int_Oint___Rings_Oidom) # label(axiom). [assumption]. 8.93/9.23 1531 idom(real) # label(arity_RealDef_Oreal___Rings_Oidom) # label(axiom). [assumption]. 8.93/9.23 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,A),C))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),B),C)) | zero_zero(real) = ti(real,A). [resolve(1529,a,1531,a)]. 8.93/9.23 1532 -idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),D)) # label(fact_512_dvd__mult__cancel__left) # label(axiom). [clausify(310)]. 8.93/9.23 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,A),C))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),C)). [resolve(1532,a,1530,a)]. 8.93/9.23 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,A),C))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),B),C)). [resolve(1532,a,1531,a)]. 8.93/9.23 1533 -idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D))) | zero_zero(A) != ti(A,B) # label(fact_512_dvd__mult__cancel__left) # label(axiom). [clausify(310)]. 8.93/9.23 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,A),C))) | zero_zero(int) != ti(int,A). [resolve(1533,a,1530,a)]. 8.93/9.23 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,A),C))) | zero_zero(real) != ti(real,A). [resolve(1533,a,1531,a)]. 8.93/9.23 1534 -idom(A) | zero_zero(A) != ti(A,B) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(A,A,times_times(A,C),B)),hAPP(A,A,times_times(A,D),B))) # label(fact_513_dvd__mult__cancel__right) # label(axiom). [clausify(461)]. 8.93/9.23 Derived: zero_zero(int) != ti(int,A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,times_times(int,B),A)),hAPP(int,int,times_times(int,C),A))). [resolve(1534,a,1530,a)]. 8.93/9.23 Derived: zero_zero(real) != ti(real,A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(real,real,times_times(real,B),A)),hAPP(real,real,times_times(real,C),A))). [resolve(1534,a,1531,a)]. 8.93/9.23 1535 -idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),D))) # label(fact_513_dvd__mult__cancel__right) # label(axiom). [clausify(461)]. 8.93/9.23 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,times_times(int,A),C)),hAPP(int,int,times_times(int,B),C))). [resolve(1535,a,1530,a)]. 8.93/9.23 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(real,real,times_times(real,A),C)),hAPP(real,real,times_times(real,B),C))). [resolve(1535,a,1531,a)]. 8.93/9.23 1536 -idom(A) | zero_zero(A) = ti(A,B) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(A,A,times_times(A,C),B)),hAPP(A,A,times_times(A,D),B))) # label(fact_513_dvd__mult__cancel__right) # label(axiom). [clausify(461)]. 8.93/9.23 Derived: zero_zero(int) = ti(int,A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),C)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,times_times(int,B),A)),hAPP(int,int,times_times(int,C),A))). [resolve(1536,a,1530,a)]. 8.93/9.23 Derived: zero_zero(real) = ti(real,A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),B),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(real,real,times_times(real,B),A)),hAPP(real,real,times_times(real,C),A))). [resolve(1536,a,1531,a)]. 9.31/9.60 1537 -one(A) | ti(A,one_one(A)) = one_one(A) # label(tsy_c_Groups_Oone__class_Oone_res) # label(axiom). [clausify(337)]. 9.31/9.60 1538 one(real) # label(arity_RealDef_Oreal___Groups_Oone) # label(axiom). [assumption]. 9.31/9.60 Derived: ti(real,one_one(real)) = one_one(real). [resolve(1537,a,1538,a)]. 9.31/9.60 1539 one(int) # label(arity_Int_Oint___Groups_Oone) # label(axiom). [assumption]. 9.31/9.60 Derived: ti(int,one_one(int)) = one_one(int). [resolve(1539,a,1537,a)]. 9.31/9.60 1540 one(nat) # label(arity_Nat_Onat___Groups_Oone) # label(axiom). [assumption]. 9.31/9.60 Derived: ti(nat,one_one(nat)) = one_one(nat). [resolve(1540,a,1537,a)]. 9.31/9.60 1541 -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_111_crossproduct__noteq) # label(axiom). [clausify(168)]. 9.31/9.60 1542 semiri456707255roduct(int) # label(arity_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc) # label(axiom). [assumption]. 9.31/9.60 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(1541,a,1542,a)]. 9.31/9.60 1543 -semiri456707255roduct(A) | ti(A,B) != ti(A,C) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,D),B)),hAPP(A,A,times_times(A,E),C)) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,D),C)),hAPP(A,A,times_times(A,E),B)) # label(fact_111_crossproduct__noteq) # label(axiom). [clausify(168)]. 9.31/9.60 Derived: ti(int,A) != ti(int,B) | hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,C),A)),hAPP(int,int,times_times(int,D),B)) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,C),B)),hAPP(int,int,times_times(int,D),A)). [resolve(1543,a,1542,a)]. 9.31/9.60 1544 -semiri456707255roduct(A) | ti(A,B) != ti(A,C) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),D)),hAPP(A,A,times_times(A,B),E)) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),E)),hAPP(A,A,times_times(A,B),D)) # label(fact_111_crossproduct__noteq) # label(axiom). [clausify(168)]. 9.31/9.60 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(1544,a,1542,a)]. 9.31/9.60 1545 -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) | ti(A,C) = ti(A,E) # label(fact_108_crossproduct__eq) # label(axiom). [clausify(484)]. 9.31/9.60 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) | ti(int,B) = ti(int,D). [resolve(1545,a,1542,a)]. 9.31/9.60 1546 -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_108_crossproduct__eq) # label(axiom). [clausify(484)]. 9.31/9.60 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(1546,a,1542,a)]. 9.31/9.60 1547 -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_108_crossproduct__eq) # label(axiom). [clausify(484)]. 9.31/9.63 1548 -semiri456707255roduct(A) | zero_zero(A) != ti(A,B) | ti(A,C) = hAPP(A,A,plus_plus(A,C),B) # label(fact_202_add__0__iff) # label(axiom). [clausify(696)]. 9.31/9.63 Derived: zero_zero(int) != ti(int,A) | ti(int,B) = hAPP(int,int,plus_plus(int,B),A). [resolve(1548,a,1542,a)]. 9.31/9.63 1549 -semiri456707255roduct(A) | zero_zero(A) = ti(A,B) | ti(A,C) != hAPP(A,A,plus_plus(A,C),B) # label(fact_202_add__0__iff) # label(axiom). [clausify(696)]. 9.31/9.63 Derived: zero_zero(int) = ti(int,A) | ti(int,B) != hAPP(int,int,plus_plus(int,B),A). [resolve(1549,a,1542,a)]. 9.31/9.63 1550 semiri456707255roduct(nat) # label(arity_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc) # label(axiom). [assumption]. 9.31/9.63 Derived: ti(nat,A) = ti(nat,B) | ti(nat,C) = ti(nat,D) | hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,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(1550,a,1541,a)]. 9.31/9.63 Derived: ti(nat,A) != ti(nat,B) | hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,C),A)),hAPP(nat,nat,times_times(nat,D),B)) = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,C),B)),hAPP(nat,nat,times_times(nat,D),A)). [resolve(1550,a,1543,a)]. 9.31/9.63 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(1550,a,1544,a)]. 9.31/9.63 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) | ti(nat,B) = ti(nat,D). [resolve(1550,a,1545,a)]. 9.31/9.63 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(1550,a,1546,a)]. 9.31/9.63 Derived: zero_zero(nat) != ti(nat,A) | ti(nat,B) = hAPP(nat,nat,plus_plus(nat,B),A). [resolve(1550,a,1548,a)]. 9.31/9.63 Derived: zero_zero(nat) = ti(nat,A) | ti(nat,B) != hAPP(nat,nat,plus_plus(nat,B),A). [resolve(1550,a,1549,a)]. 9.31/9.63 1551 semiri456707255roduct(real) # label(arity_RealDef_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossp) # label(axiom). [assumption]. 9.31/9.63 Derived: ti(real,A) = ti(real,B) | ti(real,C) = ti(real,D) | hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,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(1551,a,1541,a)]. 9.31/9.63 Derived: ti(real,A) != ti(real,B) | hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,C),A)),hAPP(real,real,times_times(real,D),B)) = hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,C),B)),hAPP(real,real,times_times(real,D),A)). [resolve(1551,a,1543,a)]. 9.31/9.63 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(1551,a,1544,a)]. 9.31/9.63 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) | ti(real,B) = ti(real,D). [resolve(1551,a,1545,a)]. 9.31/9.63 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(1551,a,1546,a)]. 9.31/9.63 Derived: zero_zero(real) != ti(real,A) | ti(real,B) = hAPP(real,real,plus_plus(real,B),A). [resolve(1551,a,1548,a)]. 9.31/9.63 Derived: zero_zero(real) = ti(real,A) | ti(real,B) != hAPP(real,real,plus_plus(real,B),A). [resolve(1551,a,1549,a)]. 9.81/10.16 1552 -semiri456707255roduct(A) | zero_zero(A) = ti(A,B) | ti(A,C) = ti(A,D) | ti(A,E) != ti(A,F) | hAPP(A,A,plus_plus(A,E),hAPP(A,A,times_times(A,B),C)) != hAPP(A,A,plus_plus(A,F),hAPP(A,A,times_times(A,B),D)) # label(fact_219_add__scale__eq__noteq) # label(axiom). [clausify(965)]. 9.81/10.16 Derived: zero_zero(int) = ti(int,A) | ti(int,B) = ti(int,C) | ti(int,D) != ti(int,E) | hAPP(int,int,plus_plus(int,D),hAPP(int,int,times_times(int,A),B)) != hAPP(int,int,plus_plus(int,E),hAPP(int,int,times_times(int,A),C)). [resolve(1552,a,1542,a)]. 9.81/10.16 Derived: zero_zero(nat) = ti(nat,A) | ti(nat,B) = ti(nat,C) | ti(nat,D) != ti(nat,E) | hAPP(nat,nat,plus_plus(nat,D),hAPP(nat,nat,times_times(nat,A),B)) != hAPP(nat,nat,plus_plus(nat,E),hAPP(nat,nat,times_times(nat,A),C)). [resolve(1552,a,1550,a)]. 9.81/10.16 Derived: zero_zero(real) = ti(real,A) | ti(real,B) = ti(real,C) | ti(real,D) != ti(real,E) | hAPP(real,real,plus_plus(real,D),hAPP(real,real,times_times(real,A),B)) != hAPP(real,real,plus_plus(real,E),hAPP(real,real,times_times(real,A),C)). [resolve(1552,a,1551,a)]. 9.81/10.16 1553 -ordere1490568538miring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,D),B)),hAPP(A,A,times_times(A,D),C))) # label(fact_477_comm__mult__left__mono) # label(axiom). [clausify(1009)]. 9.81/10.16 1554 ordere1490568538miring(real) # label(arity_RealDef_Oreal___Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 9.81/10.16 1555 ordere1490568538miring(int) # label(arity_Int_Oint___Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 9.81/10.16 1556 ordere1490568538miring(nat) # label(arity_Nat_Onat___Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 9.81/10.16 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(1553,a,1556,a)]. 9.81/10.16 1557 -ring_11004092258visors(A) | zero_zero(A) != ti(A,B) | zero_zero(A) = hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))) # label(fact_252_zero__eq__power2) # label(axiom). [clausify(189)]. 9.81/10.16 1558 ring_11004092258visors(real) # label(arity_RealDef_Oreal___Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 9.81/10.16 Derived: zero_zero(real) != ti(real,A) | zero_zero(real) = hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))). [resolve(1557,a,1558,a)]. 9.81/10.16 1559 -ring_11004092258visors(A) | zero_zero(A) = ti(A,B) | zero_zero(A) != hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))) # label(fact_252_zero__eq__power2) # label(axiom). [clausify(189)]. 9.81/10.16 Derived: zero_zero(real) = ti(real,A) | zero_zero(real) != hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))). [resolve(1559,a,1558,a)]. 9.81/10.16 1560 ring_11004092258visors(int) # label(arity_Int_Oint___Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 9.81/10.16 Derived: zero_zero(int) != ti(int,A) | zero_zero(int) = hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))). [resolve(1560,a,1557,a)]. 9.81/10.16 Derived: zero_zero(int) = ti(int,A) | zero_zero(int) != hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))). [resolve(1560,a,1559,a)]. 9.81/10.16 1561 -ring_11004092258visors(A) | zero_zero(A) = ti(A,B) | zero_zero(A) != hAPP(nat,A,power_power(A,B),C) # label(fact_194_field__power__not__zero) # label(axiom). [clausify(928)]. 9.81/10.16 Derived: zero_zero(real) = ti(real,A) | zero_zero(real) != hAPP(nat,real,power_power(real,A),B). [resolve(1561,a,1558,a)]. 9.81/10.16 Derived: zero_zero(int) = ti(int,A) | zero_zero(int) != hAPP(nat,int,power_power(int,A),B). [resolve(1561,a,1560,a)]. 9.81/10.16 1562 -linord1278240602ring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),F)) | one_one(A) != hAPP(A,A,plus_plus(A,E),F) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,E),B)),hAPP(A,A,times_times(A,F),D))),C)) # label(fact_548_convex__bound__le) # label(axiom). [clausify(520)]. 10.41/10.71 1563 linord1278240602ring_1(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring__1) # label(axiom). [assumption]. 10.41/10.71 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(1562,a,1563,a)]. 10.41/10.71 1564 linord1278240602ring_1(int) # label(arity_Int_Oint___Rings_Olinordered__semiring__1) # label(axiom). [assumption]. 10.41/10.71 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(1564,a,1562,a)]. 10.41/10.71 1565 -cancel146912293up_add(A) | hAPP(A,A,plus_plus(A,B),C) != hAPP(A,A,plus_plus(A,B),D) | ti(A,C) = ti(A,D) # label(fact_762_add__imp__eq) # label(axiom). [clausify(244)]. 10.41/10.71 1566 cancel146912293up_add(int) # label(arity_Int_Oint___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 10.41/10.71 1567 cancel146912293up_add(nat) # label(arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 10.41/10.71 1568 cancel146912293up_add(real) # label(arity_RealDef_Oreal___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 10.41/10.71 1569 -linord893533164strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,D),B)),hAPP(A,A,times_times(A,D),C))) # label(fact_490_comm__mult__strict__left__mono) # label(axiom). [clausify(404)]. 10.41/10.71 1570 linord893533164strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 10.41/10.71 1571 linord893533164strict(int) # label(arity_Int_Oint___Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 10.41/10.71 1572 linord893533164strict(nat) # label(arity_Nat_Onat___Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 10.41/10.71 1573 -ordered_ab_group_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,minus_minus(A,B),C)),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_810_le__iff__diff__le__0) # label(axiom). [clausify(360)]. 10.41/10.71 1574 ordered_ab_group_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__group__add) # label(axiom). [assumption]. 10.41/10.71 1575 -ordered_ab_group_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,minus_minus(A,B),C)),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_810_le__iff__diff__le__0) # label(axiom). [clausify(360)]. 10.41/10.71 1576 -ordered_ab_group_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,minus_minus(A,B),C)),zero_zero(A))) # label(fact_811_less__iff__diff__less__0) # label(axiom). [clausify(450)]. 10.41/10.71 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,minus_minus(real,A),B)),zero_zero(real))). [resolve(1576,a,1574,a)]. 10.41/10.76 1577 -ordered_ab_group_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,minus_minus(A,B),C)),zero_zero(A))) # label(fact_811_less__iff__diff__less__0) # label(axiom). [clausify(450)]. 10.41/10.76 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,minus_minus(real,A),B)),zero_zero(real))). [resolve(1577,a,1574,a)]. 10.41/10.76 1578 ordered_ab_group_add(int) # label(arity_Int_Oint___Groups_Oordered__ab__group__add) # label(axiom). [assumption]. 10.41/10.76 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,minus_minus(int,A),B)),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(1578,a,1573,a)]. 10.41/10.76 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,minus_minus(int,A),B)),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(1578,a,1575,a)]. 10.41/10.76 1579 -ordered_ab_group_add(A) | hAPP(A,A,minus_minus(A,B),C) != hAPP(A,A,minus_minus(A,D),E) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) # label(fact_790_diff__eq__diff__less) # label(axiom). [clausify(982)]. 10.41/10.76 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(1579,a,1574,a)]. 10.41/10.76 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(1579,a,1578,a)]. 10.41/10.76 1580 -ordered_ab_group_add(A) | hAPP(A,A,minus_minus(A,B),C) != hAPP(A,A,minus_minus(A,D),E) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) # label(fact_790_diff__eq__diff__less) # label(axiom). [clausify(982)]. 10.41/10.76 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(1580,a,1574,a)]. 10.41/10.76 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(1580,a,1578,a)]. 10.41/10.76 1581 -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),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),E)) # label(fact_789_diff__eq__diff__less__eq) # label(axiom). [clausify(1030)]. 10.41/10.76 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),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),D)). [resolve(1581,a,1574,a)]. 10.41/10.76 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),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),D)). [resolve(1581,a,1578,a)]. 10.41/10.76 1582 -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),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),E)) # label(fact_789_diff__eq__diff__less__eq) # label(axiom). [clausify(1030)]. 10.41/10.76 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),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),D)). [resolve(1582,a,1574,a)]. 10.92/11.27 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),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),D)). [resolve(1582,a,1578,a)]. 10.92/11.27 1583 ordere779506340up_add(nat) # label(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 10.92/11.27 1584 -ordere779506340up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,B),D)),hAPP(A,A,plus_plus(A,C),E))) # label(fact_775_add__mono) # label(axiom). [clausify(335)]. 10.92/11.27 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,A),C)),hAPP(nat,nat,plus_plus(nat,B),D))). [resolve(1583,a,1584,a)]. 10.92/11.27 1585 ordere779506340up_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 10.92/11.27 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(1585,a,1584,a)]. 10.92/11.27 1586 ordere779506340up_add(int) # label(arity_Int_Oint___Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 10.92/11.27 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(1586,a,1584,a)]. 10.92/11.27 1587 -ordere779506340up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,D),B)),hAPP(A,A,plus_plus(A,D),C))) # label(fact_774_add__left__mono) # label(axiom). [clausify(762)]. 10.92/11.27 1588 -ordere779506340up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,B),D)),hAPP(A,A,plus_plus(A,C),D))) # label(fact_773_add__right__mono) # label(axiom). [clausify(959)]. 10.92/11.27 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(1588,a,1583,a)]. 10.92/11.27 1589 comm_ring_1(int) # label(arity_Int_Oint___Rings_Ocomm__ring__1) # label(axiom). [assumption]. 10.92/11.27 1590 -comm_ring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,minus_minus(A,C),D))) # label(fact_461_dvd__diff) # label(axiom). [clausify(336)]. 10.92/11.27 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(1589,a,1590,a)]. 10.92/11.27 1591 comm_ring_1(real) # label(arity_RealDef_Oreal___Rings_Ocomm__ring__1) # label(axiom). [assumption]. 10.92/11.27 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(1591,a,1590,a)]. 10.92/11.27 1592 ring_1(int) # label(arity_Int_Oint___Rings_Oring__1) # label(axiom). [assumption]. 10.92/11.27 1593 -ring_1(A) | hAPP(A,A,minus_minus(A,hAPP(A,A,times_times(A,B),B)),one_one(A)) = hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),one_one(A))),hAPP(A,A,minus_minus(A,B),one_one(A))) # label(fact_547_real__squared__diff__one__factored) # label(axiom). [clausify(357)]. 10.92/11.27 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(1592,a,1593,a)]. 11.14/11.48 1594 ring_1(real) # label(arity_RealDef_Oreal___Rings_Oring__1) # label(axiom). [assumption]. 11.14/11.48 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(1594,a,1593,a)]. 11.14/11.48 1595 number_semiring(int) # label(arity_Int_Oint___Int_Onumber__semiring) # label(axiom). [assumption]. 11.14/11.48 1596 -number_semiring(A) | hAPP(A,A,plus_plus(A,B),B) = hAPP(A,A,times_times(A,B),number_number_of(A,bit0(bit1(pls)))) # label(fact_167_semiring__mult__2__right) # label(axiom). [clausify(414)]. 11.14/11.48 1597 -number_semiring(A) | hAPP(A,A,plus_plus(A,B),B) = hAPP(A,A,times_times(A,number_number_of(A,bit0(bit1(pls)))),B) # label(fact_165_semiring__mult__2) # label(axiom). [clausify(496)]. 11.14/11.48 1598 -number_semiring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),C)) | number_number_of(A,hAPP(int,int,plus_plus(int,B),C)) = hAPP(A,A,plus_plus(A,number_number_of(A,B)),number_number_of(A,C)) # label(fact_136_semiring__add__number__of) # label(axiom). [clausify(513)]. 11.14/11.48 1599 -number_semiring(A) | number_number_of(A,bit0(bit1(pls))) = hAPP(A,A,plus_plus(A,one_one(A)),one_one(A)) # label(fact_169_semiring__one__add__one__is__two) # label(axiom). [clausify(597)]. 11.14/11.48 1600 -number_semiring(A) | 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)) = hAPP(nat,A,power_power(A,hAPP(A,A,plus_plus(A,B),C)),number_number_of(nat,bit0(bit1(pls)))) # label(fact_9_power2__sum) # label(axiom). [clausify(729)]. 11.14/11.48 1601 -number_semiring(A) | zero_zero(A) = number_number_of(A,pls) # label(fact_223_semiring__numeral__0__eq__0) # label(axiom). [clausify(739)]. 11.14/11.48 1602 -number_semiring(A) | one_one(A) = number_number_of(A,bit1(pls)) # label(fact_157_semiring__numeral__1__eq__1) # label(axiom). [clausify(793)]. 11.14/11.48 Derived: hAPP(int,int,plus_plus(int,A),A) = hAPP(int,int,times_times(int,A),number_number_of(int,bit0(bit1(pls)))). [resolve(1595,a,1596,a)]. 11.14/11.48 Derived: hAPP(int,int,plus_plus(int,A),A) = hAPP(int,int,times_times(int,number_number_of(int,bit0(bit1(pls)))),A). [resolve(1595,a,1597,a)]. 11.14/11.48 Derived: number_number_of(int,bit0(bit1(pls))) = hAPP(int,int,plus_plus(int,one_one(int)),one_one(int)). [resolve(1595,a,1599,a)]. 11.14/11.48 Derived: 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)) = hAPP(nat,int,power_power(int,hAPP(int,int,plus_plus(int,A),B)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1595,a,1600,a)]. 11.14/11.48 Derived: zero_zero(int) = number_number_of(int,pls). [resolve(1595,a,1601,a)]. 11.14/11.48 Derived: one_one(int) = number_number_of(int,bit1(pls)). [resolve(1595,a,1602,a)]. 11.14/11.48 1603 number_semiring(nat) # label(arity_Nat_Onat___Int_Onumber__semiring) # label(axiom). [assumption]. 11.14/11.48 Derived: hAPP(nat,nat,plus_plus(nat,A),A) = hAPP(nat,nat,times_times(nat,A),number_number_of(nat,bit0(bit1(pls)))). [resolve(1603,a,1596,a)]. 11.14/11.48 Derived: hAPP(nat,nat,plus_plus(nat,A),A) = hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),A). [resolve(1603,a,1597,a)]. 11.14/11.48 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(1603,a,1598,a)]. 11.29/11.65 Derived: number_number_of(nat,bit0(bit1(pls))) = hAPP(nat,nat,plus_plus(nat,one_one(nat)),one_one(nat)). [resolve(1603,a,1599,a)]. 11.29/11.65 Derived: 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)) = hAPP(nat,nat,power_power(nat,hAPP(nat,nat,plus_plus(nat,A),B)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1603,a,1600,a)]. 11.29/11.65 Derived: zero_zero(nat) = number_number_of(nat,pls). [resolve(1603,a,1601,a)]. 11.29/11.65 Derived: one_one(nat) = number_number_of(nat,bit1(pls)). [resolve(1603,a,1602,a)]. 11.29/11.65 1604 -number_semiring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),C)) | number_number_of(A,hAPP(int,int,times_times(int,B),C)) = hAPP(A,A,times_times(A,number_number_of(A,B)),number_number_of(A,C)) # label(fact_135_semiring__mult__number__of) # label(axiom). [clausify(902)]. 11.29/11.65 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(1604,a,1595,a)]. 11.29/11.65 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(1604,a,1603,a)]. 11.29/11.65 1605 number_semiring(real) # label(arity_RealDef_Oreal___Int_Onumber__semiring) # label(axiom). [assumption]. 11.29/11.65 Derived: hAPP(real,real,plus_plus(real,A),A) = hAPP(real,real,times_times(real,A),number_number_of(real,bit0(bit1(pls)))). [resolve(1605,a,1596,a)]. 11.29/11.65 Derived: hAPP(real,real,plus_plus(real,A),A) = hAPP(real,real,times_times(real,number_number_of(real,bit0(bit1(pls)))),A). [resolve(1605,a,1597,a)]. 11.29/11.65 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(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(1605,a,1598,a)]. 11.29/11.65 Derived: number_number_of(real,bit0(bit1(pls))) = hAPP(real,real,plus_plus(real,one_one(real)),one_one(real)). [resolve(1605,a,1599,a)]. 11.29/11.65 Derived: 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)) = hAPP(nat,real,power_power(real,hAPP(real,real,plus_plus(real,A),B)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1605,a,1600,a)]. 11.29/11.65 Derived: zero_zero(real) = number_number_of(real,pls). [resolve(1605,a,1601,a)]. 11.29/11.65 Derived: one_one(real) = number_number_of(real,bit1(pls)). [resolve(1605,a,1602,a)]. 11.29/11.65 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(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(1605,a,1604,a)]. 11.29/11.65 1606 -zero(A) | zero_zero(A) = ti(A,zero_zero(A)) # label(tsy_c_Groups_Ozero__class_Ozero_res) # label(axiom). [clausify(776)]. 11.29/11.65 1607 zero(int) # label(arity_Int_Oint___Groups_Ozero) # label(axiom). [assumption]. 11.29/11.65 1608 zero(nat) # label(arity_Nat_Onat___Groups_Ozero) # label(axiom). [assumption]. 11.29/11.65 1609 zero(real) # label(arity_RealDef_Oreal___Groups_Ozero) # label(axiom). [assumption]. 11.29/11.65 Derived: zero_zero(int) = ti(int,zero_zero(int)). [resolve(1606,a,1607,a)]. 11.29/11.65 Derived: zero_zero(nat) = ti(nat,zero_zero(nat)). [resolve(1606,a,1608,a)]. 14.18/14.48 Derived: zero_zero(real) = ti(real,zero_zero(real)). [resolve(1606,a,1609,a)]. 14.18/14.48 1610 comm_monoid_mult(real) # label(arity_RealDef_Oreal___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 14.18/14.48 1611 -comm_monoid_mult(A) | hAPP(A,A,times_times(A,hAPP(nat,A,power_power(A,B),C)),hAPP(nat,A,power_power(A,D),C)) = hAPP(nat,A,power_power(A,hAPP(A,A,times_times(A,B),D)),C) # label(fact_260_power__mult__distrib) # label(axiom). [clausify(529)]. 14.18/14.48 1612 comm_monoid_mult(int) # label(arity_Int_Oint___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 14.18/14.48 1613 -comm_monoid_mult(A) | ti(A,B) = hAPP(A,A,times_times(A,one_one(A)),B) # label(fact_792_mult__1) # label(axiom). [clausify(749)]. 14.18/14.48 1614 comm_monoid_mult(nat) # label(arity_Nat_Onat___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 14.18/14.48 1615 -comm_monoid_mult(A) | ti(A,B) = hAPP(A,A,times_times(A,B),one_one(A)) # label(fact_794_mult_Ocomm__neutral) # label(axiom). [clausify(1036)]. 14.18/14.48 1616 -comm_semiring(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C)) = hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),D)),C) # label(fact_448_comm__semiring__class_Odistrib) # label(axiom). [clausify(932)]. 14.18/14.48 1617 comm_semiring(int) # label(arity_Int_Oint___Rings_Ocomm__semiring) # label(axiom). [assumption]. 14.18/14.48 1618 comm_semiring(nat) # label(arity_Nat_Onat___Rings_Ocomm__semiring) # label(axiom). [assumption]. 14.18/14.48 1619 comm_semiring(real) # label(arity_RealDef_Oreal___Rings_Ocomm__semiring) # label(axiom). [assumption]. 14.18/14.48 14.18/14.48 ============================== end predicate elimination ============= 14.18/14.48 14.18/14.48 Auto_denials: (non-Horn, no changes). 14.18/14.48 14.18/14.48 Term ordering decisions: 14.18/14.48 Function symbol KB weights: int=1. nat=1. bool=1. real=1. pls=1. min=1. zprime=1. m=1. s=1. t=1. s1=1. twoSqu658283162sum2sq=1. fconj=1. c1=1. c2=1. c3=1. c4=1. c5=1. c6=1. c7=1. c8=1. fun=1. times_times=1. plus_plus=1. ti=1. number_number_of=1. power_power=1. minus_minus=1. div_mod=1. zcong=1. multInv=1. member=1. standardRes=1. wset=1. inv=1. legendre=1. collect=1. product_prod=1. f2=1. f3=1. f9=1. f11=1. f12=1. f21=1. f23=1. f27=1. f28=1. f29=1. f31=1. f35=1. zero_zero=1. ord_less=1. ord_less_eq=1. one_one=1. dvd_dvd=1. bit1=1. bit0=1. quadRes=1. twoSqu1929807760sum2sq=1. d22set=1. zfact=1. sr=1. undefined=1. f10=1. f17=1. f18=1. f22=1. f1=1. f4=1. f5=1. f7=1. f8=1. f13=1. f14=1. f15=1. f16=1. f20=1. f24=1. f30=1. f32=1. f33=1. f34=1. f36=1. f37=1. f38=1. f39=1. hAPP=1. product_Pair=1. combb=1. combc=1. combs=1. 14.18/14.48 14.18/14.48 ============================== end of process initial clauses ======== 14.18/14.48 14.18/14.48 ============================== CLAUSES FOR SEARCH ==================== 14.18/14.48 14.18/14.48 ============================== end of clauses for search ============= 14.18/14.48 14.18/14.48 ============================== SEARCH ================================ 14.18/14.48 14.18/14.48 % Starting search at 10.09 seconds. 14.18/14.48 14.18/14.48 Low Water (keep): wt=97.000, iters=3404 14.18/14.48 14.18/14.48 Low Water (keep): wt=95.000, iters=3389 14.18/14.48 14.18/14.48 Low Water (keep): wt=94.000, iters=3369 14.18/14.48 14.18/14.48 Low Water (keep): wt=90.000, iters=3341 14.18/14.48 14.18/14.48 Low Water (keep): wt=89.000, iters=3333 14.18/14.48 14.18/14.48 Low Water (keep): wt=87.000, iters=3378 14.18/14.48 14.18/14.48 Low Water (keep): wt=86.000, iters=3349 14.18/14.48 14.18/14.48 Low Water (keep): wt=84.000, iters=3349 14.18/14.48 14.18/14.48 Low Water (keep): wt=83.000, iters=3353 14.18/14.48 14.18/14.48 Low Water (keep): wt=82.000, iters=3392 14.18/14.48 14.18/14.48 Low Water (keep): wt=81.000, iters=3359 14.18/14.48 14.18/14.48 Low Water (keep): wt=77.000, iters=3374 14.18/14.48 14.18/14.48 Low Water (keep): wt=76.000, iters=3374 14.18/14.48 14.18/14.48 Low Water (keep): wt=74.000, iters=3339 14.18/14.48 14.18/14.48 Low Water (keep): wt=72.000, iters=3341 14.18/14.48 14.18/14.48 Low Water (keep): wt=71.000, iters=3395 14.18/14.48 14.18/14.48 Low Water (keep): wt=70.000, iters=3367 14.18/14.48 14.18/14.48 Low Water (keep): wt=69.000, iters=3337 14.18/14.48 14.18/14.48 Low Water (keep): wt=67.000, iters=3373 14.18/14.48 14.18/14.48 Low Water (keep): wt=66.000, iters=3431 14.18/14.48 14.18/14.48 Low Water (keep): wt=64.000, iters=3353 14.18/14.48 14.18/14.48 Low Water (keep): wt=63.000, iters=3347 14.18/14.48 14.18/14.48 Low Water (keep): wt=62.000, iters=3349 14.18/14.48 14.18/14.48 Low Water (keep): wt=61.000, iters=3347 14.18/14.48 14.18/14.48 Low Water (keep): wt=60.000, iters=3431 14.18/14.48 14.18/14.48 Low Water (keep): wt=58.000, iters=3489 14.18/14.48 14.18/14.48 Low Water (keep): wt=57.000, iters=3512 14.18/14.48 14.18/14.48 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 43 (0.00 of 11.80 sec). 14.18/14.48 14.18/14.48 Low Water (keep): wt=56.000, iters=3490 17.90/18.23 17.90/18.23 Low Water (keep): wt=55.000, iters=3422 17.90/18.23 17.90/18.23 Low Water (keep): wt=54.000, iters=3359 17.90/18.23 17.90/18.23 Low Water (keep): wt=53.000, iters=3348 17.90/18.23 17.90/18.23 Low Water (keep): wt=52.000, iters=3349 17.90/18.23 17.90/18.23 Low Water (keep): wt=51.000, iters=3354 17.90/18.23 17.90/18.23 Low Water (keep): wt=50.000, iters=3334 17.90/18.23 17.90/18.23 Low Water (keep): wt=49.000, iters=3393 17.90/18.23 17.90/18.23 Low Water (keep): wt=48.000, iters=3415 17.90/18.23 17.90/18.23 Low Water (keep): wt=47.000, iters=3366 17.90/18.23 17.90/18.23 Low Water (keep): wt=46.000, iters=3350 17.90/18.23 17.90/18.23 Low Water (keep): wt=45.000, iters=3339 17.90/18.23 17.90/18.23 Low Water (keep): wt=44.000, iters=3447 17.90/18.23 17.90/18.23 Low Water (keep): wt=43.000, iters=3333 17.90/18.23 17.90/18.23 Low Water (keep): wt=42.000, iters=3459 17.90/18.23 17.90/18.23 Low Water (keep): wt=41.000, iters=3334 17.90/18.23 17.90/18.23 Low Water (keep): wt=40.000, iters=3375 17.90/18.23 17.90/18.23 Low Water (keep): wt=39.000, iters=3366 17.90/18.23 17.90/18.23 Low Water (keep): wt=38.000, iters=3335 17.90/18.23 17.90/18.23 Low Water (keep): wt=37.000, iters=3379 17.90/18.23 17.90/18.23 Low Water (keep): wt=36.000, iters=3351 17.90/18.23 17.90/18.23 Low Water (keep): wt=35.000, iters=3352 17.90/18.23 17.90/18.23 Low Water (keep): wt=34.000, iters=3349 17.90/18.23 17.90/18.23 Low Water (keep): wt=33.000, iters=3341 17.90/18.23 17.90/18.23 ============================== PROOF ================================= 17.90/18.23 % SZS status Theorem 17.90/18.23 % SZS output start Refutation 17.90/18.23 17.90/18.23 % Proof 1 at 15.47 (+ 0.11) seconds. 17.90/18.23 % Length of proof is 62. 17.90/18.23 % Level of proof is 7. 17.90/18.23 % Maximum clause weight is 90.000. 17.90/18.23 % Given clauses 1639. 17.90/18.23 17.90/18.23 55 (all K_1 bit1(K_1) = hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,one_one(int)),K_1)),K_1)) # label(fact_153_Bit1__def) # label(axiom) # label(non_clause). [assumption]. 17.90/18.23 77 (all Z_1 ti(int,Z_1) = hAPP(int,int,plus_plus(int,Z_1),zero_zero(int))) # label(fact_207_zadd__0__right) # label(axiom) # label(non_clause). [assumption]. 17.90/18.23 108 t = one_one(int) -> (exists X exists Y hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,X),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,Y),number_number_of(nat,bit0(bit1(pls))))) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int))) # label(fact_1__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06) # label(axiom) # label(non_clause). [assumption]. 17.90/18.23 337 (all X_a (one(X_a) -> ti(X_a,one_one(X_a)) = one_one(X_a))) # label(tsy_c_Groups_Oone__class_Oone_res) # label(axiom) # label(non_clause). [assumption]. 17.90/18.23 373 (all K_1 hAPP(int,int,plus_plus(int,K_1),K_1) = bit0(K_1)) # label(fact_128_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 17.90/18.23 472 (all X_c all X_a all B_1_1 all B_2_1 ti(X_c,hAPP(X_a,X_c,B_1_1,B_2_1)) = hAPP(X_a,X_c,B_1_1,B_2_1)) # label(tsy_c_hAPP_res) # label(axiom) # label(non_clause). [assumption]. 17.90/18.23 574 (all Z1 all Z2 all Z3 hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,Z1),Z2)),Z3) = hAPP(int,int,plus_plus(int,Z1),hAPP(int,int,plus_plus(int,Z2),Z3))) # label(fact_89_zadd__assoc) # label(axiom) # label(non_clause). [assumption]. 17.90/18.23 597 (all X_a (number_semiring(X_a) -> hAPP(X_a,X_a,plus_plus(X_a,one_one(X_a)),one_one(X_a)) = number_number_of(X_a,bit0(bit1(pls))))) # label(fact_169_semiring__one__add__one__is__two) # label(axiom) # label(non_clause). [assumption]. 17.90/18.23 658 (all T all A ti(T,ti(T,A)) = ti(T,A)) # label(help_ti_idem) # label(axiom) # label(non_clause). [assumption]. 17.90/18.23 700 (all K_1 ti(int,K_1) = number_number_of(int,K_1)) # label(fact_88_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 17.90/18.23 831 (all X_a all X_c all B_1_1 all B_2_1 hAPP(X_a,X_c,B_1_1,ti(X_a,B_2_1)) = hAPP(X_a,X_c,B_1_1,B_2_1)) # label(tsy_c_hAPP_arg2) # label(axiom) # label(non_clause). [assumption]. 17.90/18.23 877 (all V all W hAPP(int,int,plus_plus(int,number_number_of(int,V)),number_number_of(int,W)) = number_number_of(int,hAPP(int,int,plus_plus(int,V),W))) # label(fact_134_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption]. 17.90/18.23 917 (all Z_2 all W_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_2),W_1)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_2),W_1)) & ti(int,W_1) != ti(int,Z_2))) # label(fact_22_zless__le) # label(axiom) # label(non_clause). [assumption]. 17.90/18.23 931 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),t)) -> (exists X exists Y hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,X),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,Y),number_number_of(nat,bit0(bit1(pls))))) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int))) # label(fact_2__0961_A_060_At_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06) # label(axiom) # label(non_clause). [assumption]. 17.90/18.24 1043 -(exists X exists Y hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,X),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,Y),number_number_of(nat,bit0(bit1(pls))))) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int))) # label(conj_0) # label(negated_conjecture) # label(non_clause). [assumption]. 17.90/18.24 1537 -one(A) | ti(A,one_one(A)) = one_one(A) # label(tsy_c_Groups_Oone__class_Oone_res) # label(axiom). [clausify(337)]. 17.90/18.24 1539 one(int) # label(arity_Int_Oint___Groups_Oone) # label(axiom). [assumption]. 17.90/18.24 1599 -number_semiring(A) | number_number_of(A,bit0(bit1(pls))) = hAPP(A,A,plus_plus(A,one_one(A)),one_one(A)) # label(fact_169_semiring__one__add__one__is__two) # label(axiom). [clausify(597)]. 17.90/18.24 1603 number_semiring(nat) # label(arity_Nat_Onat___Int_Onumber__semiring) # label(axiom). [assumption]. 17.90/18.24 1678 bit1(A) = hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,one_one(int)),A)),A) # label(fact_153_Bit1__def) # label(axiom). [clausify(55)]. 17.90/18.24 1713 ti(int,A) = hAPP(int,int,plus_plus(int,A),zero_zero(int)) # label(fact_207_zadd__0__right) # label(axiom). [clausify(77)]. 17.90/18.24 1714 hAPP(int,int,plus_plus(int,A),zero_zero(int)) = ti(int,A). [copy(1713),flip(a)]. 17.90/18.24 1760 one_one(int) != t | hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,c1),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,c2),number_number_of(nat,bit0(bit1(pls))))) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int)) # label(fact_1__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06) # label(axiom). [clausify(108)]. 17.90/18.24 1761 one_one(int) != t | hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,c1),number_number_of(nat,bit0(hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,one_one(int)),pls)),pls))))),hAPP(nat,int,power_power(int,c2),number_number_of(nat,bit0(hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,one_one(int)),pls)),pls))))) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,one_one(int)),pls)),pls))))),m)),one_one(int)). [copy(1760),rewrite([1678(15),1678(39),1678(64)])]. 17.90/18.24 1764 zero_zero(int) = pls # label(fact_204_Pls__def) # label(axiom). [assumption]. 17.90/18.24 1765 pls = zero_zero(int). [copy(1764),flip(a)]. 17.90/18.24 1789 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),t)) # label(fact_0_tpos) # label(axiom). [assumption]. 17.90/18.24 2014 number_number_of(int,bit1(pls)) = one_one(int) # label(fact_160_one__is__num__one) # label(axiom). [assumption]. 17.90/18.24 2015 number_number_of(int,ti(int,ti(int,one_one(int)))) = one_one(int). [copy(2014),rewrite([1765(2),1678(4),1714(13),1714(12)])]. 17.90/18.24 2093 zero_zero(int) != one_one(int) # label(fact_205_int__0__neq__1) # label(axiom). [assumption]. 17.90/18.24 2094 one_one(int) != zero_zero(int). [copy(2093),flip(a)]. 17.90/18.24 2099 bit0(A) = hAPP(int,int,plus_plus(int,A),A) # label(fact_128_Bit0__def) # label(axiom). [clausify(373)]. 17.90/18.24 2308 ti(A,hAPP(B,A,C,D)) = hAPP(B,A,C,D) # label(tsy_c_hAPP_res) # label(axiom). [clausify(472)]. 17.90/18.24 2414 hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,A),B)),C) = hAPP(int,int,plus_plus(int,A),hAPP(int,int,plus_plus(int,B),C)) # label(fact_89_zadd__assoc) # label(axiom). [clausify(574)]. 17.90/18.24 2521 ti(A,ti(A,B)) = ti(A,B) # label(help_ti_idem) # label(axiom). [clausify(658)]. 17.90/18.24 2585 ti(int,A) = number_number_of(int,A) # label(fact_88_number__of__is__id) # label(axiom). [clausify(700)]. 17.90/18.24 2586 number_number_of(int,A) = ti(int,A). [copy(2585),flip(a)]. 17.90/18.24 2734 hAPP(A,B,C,ti(A,D)) = hAPP(A,B,C,D) # label(tsy_c_hAPP_arg2) # label(axiom). [clausify(831)]. 17.90/18.24 2737 number_number_of(nat,bit0(bit1(pls))) = hAPP(nat,nat,plus_plus(nat,one_one(nat)),one_one(nat)) # label(fact_37_nat__1__add__1) # label(axiom). [assumption]. 17.90/18.24 2738 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(2737),rewrite([1765(2),1678(4),1714(13),1714(12),2521(7),2099(6),2734(14)])]. 17.90/18.24 2805 number_number_of(int,hAPP(int,int,plus_plus(int,A),B)) = hAPP(int,int,plus_plus(int,number_number_of(int,A)),number_number_of(int,B)) # label(fact_134_plus__numeral__code_I9_J) # label(axiom). [clausify(877)]. 17.90/18.24 2806 hAPP(int,int,plus_plus(int,ti(int,A)),B) = hAPP(int,int,plus_plus(int,A),B). [copy(2805),rewrite([2586(7),2308(7),2586(10),2586(13),2734(14)]),flip(a)]. 17.90/18.24 2844 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),A),B)) | ti(int,A) = ti(int,B) # label(fact_22_zless__le) # label(axiom). [clausify(917)]. 17.90/18.24 2858 -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),t)) | hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,c6),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,c7),number_number_of(nat,bit0(bit1(pls))))) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int)) # label(fact_2__0961_A_060_At_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06) # label(axiom). [clausify(931)]. 17.90/18.24 2859 -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),t)) | hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,c6),number_number_of(nat,hAPP(int,int,plus_plus(int,one_one(int)),one_one(int))))),hAPP(nat,int,power_power(int,c7),number_number_of(nat,hAPP(int,int,plus_plus(int,one_one(int)),one_one(int))))) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,one_one(int)),hAPP(int,int,plus_plus(int,one_one(int)),hAPP(int,int,plus_plus(int,one_one(int)),one_one(int))))),m)),one_one(int)). [copy(2858),rewrite([1765(24),1678(26),1714(35),1714(34),2521(29),2099(28),2734(36),2806(34),1765(42),1678(44),1714(53),1714(52),2521(47),2099(46),2734(54),2806(52),1765(61),1678(63),1714(72),1714(71),2521(66),2099(65),2734(73),2806(71),2099(70),2414(83),2586(84),2308(84)])]. 17.90/18.24 2937 t = ti(int,t) # label(tsy_v_t_____res) # label(axiom). [assumption]. 17.90/18.24 2938 ti(int,t) = t. [copy(2937),flip(a)]. 17.90/18.24 2999 hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,B),number_number_of(nat,bit0(bit1(pls))))) != hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int)) # label(conj_0) # label(negated_conjecture). [clausify(1043)]. 17.90/18.24 3000 hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,A),number_number_of(nat,hAPP(int,int,plus_plus(int,one_one(int)),one_one(int))))),hAPP(nat,int,power_power(int,B),number_number_of(nat,hAPP(int,int,plus_plus(int,one_one(int)),one_one(int))))) != hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,one_one(int)),hAPP(int,int,plus_plus(int,one_one(int)),hAPP(int,int,plus_plus(int,one_one(int)),one_one(int))))),m)),one_one(int)). [copy(2999),rewrite([1765(9),1678(11),1714(20),1714(19),2521(14),2099(13),2734(21),2806(19),1765(26),1678(28),1714(37),1714(36),2521(31),2099(30),2734(38),2806(36),1765(45),1678(47),1714(56),1714(55),2521(50),2099(49),2734(57),2806(55),2099(54),2414(67),2586(68),2308(68)])]. 17.90/18.24 3947 ti(int,one_one(int)) = one_one(int). [resolve(1539,a,1537,a)]. 17.90/18.24 4038 number_number_of(nat,bit0(bit1(pls))) = hAPP(nat,nat,plus_plus(nat,one_one(nat)),one_one(nat)). [resolve(1603,a,1599,a)]. 17.90/18.24 4039 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)). [copy(4038),rewrite([1765(2),1678(4),1714(13),3947(8),1714(10),3947(5),2099(4)])]. 17.90/18.24 4074 one_one(int) != t | hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,one_one(int)),hAPP(int,int,plus_plus(int,one_one(int)),hAPP(int,int,plus_plus(int,one_one(int)),one_one(int))))),m)),one_one(int)) = hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,c1),hAPP(nat,nat,plus_plus(nat,one_one(nat)),one_one(nat)))),hAPP(nat,int,power_power(int,c2),hAPP(nat,nat,plus_plus(nat,one_one(nat)),one_one(nat)))). [back_rewrite(1761),rewrite([1765(23),1714(25),3947(20),1765(20),1714(22),3947(17),2099(16),4039(23),1765(39),1714(41),3947(36),1765(36),1714(38),3947(33),2099(32),4039(39),1765(56),1714(58),3947(53),1765(53),1714(55),3947(50),2099(49),2099(56),2414(69),2586(70),2308(70)]),flip(b)]. 17.90/18.24 4096 one_one(int) = c_0. [new_symbol(2094)]. 17.90/18.24 4200 ti(int,c_0) = c_0. [back_rewrite(2015),rewrite([4096(5),2521(6),2586(5),2521(5),4096(5)])]. 17.90/18.24 4235 hAPP(nat,nat,plus_plus(nat,one_one(nat)),one_one(nat)) = number_number_of(nat,hAPP(int,int,plus_plus(int,c_0),c_0)). [back_rewrite(2738),rewrite([4096(7),4200(7),4096(8)]),flip(a)]. 17.90/18.24 4249 -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),c_0),t)) | hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,c6),number_number_of(nat,hAPP(int,int,plus_plus(int,c_0),c_0)))),hAPP(nat,int,power_power(int,c7),number_number_of(nat,hAPP(int,int,plus_plus(int,c_0),c_0)))) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,c_0),hAPP(int,int,plus_plus(int,c_0),hAPP(int,int,plus_plus(int,c_0),c_0)))),m)),c_0). [back_unit_del(2859),rewrite([4096(10),4096(27),4096(29),4096(43),4096(45),4096(59),4096(64),4096(69),4096(71),4096(79)])]. 17.90/18.24 4410 hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,A),number_number_of(nat,hAPP(int,int,plus_plus(int,c_0),c_0)))),hAPP(nat,int,power_power(int,B),number_number_of(nat,hAPP(int,int,plus_plus(int,c_0),c_0)))) != hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,c_0),hAPP(int,int,plus_plus(int,c_0),hAPP(int,int,plus_plus(int,c_0),c_0)))),m)),c_0). [back_rewrite(3000),rewrite([4096(13),4096(15),4096(28),4096(30),4096(44),4096(49),4096(54),4096(56),4096(64)])]. 17.90/18.24 4429 c_0 != t. [back_rewrite(4074),rewrite([4096(2),4096(14),4096(19),4096(24),4096(26),4096(34),4235(51),4235(67)]),flip(b),unit_del(b,4410)]. 17.90/18.24 4574 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),c_0),t)). [back_rewrite(1789),rewrite([4096(10)])]. 17.90/18.24 4596 -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),c_0),t)). [back_unit_del(4249),unit_del(b,4410)]. 17.90/18.24 13002 $F. [resolve(4574,a,2844,b),rewrite([4200(16),2938(17)]),unit_del(a,4596),unit_del(b,4429)]. 17.90/18.24 17.90/18.24 % SZS output end Refutation 17.90/18.24 ============================== end of proof ========================== 17.90/18.24 17.90/18.24 ============================== STATISTICS ============================ 17.90/18.24 17.90/18.24 Given=1639. Generated=70626. Kept=10555. proofs=1. 17.90/18.24 Usable=1625. Sos=8041. Demods=854. Limbo=17, Disabled=3382. Hints=0. 17.90/18.24 Megabytes=49.48. 17.90/18.24 User_CPU=15.47, System_CPU=0.11, Wall_clock=16. 17.90/18.24 17.90/18.24 ============================== end of statistics ===================== 17.90/18.24 17.90/18.24 ============================== end of search ========================= 17.90/18.24 17.90/18.24 THEOREM PROVED 17.90/18.24 % SZS status Theorem 17.90/18.24 17.90/18.24 Exiting with 1 proof. 17.90/18.24 17.90/18.24 Process 14999 exit (max_proofs) Mon Jul 3 08:39:11 2023 17.90/18.24 Prover9 interrupted 17.90/18.24 EOF