0.04/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.12/0.12 % Command : tptp2X_and_run_prover9 %d %s 0.12/0.33 % Computer : n010.cluster.edu 0.12/0.33 % Model : x86_64 x86_64 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.12/0.33 % Memory : 8042.1875MB 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64 0.12/0.33 % CPULimit : 1200 0.12/0.33 % DateTime : Tue Jul 13 16:30:21 EDT 2021 0.12/0.33 % CPUTime : 1.52/1.79 ============================== Prover9 =============================== 1.52/1.79 Prover9 (32) version 2009-11A, November 2009. 1.52/1.79 Process 32689 was started by sandbox2 on n010.cluster.edu, 1.52/1.79 Tue Jul 13 16:30:22 2021 1.52/1.79 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 1200 -f /tmp/Prover9_32534_n010.cluster.edu". 1.52/1.79 ============================== end of head =========================== 1.52/1.79 1.52/1.79 ============================== INPUT ================================= 1.52/1.79 1.52/1.79 % Reading from file /tmp/Prover9_32534_n010.cluster.edu 1.52/1.79 1.52/1.79 set(prolog_style_variables). 1.52/1.79 set(auto2). 1.52/1.79 % set(auto2) -> set(auto). 1.52/1.79 % set(auto) -> set(auto_inference). 1.52/1.79 % set(auto) -> set(auto_setup). 1.52/1.79 % set(auto_setup) -> set(predicate_elim). 1.52/1.79 % set(auto_setup) -> assign(eq_defs, unfold). 1.52/1.79 % set(auto) -> set(auto_limits). 1.52/1.79 % set(auto_limits) -> assign(max_weight, "100.000"). 1.52/1.79 % set(auto_limits) -> assign(sos_limit, 20000). 1.52/1.79 % set(auto) -> set(auto_denials). 1.52/1.79 % set(auto) -> set(auto_process). 1.52/1.79 % set(auto2) -> assign(new_constants, 1). 1.52/1.79 % set(auto2) -> assign(fold_denial_max, 3). 1.52/1.79 % set(auto2) -> assign(max_weight, "200.000"). 1.52/1.79 % set(auto2) -> assign(max_hours, 1). 1.52/1.79 % assign(max_hours, 1) -> assign(max_seconds, 3600). 1.52/1.79 % set(auto2) -> assign(max_seconds, 0). 1.52/1.79 % set(auto2) -> assign(max_minutes, 5). 1.52/1.79 % assign(max_minutes, 5) -> assign(max_seconds, 300). 1.52/1.79 % set(auto2) -> set(sort_initial_sos). 1.52/1.79 % set(auto2) -> assign(sos_limit, -1). 1.52/1.79 % set(auto2) -> assign(lrs_ticks, 3000). 1.52/1.79 % set(auto2) -> assign(max_megs, 400). 1.52/1.79 % set(auto2) -> assign(stats, some). 1.52/1.79 % set(auto2) -> clear(echo_input). 1.52/1.79 % set(auto2) -> set(quiet). 1.52/1.79 % set(auto2) -> clear(print_initial_clauses). 1.52/1.79 % set(auto2) -> clear(print_given). 1.52/1.79 assign(lrs_ticks,-1). 1.52/1.79 assign(sos_limit,10000). 1.52/1.79 assign(order,kbo). 1.52/1.79 set(lex_order_vars). 1.52/1.79 clear(print_given). 1.52/1.79 1.52/1.79 % formulas(sos). % not echoed (1254 formulas) 1.52/1.79 1.52/1.79 ============================== end of input ========================== 1.52/1.79 1.52/1.79 % From the command line: assign(max_seconds, 1200). 1.52/1.79 1.52/1.79 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 1.52/1.79 1.52/1.79 % Formulas that are not ordinary clauses: 1.52/1.79 1 (all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),plus_plus(real,one_one(real),abs_abs(real,X))))) # label(fact_974_abs__add__one__gt__zero) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 2 (all B_1_1 all B_2_1 all X_a (dvd(X_a) -> hAPP(X_a,bool,B_1_1,B_2_1) = ti(bool,hAPP(X_a,bool,B_1_1,B_2_1)))) # label(tsy_c_hAPP_3_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 3 (all N hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),N))) # label(fact_572_le__refl) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 4 (all B_1_1 all X_a (number(X_a) -> number_number_of(X_a,B_1_1) = number_number_of(X_a,ti(int,B_1_1)))) # label(tsy_c_Int_Onumber__class_Onumber__of_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 5 (all L all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),minus_minus(nat,L,N)),minus_minus(nat,L,M))))) # label(fact_638_diff__le__mono2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 6 (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_178_zero__less__one) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 7 (all X_a (ordere453448008miring(X_a) -> (all B_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),B_1),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,B_1,A_1)),zero_zero(X_a)))))))) # label(fact_280_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 8 (all B_1_1 all B_2_1 all X_a (dvd(X_a) -> times_times(X_a,ti(X_a,B_1_1),B_2_1) = times_times(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_0_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 9 (all X_a (ordere216010020id_add(X_a) -> (all B_1 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_1),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,A_1,B_1)),zero_zero(X_a)))))))) # label(fact_197_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 10 (all X_a (linordered_semidom(X_a) -> (all N 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)) -> 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)))))))) # label(fact_108_one__less__power) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 11 (all M all N hAPP(nat,int,semiring_1_of_nat(int),times_times(nat,M,N)) = times_times(int,hAPP(nat,int,semiring_1_of_nat(int),M),hAPP(nat,int,semiring_1_of_nat(int),N))) # label(fact_665_int__mult) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 12 (all C all A_1 all B_1 all M (hBOOL(hAPP(int,bool,zcong(A_1,B_1),M)) -> hBOOL(hAPP(int,bool,zcong(plus_plus(int,A_1,C),plus_plus(int,B_1,C)),M)))) # label(fact_817_zcong__shift) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 13 (all X_a (linordered_idom(X_a) -> (all A_1 hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,bit0(bit1(pls)))) = abs_abs(X_a,hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,bit0(bit1(pls)))))))) # label(fact_964_abs__power2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 14 (all Na all Ma (ti(int,Ma) != zero_zero(int) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),times_times(int,Ma,Na)),Ma)) <-> one_one(int) = abs_abs(int,Na)))) # label(fact_961_zdvd__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 15 (all X all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) -> Y != X)) # label(fact_850_dvd_Oless__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 16 (all X hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X),hAPP(nat,int,power_power(int,X),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_544_power2__ge__self) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 17 (all M all N all K_1 times_times(nat,minus_minus(nat,M,N),K_1) = minus_minus(nat,times_times(nat,M,K_1),times_times(nat,N,K_1))) # label(fact_630_diff__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 18 (all X_a (linordered_semidom(X_a) -> (all N 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))))))) # label(fact_449_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 19 (all X_a (number_semiring(X_a) -> (all V_2 all V_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_2)) -> times_times(X_a,number_number_of(X_a,V_1),number_number_of(X_a,V_2)) = number_number_of(X_a,times_times(int,V_1,V_2))))))) # label(fact_250_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 20 (all X -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),plus_plus(real,abs_abs(real,X),one_one(real))),X))) # label(fact_973_abs__add__one__not__less__self) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 21 (all N -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat)))) # label(fact_146_not__less0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 22 (all X_a (cancel_semigroup_add(X_a) -> (all A_1 all B_1 all C (plus_plus(X_a,A_1,C) = plus_plus(X_a,A_1,B_1) -> ti(X_a,B_1) = ti(X_a,C))))) # label(fact_201_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 23 (all W nat_1(number_number_of(int,W)) = number_number_of(nat,W)) # label(fact_426_nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 24 (all X_a (group_add(X_a) -> (all A_1 minus_minus(X_a,A_1,A_1) = zero_zero(X_a)))) # label(fact_322_diff__self) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 25 (all B_1_1 succ(B_1_1) = ti(int,succ(B_1_1))) # label(tsy_c_Int_Osucc_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 26 (exists 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),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))) & hBOOL(hAPP(int,bool,zcong(s1,X_1),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))) & (all Y_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_1)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y_1),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))) & hBOOL(hAPP(int,bool,zcong(s1,Y_1),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))) -> ti(int,Y_1) = ti(int,X_1))))) # label(fact_841__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 27 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B_1 all C plus_plus(X_a,plus_plus(X_a,A_1,B_1),C) = plus_plus(X_a,A_1,plus_plus(X_a,B_1,C))))) # label(fact_207_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 28 (all M all K_1 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),minus_minus(nat,plus_plus(nat,N,M),K_1))))) # label(fact_668_le__add__diff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 29 (all M all N all P_2 (hBOOL(hAPP(int,bool,zprime,P_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),times_times(int,M,N))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),N)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),M))))) # label(fact_727_zprime__zdvd__zmult__better) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 30 (all C all A_1 all B_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1)) -> (C = B_1 -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C))))) # label(fact_861_dvd_Oord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 31 (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)),times_times(X_a,A_1,A_1)))))) # label(fact_294_zero__le__square) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 32 (all B_1_1 all X_a (power(X_a) -> power_power(X_a,B_1_1) = power_power(X_a,ti(X_a,B_1_1)))) # label(tsy_c_Power_Opower__class_Opower_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 33 (all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) <-> (all M_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M_1)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M_1),P_3)) -> ti(int,M_1) = ti(int,P_3) | one_one(int) = ti(int,M_1))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),P_3)))) # label(fact_901_zprime__def) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 34 (all X_a (ordere223160158up_add(X_a) -> (all C all D_1 all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,A_1,C)),plus_plus(X_a,B_1,D_1)))))))) # label(fact_222_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 35 (all Va all V_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),number_number_of(nat,Va)),number_number_of(nat,V_3))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Va),V_3)) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Va),V_3)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),V_3))))) # label(fact_151_less__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 36 (all X_a (ordere236663937imp_le(X_a) -> (all C all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,C,A_1)),plus_plus(X_a,C,B_1))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1)))))) # label(fact_353_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 37 (all M all N (minus_minus(nat,M,N) = zero_zero(nat) -> (zero_zero(nat) = minus_minus(nat,N,M) -> N = M))) # label(fact_335_diffs0__imp__equal) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 38 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx times_times(X_a,times_times(X_a,Lx,Ly),Rx) = times_times(X_a,Lx,times_times(X_a,Ly,Rx))))) # label(fact_243_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 39 (all K_1 all I_2 all J_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_2),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),times_times(int,K_1,I_2)),times_times(int,K_1,J_1)))))) # label(fact_467_zmult__zless__mono2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 40 (all X_a (group_add(X_a) -> (all A_1 all B_1 ti(X_a,A_1) = minus_minus(X_a,plus_plus(X_a,A_1,B_1),B_1)))) # label(fact_326_add__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 41 (all X_a (ordere216010020id_add(X_a) -> (all B_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_eq(X_a),zero_zero(X_a)),B_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),plus_plus(X_a,A_1,B_1)))))))) # label(fact_488_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 42 (all K_1 all I_2 all J_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_2),J_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,I_2,K_1)),plus_plus(int,J_1,K_1))))) # label(fact_35_zadd__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 43 (all K_1 all L bit0(plus_plus(int,K_1,L)) = plus_plus(int,bit0(K_1),bit0(L))) # label(fact_77_add__Bit0__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 44 (all Xa all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya)) & Ya != Xa <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya)))) # label(fact_610_real__less__def) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 45 (all Xa all Ya 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),times_times(real,Xa,Z_1)),times_times(real,Ya,Z_1))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya))))) # label(fact_656_real__mult__less__iff1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 46 (all B_1_1 all B_2_1 minus_minus(nat,B_1_1,ti(nat,B_2_1)) = minus_minus(nat,B_1_1,B_2_1)) # label(tsy_c_Groups_Ominus__class_Ominus_1_arg2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 47 (all M all N (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) -> M = plus_plus(nat,N,minus_minus(nat,M,N)))) # label(fact_456_add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 48 (all X_a (semiring_1(X_a) -> one_one(X_a) = hAPP(nat,X_a,power_power(X_a,one_one(X_a)),number_number_of(nat,bit0(bit1(pls)))))) # label(fact_3_one__power2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 49 (all P_1 all Xa all Ya (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)) -> hBOOL(P_1)))) # label(fact_849_dvd_Oless__imp__triv) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 50 (all K_1 all I_2 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),J_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,I_2,K_1)),plus_plus(nat,J_1,K_1))))) # label(fact_134_add__less__mono1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 51 (all X_a (comm_semiring_1(X_a) -> (all C all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),times_times(X_a,B_1,C))))))) # label(fact_733_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 52 (all X_a (linordered_semidom(X_a) -> (all Ma all Na (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Ma)),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)))))) # label(fact_266_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 53 (all X_a (comm_semiring_1(X_a) -> (all A_1 all C all D_1 plus_plus(X_a,plus_plus(X_a,A_1,C),D_1) = plus_plus(X_a,A_1,plus_plus(X_a,C,D_1))))) # label(fact_208_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 54 (all M all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),M)) -> (hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),M)) -> ti(int,A_1) = zero_zero(int))))) # label(fact_898_zcong__zless__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 55 (all P_1 all A_2 all B (-((exists D_3 (-hBOOL(hAPP(nat,bool,P_1,D_3)) & A_2 = plus_plus(nat,B,D_3))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_2),B)) & -hBOOL(hAPP(nat,bool,P_1,zero_zero(nat)))) <-> hBOOL(hAPP(nat,bool,P_1,minus_minus(nat,A_2,B))))) # label(fact_499_nat__diff__split__asm) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 56 (all C all D_1 all A_1 all B_1 all M (hBOOL(hAPP(int,bool,zcong(A_1,B_1),M)) -> (hBOOL(hAPP(int,bool,zcong(C,D_1),M)) -> hBOOL(hAPP(int,bool,zcong(minus_minus(int,A_1,C),minus_minus(int,B_1,D_1)),M))))) # label(fact_892_zcong__zdiff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 57 (all A_1 all P_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),P_2)) -> times_times(int,A_1,hAPP(nat,int,power_power(int,A_1),minus_minus(nat,nat_1(P_2),one_one(nat)))) = hAPP(nat,int,power_power(int,A_1),nat_1(P_2)))) # label(fact_555_Euler_Oaux__1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 58 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M)) -> N = M))) # label(fact_582_le__antisym) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 59 (all X_a (cancel_semigroup_add(X_a) -> (all B_1 all A_1 all C (plus_plus(X_a,C,A_1) = plus_plus(X_a,B_1,A_1) -> ti(X_a,C) = ti(X_a,B_1))))) # label(fact_199_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 60 (all Z all N (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Z),N)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),N))))) # label(fact_778_zdvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 61 (all B_1_1 all B_2_1 all X_a (order(X_a) -> hAPP(X_a,fun(X_a,bool),ti(fun(X_a,fun(X_a,bool)),B_1_1),B_2_1) = hAPP(X_a,fun(X_a,bool),B_1_1,B_2_1))) # label(tsy_c_hAPP_0_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 62 (all X_a (comm_semiring_1(X_a) -> (all C all D_1 all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),C),D_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),times_times(X_a,A_1,C)),times_times(X_a,B_1,D_1)))))))) # label(fact_731_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 63 (all X_a (ordere223160158up_add(X_a) -> (all C all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,A_1,C)),plus_plus(X_a,B_1,C))))))) # label(fact_224_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 64 (all B_1_1 all B_2_1 all X_a (group_add(X_a) -> minus_minus(X_a,B_1_1,B_2_1) = minus_minus(X_a,ti(X_a,B_1_1),B_2_1))) # label(tsy_c_Groups_Ominus__class_Ominus_0_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 65 (all X_a (number_ring(X_a) -> (all W number_number_of(X_a,bit0(W)) = plus_plus(X_a,plus_plus(X_a,zero_zero(X_a),number_number_of(X_a,W)),number_number_of(X_a,W))))) # label(fact_99_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 66 (all Z_1 all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),nat_1(Z_1)),Ma)) <-> (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),dvd_dvd(int),Z_1),hAPP(nat,int,semiring_1_of_nat(int),Ma)))) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1)) -> zero_zero(nat) = Ma))) # label(fact_835_nat__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 67 (all X_a (group_add(X_a) -> (all A_2 all B (ti(X_a,A_2) = ti(X_a,B) <-> minus_minus(X_a,A_2,B) = zero_zero(X_a))))) # label(fact_320_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 68 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),K)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),bit0(K))))) # label(fact_780_rel__simps_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 69 (all X_a (linord20386208strict(X_a) -> (all C all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_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),times_times(X_a,C,A_1)),times_times(X_a,C,B_1)))))))) # label(fact_422_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 70 (all Y all X all P_2 (hBOOL(hAPP(int,bool,zprime,P_2)) -> (-hBOOL(hAPP(int,bool,zcong(X,zero_zero(int)),P_2)) -> (-hBOOL(hAPP(int,bool,zcong(Y,zero_zero(int)),P_2)) -> -hBOOL(hAPP(int,bool,zcong(times_times(int,X,Y),zero_zero(int)),P_2)))))) # label(fact_825_zcong__zmult__prop3) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 71 (all P_4 all P_1 all Xa ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> (hBOOL(P_4) <-> hBOOL(P_1))) -> (hBOOL(P_1) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) & hBOOL(P_4)))) # label(fact_905_conj__le__cong) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 72 (all X_a (comm_monoid_mult(X_a) -> (all A_1 times_times(X_a,A_1,one_one(X_a)) = ti(X_a,A_1)))) # label(fact_347_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 73 (all X_a (ab_group_add(X_a) -> (all A_2 all B (zero_zero(X_a) = minus_minus(X_a,A_2,B) <-> ti(X_a,B) = ti(X_a,A_2))))) # label(fact_321_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 74 (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_503_rel__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 75 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B_1 all C (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),times_times(X_a,A_1,B_1)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B_1),C)))))) # label(fact_728_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 76 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),bit1(K))))) # label(fact_461_rel__simps_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 77 -(all T_1 times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),T_1) != plus_plus(int,hAPP(nat,int,power_power(int,s),number_number_of(nat,bit0(bit1(pls)))),one_one(int))) # label(fact_711__096_B_Bthesis_O_A_I_B_Bt_O_As_____A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 78 (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_55_rel__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 79 (all X_a (ordere453448008miring(X_a) -> (all B_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),B_1),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,A_1,B_1)),zero_zero(X_a)))))))) # label(fact_281_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 80 (all Ma all Z_1 (ti(int,Z_1) = hAPP(nat,int,semiring_1_of_nat(int),Ma) <-> Ma = nat_1(Z_1) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1)))) # label(fact_318_int__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 81 (all Ma all Na all K (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K)) -> (times_times(nat,K,Ma) = times_times(nat,K,Na) <-> Na = Ma))) # label(fact_983_nat__mult__eq__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 82 (all M all N (times_times(int,M,N) = one_one(int) -> number_number_of(int,min) = ti(int,M) | one_one(int) = ti(int,M))) # label(fact_786_pos__zmult__eq__1__iff__lemma) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 83 (all N all M ((M != zero_zero(nat) -> times_times(nat,M,N) = plus_plus(nat,N,times_times(nat,minus_minus(nat,M,one_one(nat)),N))) & (zero_zero(nat) = M -> zero_zero(nat) = times_times(nat,M,N)))) # label(fact_687_mult__eq__if) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 84 (all K all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,K,Ma)),plus_plus(nat,K,Na))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)))) # label(fact_127_nat__add__left__cancel__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 85 (all L all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),L)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),minus_minus(nat,L,N)),minus_minus(nat,L,M)))))) # label(fact_361_diff__less__mono2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 86 (all Xa all Ya (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)) & Xa != Ya)) # label(fact_869_dvd_Oless__le) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 87 (all N all A_1 all B_1 all P_2 (hBOOL(hAPP(int,bool,zprime,P_2)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),B_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,P_2),N)),times_times(int,A_1,B_1))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,P_2),N)),A_1)))))) # label(fact_787_zprime__power__zdvd__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 88 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K1)),bit0(K2))))) # label(fact_52_less__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 89 (all X_a (ab_semigroup_mult(X_a) -> (all A_1 all B_1 all C times_times(X_a,A_1,times_times(X_a,B_1,C)) = times_times(X_a,times_times(X_a,A_1,B_1),C)))) # label(fact_242_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 90 (all B_1 all Q all R_1 all B_2 all Q_1 all R_2 (plus_plus(int,times_times(int,B_2,Q_1),R_2) = plus_plus(int,times_times(int,B_1,Q),R_1) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,times_times(int,B_2,Q_1),R_2)),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_1),B_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),B_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_1),Q))))))))) # label(fact_704_zdiv__mono2__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 91 (all B_1_1 all B_2_1 all X_a (semiring(X_a) -> times_times(X_a,B_1_1,ti(X_a,B_2_1)) = times_times(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_2_arg2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 92 (all Y all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),X)) -> (exists N_1 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),times_times(int,number_number_of(int,bit0(bit1(pls))),abs_abs(int,minus_minus(int,Y,times_times(int,N_1,X))))),X))))) # label(fact_971_best__division__abs) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 93 (all X_a (comm_semiring_1(X_a) -> (all A_1 times_times(X_a,one_one(X_a),A_1) = ti(X_a,A_1)))) # label(fact_351_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 94 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),N)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N),M)) -> N = M))) # label(fact_860_dvd__antisym) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 95 (all A_2 all B all C_1 (zero_zero(real) != C_1 -> (B = A_2 <-> times_times(real,B,C_1) = times_times(real,A_2,C_1)))) # label(fact_604_real__mult__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 96 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),bit0(K))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K)))) # label(fact_464_rel__simps_I21_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 97 (all X_a (linord219039673up_add(X_a) -> (all A_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,A_2,A_2)),zero_zero(X_a))))))) # label(fact_433_double__add__le__zero__iff__single__add__le__zero) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 98 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),min)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K)),min)))) # label(fact_781_rel__simps_I11_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 99 (all A_1 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1)) -> (exists R (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R)) & hAPP(nat,real,power_power(real,R),N) = A_1))))) # label(fact_230_realpow__pos__nth) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 100 (all I_2 all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,I_2),M)),hAPP(nat,nat,power_power(nat,I_2),N))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),I_2)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))))) # label(fact_834_power__dvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 101 (all K_1 all I_2 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),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),times_times(nat,I_2,K_1)),times_times(nat,J_1,K_1)))))) # label(fact_649_mult__less__mono1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 102 (all N all P_2 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_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),times_times(int,M,N))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),N)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),M)))))) # label(fact_900_zprime__zdvd__zmult) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 103 (all A_2 all B all Ma ((exists K_2 plus_plus(int,A_2,times_times(int,Ma,K_2)) = ti(int,B)) <-> hBOOL(hAPP(int,bool,zcong(A_2,B),Ma)))) # label(fact_895_zcong__iff__lin) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 104 (all K_1 plus_plus(int,K_1,pls) = ti(int,K_1)) # label(fact_75_add__Pls__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 105 (all X all Y plus_plus(real,plus_plus(real,hAPP(nat,real,power_power(real,X),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,real,power_power(real,Y),number_number_of(nat,bit0(bit1(pls))))),times_times(real,times_times(real,number_number_of(real,bit0(bit1(pls))),X),Y)) = hAPP(nat,real,power_power(real,plus_plus(real,X,Y)),number_number_of(nat,bit0(bit1(pls))))) # label(fact_721_real__sum__squared__expand) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 106 (all N -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N))) # label(fact_117_less__not__refl) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 107 (all X_a (ordere216010020id_add(X_a) -> (all B_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)),B_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),plus_plus(X_a,A_1,B_1)))))))) # label(fact_434_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 108 (all X_a (number_ring(X_a) -> (all W number_number_of(X_a,bit1(W)) = plus_plus(X_a,plus_plus(X_a,one_one(X_a),number_number_of(X_a,W)),number_number_of(X_a,W))))) # label(fact_100_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 109 (all V_1 ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls)) -> plus_plus(nat,number_number_of(nat,V_1),one_one(nat)) = number_number_of(nat,succ(V_1))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls)) -> plus_plus(nat,number_number_of(nat,V_1),one_one(nat)) = one_one(nat)))) # label(fact_232_nat__number__of__add__1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 110 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K1)),bit0(K2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)))) # label(fact_33_less__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 111 (all Na all P_1 (-hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) -> (hBOOL(hAPP(nat,bool,P_1,Na)) -> (exists K_2 ((all I (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),K_2)) -> -hBOOL(hAPP(nat,bool,P_1,I)))) & hBOOL(hAPP(nat,bool,P_1,plus_plus(nat,K_2,one_one(nat)))) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_2),Na))))))) # label(fact_976_ex__least__nat__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 112 (all X_a (semiring_1(X_a) -> zero_zero(X_a) = hAPP(nat,X_a,semiring_1_of_nat(X_a),zero_zero(nat)))) # label(fact_188_of__nat__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 113 (all M (hBOOL(hAPP(int,bool,zprime,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),M),one_one(int)))) -> one_one(int) = legendre(number_number_of(int,min),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),M),one_one(int))))) # label(fact_839_Legendre__1mod4) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 114 (all M all N all K_1 plus_plus(nat,plus_plus(nat,M,N),K_1) = plus_plus(nat,M,plus_plus(nat,N,K_1))) # label(fact_124_nat__add__assoc) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 115 (all X_a (linordered_semidom(X_a) -> (all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na))))))) # label(fact_107_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 116 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)))) # label(fact_607_less__imp__le__nat) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 117 (all Xa all Ya all P_1 ((-hBOOL(P_1) -> nat_1(if(int,P_1,Xa,Ya)) = nat_1(Ya)) & (hBOOL(P_1) -> nat_1(Xa) = nat_1(if(int,P_1,Xa,Ya))))) # label(fact_268_nat__if__cong) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 118 (all N N = times_times(nat,one_one(nat),N)) # label(fact_629_nat__mult__1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 119 (all Z plus_plus(nat,Z,Z) = times_times(nat,Z,number_number_of(nat,bit0(bit1(pls))))) # label(fact_692_nat__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 120 (all X_a (number_ring(X_a) -> (all V_1 all W all Z plus_plus(X_a,number_number_of(X_a,V_1),plus_plus(X_a,number_number_of(X_a,W),Z)) = plus_plus(X_a,number_number_of(X_a,plus_plus(int,V_1,W)),Z)))) # label(fact_87_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 121 (all X_a (monoid_mult(X_a) -> (all A_1 all M all N hAPP(nat,X_a,power_power(X_a,A_1),times_times(nat,M,N)) = hAPP(nat,X_a,power_power(X_a,hAPP(nat,X_a,power_power(X_a,A_1),M)),N)))) # label(fact_600_power__mult) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 122 (all B_1_1 all B_2_1 all X_a (monoid_add(X_a) -> plus_plus(X_a,ti(X_a,B_1_1),B_2_1) = plus_plus(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Oplus__class_Oplus_0_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 123 (all N all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M)) -> M != N)) # label(fact_130_less__not__refl2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 124 (all K (min = bit1(K) <-> min = ti(int,K))) # label(fact_771_rel__simps_I47_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 125 (all N all M minus_minus(nat,plus_plus(nat,N,M),N) = M) # label(fact_366_diff__add__inverse) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 126 (all B_1_1 all B_2_1 all X_a (cancel_semigroup_add(X_a) -> plus_plus(X_a,ti(X_a,B_1_1),B_2_1) = plus_plus(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Oplus__class_Oplus_2_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 127 (all X_a (linordered_idom(X_a) -> (all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,Xa),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,X_a,power_power(X_a,Ya),number_number_of(nat,bit0(bit1(pls))))))) <-> zero_zero(X_a) != ti(X_a,Ya) | ti(X_a,Xa) != zero_zero(X_a))))) # label(fact_97_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 128 (all K_1 ti(int,K_1) = plus_plus(int,pls,K_1)) # label(fact_76_add__Pls) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 129 (all X_a (semiring_0(X_a) & power(X_a) -> (all W ((zero_zero(nat) = number_number_of(nat,W) -> hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),number_number_of(nat,W)) = one_one(X_a)) & (zero_zero(nat) != number_number_of(nat,W) -> zero_zero(X_a) = hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),number_number_of(nat,W))))))) # label(fact_11_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 130 (all Ma all Xa ((exists Y_1 hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,Y_1),number_number_of(nat,bit0(bit1(pls)))),Xa),Ma))) <-> hBOOL(hAPP(int,bool,quadRes(Ma),Xa)))) # label(fact_878_QuadRes__def) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 131 (all A_1 all B_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B_1),A_1)) -> -(-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B_1),A_1))))) # label(fact_846_dvd_Oless__asym_H) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 132 (all Va (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Va),pls)) <-> zero_zero(nat) = number_number_of(nat,Va))) # label(fact_502_eq__number__of__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 133 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) -> zero_zero(nat) != N)) # label(fact_140_gr__implies__not0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 134 (all P_1 ((exists X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) & hBOOL(hAPP(nat,bool,P_1,nat_1(X_1))))) <-> (exists X1 hBOOL(hAPP(nat,bool,P_1,X1))))) # label(fact_293_ex__nat) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 135 (all X all Q all N all R_1 (plus_plus(nat,times_times(nat,Q,N),R_1) = X -> (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)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N),X)))))) # label(fact_830_divides__div__not) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 136 (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]. 1.52/1.79 137 (all X_a (number_ring(X_a) -> (all A_1 times_times(X_a,A_1,number_number_of(X_a,bit1(pls))) = ti(X_a,A_1)))) # label(fact_497_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 138 (all Ya all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Ya)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),Ya)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),nat_1(Xa)),nat_1(Ya))))))) # label(fact_683_transfer__nat__int__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.79 139 (all N (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,N))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,bit0(N)))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,bit1(N)))))) # label(fact_914_number__of1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 140 (all Ma all Wa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),nat_1(Wa)),Ma)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),hAPP(nat,int,semiring_1_of_nat(int),Ma)))))) # label(fact_402_nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 141 -(all R all V -(hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),times_times(int,number_number_of(int,bit0(bit1(pls))),abs_abs(int,V))),plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n)))) & minus_minus(int,x,times_times(int,R,plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n)))) = ti(int,V))) # label(fact_916__096_B_Bthesis_O_A_I_B_Br_Av_O_Av_A_061_Ax_A_N_Ar_A_K_A_I1_A_L_Aint_An_) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 142 (all X_a (linord219039673up_add(X_a) -> (all A_2 (zero_zero(X_a) = ti(X_a,A_2) <-> plus_plus(X_a,A_2,A_2) = zero_zero(X_a))))) # label(fact_65_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 143 (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_4_zero__power2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 144 (all X_a all B_1_1 all B_2_1 all B_3 ti(X_a,if(X_a,B_1_1,B_2_1,B_3)) = if(X_a,B_1_1,B_2_1,B_3)) # label(tsy_c_If_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 145 (all M zero_zero(nat) = minus_minus(nat,M,M)) # label(fact_336_diff__self__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 146 (all Z all X all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),plus_plus(real,Z,X)),plus_plus(real,Z,Y))))) # label(fact_642_real__add__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 147 (all X_a (monoid_mult(X_a) -> (all X all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> times_times(X_a,hAPP(nat,X_a,power_power(X_a,X),minus_minus(nat,N,one_one(nat))),X) = hAPP(nat,X_a,power_power(X_a,X),N))))) # label(fact_405_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 148 (all X_a (linordered_semidom(X_a) -> (all X all Y (hAPP(nat,X_a,power_power(X_a,Y),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,X_a,power_power(X_a,X),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)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y)) -> ti(X_a,Y) = ti(X_a,X))))))) # label(fact_548_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 149 (all I_2 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),J_1)) -> (exists K_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_2)) & J_1 = plus_plus(nat,I_2,K_2))))) # label(fact_191_less__imp__add__positive) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 150 (all X_a (linordered_idom(X_a) -> (all A_1 all N 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,abs_abs(X_a,A_1)),N)))))) # label(fact_926_zero__le__power__abs) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 151 (all B_1_1 all B_2_1 all X_a (mult_zero(X_a) -> ti(X_a,times_times(X_a,B_1_1,B_2_1)) = times_times(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_4_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 152 (all X_a all X all Y if(X_a,fTrue,X,Y) = ti(X_a,X)) # label(help_If_1_1_T) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 153 (all X_a (linordered_idom(X_a) -> (all Xa all Ya (zero_zero(X_a) = ti(X_a,Ya) & ti(X_a,Xa) = zero_zero(X_a) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,Xa),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,X_a,power_power(X_a,Ya),number_number_of(nat,bit0(bit1(pls)))))),zero_zero(X_a))))))) # label(fact_553_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 154 (all P_2 (hBOOL(hAPP(int,bool,zprime,P_2)) -> (ti(int,P_2) != number_number_of(int,bit0(bit1(pls))) -> (number_number_of(int,bit1(bit1(pls))) != ti(int,P_2) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,bit1(bit0(bit1(pls))))),P_2)))))) # label(fact_904_prime__g__5) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 155 (all X_a (ordere142940540dd_abs(X_a) -> (all A_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),abs_abs(X_a,A_2))) <-> ti(X_a,A_2) != zero_zero(X_a))))) # label(fact_945_zero__less__abs__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 156 (all X_a (linordered_idom(X_a) -> (all X all Y (ti(X_a,X) != ti(X_a,Y) -> (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X))))))) # label(fact_152_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 157 (all Xa all Ya (plus_plus(real,times_times(real,Xa,Xa),times_times(real,Ya,Ya)) = zero_zero(real) <-> Ya = zero_zero(real) & Xa = zero_zero(real))) # label(fact_660_real__two__squares__add__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 158 (all X_a (ordere216010020id_add(X_a) -> (all B_1 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_1),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_1),plus_plus(X_a,A_1,C)))))))) # label(fact_491_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 159 (all K all L_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L_1)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K)),bit1(L_1))))) # label(fact_462_rel__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 160 (all N all M hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),plus_plus(nat,M,N)))) # label(fact_624_le__add2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 161 (all X_a (linord581940658strict(X_a) -> (all A_2 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,A_2,B)),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_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),A_2),zero_zero(X_a))))))) # label(fact_290_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 162 (all K_1 minus_minus(int,K_1,pls) = ti(int,K_1)) # label(fact_590_diff__bin__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 163 (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_875_one__not__neg__one__mod__m) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 164 (all X_a (number_ring(X_a) -> (all V_1 all W number_number_of(X_a,plus_plus(int,V_1,W)) = plus_plus(X_a,number_number_of(X_a,V_1),number_number_of(X_a,W))))) # label(fact_88_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 165 (all Ya all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Ya)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Xa),Ya)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),nat_1(Xa)),nat_1(Ya))))))) # label(fact_833_transfer__nat__int__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 166 (all X all Y (ti(int,X) = ti(int,Y) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y),X)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),Y)))) # label(fact_15_zless__linear) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 167 (all Ma all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,semiring_1_of_nat(int),Ma)),Z_1)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),nat_1(abs_abs(int,Z_1)))))) # label(fact_962_int__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 168 (all C_1 all D all A_2 all B all Ma (hBOOL(hAPP(int,bool,zcong(A_2,B),Ma)) -> (hBOOL(hAPP(int,bool,zcong(C_1,times_times(int,B,D)),Ma)) <-> hBOOL(hAPP(int,bool,zcong(C_1,times_times(int,A_2,D)),Ma))))) # label(fact_822_zcong__zmult__prop1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 169 (all Z1 all Z2 all W times_times(int,minus_minus(int,Z1,Z2),W) = minus_minus(int,times_times(int,Z1,W),times_times(int,Z2,W))) # label(fact_592_zdiff__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 170 (all X_a (semiri456707255roduct(X_a) -> (all C all D_1 all A_1 all B_1 all R_1 (ti(X_a,R_1) != zero_zero(X_a) -> (ti(X_a,C) != ti(X_a,D_1) & ti(X_a,B_1) = ti(X_a,A_1) -> plus_plus(X_a,B_1,times_times(X_a,R_1,D_1)) != plus_plus(X_a,A_1,times_times(X_a,R_1,C))))))) # label(fact_429_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 171 (all B_1_1 all X_a (ordere142940540dd_abs(X_a) -> ti(X_a,abs_abs(X_a,B_1_1)) = abs_abs(X_a,B_1_1))) # label(tsy_c_Groups_Oabs__class_Oabs_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 172 (all X_a (comm_semiring_1(X_a) -> (all M plus_plus(X_a,M,M) = times_times(X_a,plus_plus(X_a,one_one(X_a),one_one(X_a)),M)))) # label(fact_445_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 173 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B_1 all C (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),times_times(X_a,A_1,B_1)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C)))))) # label(fact_729_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 174 (all Z hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),Z)))) # label(fact_472_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 175 (all Ma all K all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),times_times(nat,Ma,K)),times_times(nat,Na,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),Na))))) # label(fact_585_mult__le__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 176 (all X_a (number_ring(X_a) -> (all V_1 all W times_times(X_a,number_number_of(X_a,V_1),number_number_of(X_a,W)) = number_number_of(X_a,times_times(int,V_1,W))))) # label(fact_259_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 177 (all M all N (N = M -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)))) # label(fact_578_eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 178 (all Xa all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),minus_minus(real,Xa,Ya)),zero_zero(real))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya)))) # label(fact_584_real__le__eq__diff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 179 (all Ma all Z_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),nat_1(Z_1))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),Ma)),Z_1)))) # label(fact_487_zless__nat__eq__int__zless) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 180 (all N -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N))) # label(fact_129_less__irrefl__nat) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 181 (all Z1 all Z2 all W times_times(int,plus_plus(int,Z1,Z2),W) = plus_plus(int,times_times(int,Z1,W),times_times(int,Z2,W))) # label(fact_382_zadd__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 182 (all X_a (comm_monoid_add(X_a) -> (all A_1 plus_plus(X_a,zero_zero(X_a),A_1) = ti(X_a,A_1)))) # label(fact_217_add__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 183 (all B_1 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_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_1),A_1)) -> -hBOOL(hAPP(int,bool,zcong(A_1,B_1),M))))))) # label(fact_893_zcong__not) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 184 (all X all Y (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) -> -(hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))))) # label(fact_852_dvd_Oless__imp__not__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 185 (all M all N all K_1 all L (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_1),L)) -> (plus_plus(nat,M,L) = plus_plus(nat,K_1,N) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))))) # label(fact_136_less__add__eq__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 186 (all X_a (number_ring(X_a) -> plus_plus(X_a,one_one(X_a),one_one(X_a)) = number_number_of(X_a,bit0(bit1(pls))))) # label(fact_8_one__add__one__is__two) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 187 (all X_a (ordere779506340up_add(X_a) -> (all C all D_1 all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,A_1,C)),plus_plus(X_a,B_1,D_1)))))))) # label(fact_355_add__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 188 (all N -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat)))) # label(fact_142_less__nat__zero__code) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 189 (all X_a (idom(X_a) -> (all A_2 all C_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),times_times(X_a,A_2,C_1)),times_times(X_a,B,C_1))) <-> ti(X_a,C_1) = zero_zero(X_a) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_2),B)))))) # label(fact_741_dvd__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 190 (all K all F all Na ((all I (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),Na)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),abs_abs(int,minus_minus(int,hAPP(nat,int,F,plus_plus(nat,I,one_one(nat))),hAPP(nat,int,F,I)))),one_one(int))))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,F,zero_zero(nat))),K)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),hAPP(nat,int,F,Na))) -> (exists I (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),Na)) & ti(int,K) = hAPP(nat,int,F,I))))))) # label(fact_994_int__val__lemma) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 191 (all Ma all Na (times_times(nat,Ma,Na) = zero_zero(nat) <-> Ma = zero_zero(nat) | Na = zero_zero(nat))) # label(fact_596_mult__is__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 192 (all X_a (monoid_mult(X_a) -> (all A_1 all N hAPP(nat,X_a,power_power(X_a,hAPP(nat,X_a,power_power(X_a,A_1),N)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,X_a,power_power(X_a,A_1),times_times(nat,number_number_of(nat,bit0(bit1(pls))),N))))) # label(fact_690_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 193 (all X_a (ab_group_add(X_a) -> (all A_2 all B all C_1 all D (minus_minus(X_a,A_2,B) = minus_minus(X_a,C_1,D) -> (ti(X_a,A_2) = ti(X_a,B) <-> ti(X_a,C_1) = ti(X_a,D)))))) # label(fact_248_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 194 (all Z_2 all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_2)) -> plus_plus(nat,nat_1(Z),nat_1(Z_2)) = nat_1(plus_plus(int,Z,Z_2))))) # label(fact_397_nat__add__distrib) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 195 (all X_a (ordere142940540dd_abs(X_a) -> zero_zero(X_a) = abs_abs(X_a,zero_zero(X_a)))) # label(fact_930_abs__zero) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 196 (all B_1_1 all B_2_1 all X_a (ab_semigroup_mult(X_a) -> times_times(X_a,B_1_1,B_2_1) = times_times(X_a,ti(X_a,B_1_1),B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_5_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 197 (all Wa all Z_1 (ti(int,Z_1) = ti(int,Wa) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z_1)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),plus_plus(int,Z_1,one_one(int)))))) # label(fact_60_zless__add1__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 198 (all X_a (linord581940658strict(X_a) -> (all A_2 all C_1 all B (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),A_2)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B)) & 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),times_times(X_a,A_2,C_1)),times_times(X_a,B,C_1))))))) # label(fact_410_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 199 (all W hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),W),W))) # label(fact_712_real__le__refl) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 200 (all Ma all Na (times_times(nat,Ma,Na) = one_one(nat) <-> Na = one_one(nat) & Ma = one_one(nat))) # label(fact_628_nat__1__eq__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 201 (all X_a (linord581940658strict(X_a) -> (all A_2 all B 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),times_times(X_a,C_1,A_2)),times_times(X_a,C_1,B))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B))))))) # label(fact_412_mult__less__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 202 (all B_1 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_1),C)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_1),A_1)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),B_1))))) # label(fact_697_Euler_Oaux2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 203 (all K_1 all I_2 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),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_2),K_1))))) # label(fact_581_le__trans) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 204 (all K all L_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(K)),bit1(L_1))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L_1)))) # label(fact_372_rel__simps_I34_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 205 (all Z zero_zero(int) != plus_plus(int,plus_plus(int,one_one(int),Z),Z)) # label(fact_93_odd__nonzero) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 206 (all X hAPP(nat,int,power_power(int,abs_abs(int,X)),number_number_of(nat,bit1(bit1(pls)))) = abs_abs(int,hAPP(nat,int,power_power(int,X),number_number_of(nat,bit1(bit1(pls)))))) # label(fact_959_abs__power3__distrib) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 207 (all N 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)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),minus_minus(nat,M,N)))))) # label(fact_800_dvd__diff__nat) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 208 (all X_a (linordered_semidom(X_a) -> (all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,X),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,X_a,power_power(X_a,Y),number_number_of(nat,bit0(bit1(pls)))))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))))))) # label(fact_551_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 209 (all Z1 all Z2 all Z3 times_times(int,times_times(int,Z1,Z2),Z3) = times_times(int,Z1,times_times(int,Z2,Z3))) # label(fact_254_zmult__assoc) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 210 (all X_a (linordered_idom(X_a) -> (all Xa all Ya (zero_zero(X_a) = plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,Xa),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,X_a,power_power(X_a,Ya),number_number_of(nat,bit0(bit1(pls))))) <-> ti(X_a,Xa) = zero_zero(X_a) & zero_zero(X_a) = ti(X_a,Ya))))) # label(fact_2_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 211 (all X all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) | X = Y)) # label(fact_864_dvd_Ole__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 212 (all B_1_1 nat_1(B_1_1) = nat_1(ti(int,B_1_1))) # label(tsy_c_Int_Onat_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 213 (all K_1 all L all I_2 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),L)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),times_times(nat,I_2,K_1)),times_times(nat,J_1,L)))))) # label(fact_583_mult__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 214 (all X_a (comm_semiring_1(X_a) -> (all M all A_1 times_times(X_a,plus_plus(X_a,A_1,one_one(X_a)),M) = plus_plus(X_a,M,times_times(X_a,A_1,M))))) # label(fact_446_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 215 -hBOOL(hAPP(int,bool,quadRes(plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))),number_number_of(int,min))) -> legendre(number_number_of(int,min),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) != one_one(int) # label(fact_794__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_N) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 216 (all K_1 bit1(K_1) != pls) # label(fact_66_rel__simps_I46_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 217 (all P_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_2)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),nat_1(minus_minus(int,P_2,number_number_of(int,bit0(bit1(pls))))))))) # label(fact_710_Int2_Oaux__2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 218 (all X_a (number_semiring(X_a) -> (all V_2 all V_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_2)) -> plus_plus(X_a,number_number_of(X_a,V_1),number_number_of(X_a,V_2)) = number_number_of(X_a,plus_plus(int,V_1,V_2))))))) # label(fact_528_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 219 (all Z_2 all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) -> times_times(nat,nat_1(Z),nat_1(Z_2)) = nat_1(times_times(int,Z,Z_2)))) # label(fact_685_nat__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 220 (all B_1_1 all B_2_1 minus_minus(nat,B_1_1,B_2_1) = minus_minus(nat,ti(nat,B_1_1),B_2_1)) # label(tsy_c_Groups_Ominus__class_Ominus_1_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 221 (all Z_1 (one_one(int) = abs_abs(int,Z_1) <-> number_number_of(int,min) = ti(int,Z_1) | ti(int,Z_1) = one_one(int))) # label(fact_957_abs__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 222 (all X_a all B_1_1 all B_2_1 all B_3 if(X_a,B_1_1,B_2_1,B_3) = if(X_a,B_1_1,B_2_1,ti(X_a,B_3))) # label(tsy_c_If_arg3) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 223 (all Wa all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z_1)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),nat_1(Wa)),nat_1(Z_1)))))) # label(fact_483_nat__mono__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 224 (all Z ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) -> ti(int,Z) = hAPP(nat,int,semiring_1_of_nat(int),nat_1(Z))) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) -> hAPP(nat,int,semiring_1_of_nat(int),nat_1(Z)) = zero_zero(int)))) # label(fact_317_int__nat__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 225 (all K all L_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K)),bit0(L_1))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L_1)))) # label(fact_53_rel__simps_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 226 (all X_a (linordered_semidom(X_a) -> (all M all N 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))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))))))) # label(fact_183_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 227 (all X_a (no_zero_divisors(X_a) -> (all A_1 all B_1 (times_times(X_a,A_1,B_1) = zero_zero(X_a) -> zero_zero(X_a) = ti(X_a,B_1) | zero_zero(X_a) = ti(X_a,A_1))))) # label(fact_332_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 228 (all X_a (linordered_semidom(X_a) -> (all Xa all Ya all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),B)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Xa),Ya)) <-> 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),Xa)),hAPP(nat,X_a,power_power(X_a,B),Ya)))))))) # label(fact_392_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 229 (all M all I_2 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),plus_plus(nat,M,J_1))))) # label(fact_619_trans__le__add2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 230 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) | Na = Ma <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)))) # label(fact_608_le__eq__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 231 (all X_a (linordered_semidom(X_a) -> (all Xa all Ya all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),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,B),Xa)),hAPP(nat,X_a,power_power(X_a,B),Ya))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Xa),Ya))))))) # label(fact_182_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 232 (all M hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),times_times(nat,M,times_times(nat,M,M))))) # label(fact_574_le__cube) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 233 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),Ma)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) <-> Na != Ma)) # label(fact_121_nat__neq__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 234 (all K_1 all M all N minus_minus(nat,times_times(nat,K_1,M),times_times(nat,K_1,N)) = times_times(nat,K_1,minus_minus(nat,M,N))) # label(fact_631_diff__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 235 (all X_a (linordered_idom(X_a) -> (all Ma all K (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Ma),K)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Ma),abs_abs(X_a,K))))))) # label(fact_921_dvd__abs__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 236 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all Xa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),number_number_of(X_a,Xa)),zero_zero(X_a))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Xa),pls)))))) # label(fact_49_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 237 (all B_1_1 all B_2_1 ti(int,legendre(B_1_1,B_2_1)) = legendre(B_1_1,B_2_1)) # label(tsy_c_Residues_OLegendre_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 238 (all X_a (number_ring(X_a) -> (all W number_number_of(X_a,bit0(W)) = times_times(X_a,plus_plus(X_a,one_one(X_a),one_one(X_a)),number_number_of(X_a,W))))) # label(fact_525_double__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 239 (all X_a (semiri456707255roduct(X_a) -> (all C_1 all D all A_2 all B (ti(X_a,A_2) != ti(X_a,B) & ti(X_a,C_1) != ti(X_a,D) <-> plus_plus(X_a,times_times(X_a,A_2,D),times_times(X_a,B,C_1)) != plus_plus(X_a,times_times(X_a,A_2,C_1),times_times(X_a,B,D)))))) # label(fact_344_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 240 (all X_a (comm_semiring_1(X_a) -> (all X all N times_times(X_a,hAPP(nat,X_a,power_power(X_a,X),N),hAPP(nat,X_a,power_power(X_a,X),N)) = hAPP(nat,X_a,power_power(X_a,X),times_times(nat,number_number_of(nat,bit0(bit1(pls))),N))))) # label(fact_481_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 241 (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),plus_plus(X_a,A_1,one_one(X_a))))))) # label(fact_180_less__add__one) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 242 (all X_a (order(X_a) -> ord_less(X_a) = ti(fun(X_a,fun(X_a,bool)),ord_less(X_a)))) # label(tsy_c_Orderings_Oord__class_Oless_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 243 (all X_a (ring_11004092258visors(X_a) -> (all N all A_1 (ti(X_a,A_1) != zero_zero(X_a) -> hAPP(nat,X_a,power_power(X_a,A_1),N) != zero_zero(X_a))))) # label(fact_165_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 244 (all X all Y times_times(nat,plus_plus(nat,X,Y),minus_minus(nat,X,Y)) = minus_minus(nat,hAPP(nat,nat,power_power(nat,X),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,nat,power_power(nat,Y),number_number_of(nat,bit0(bit1(pls)))))) # label(fact_701_diff__square) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 245 (all X_a (linordered_semidom(X_a) -> (all M -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),zero_zero(X_a)))))) # label(fact_186_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 246 (all B_1_1 all B_2_1 all X_a (dvd(X_a) -> times_times(X_a,B_1_1,B_2_1) = times_times(X_a,B_1_1,ti(X_a,B_2_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_0_arg2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 247 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 all B_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),abs_abs(X_a,minus_minus(X_a,A_1,B_1))),plus_plus(X_a,abs_abs(X_a,A_1),abs_abs(X_a,B_1))))))) # label(fact_928_abs__triangle__ineq4) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 248 (all X_a (linord581940658strict(X_a) -> (all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,times_times(X_a,Xa,Xa),times_times(X_a,Ya,Ya))),zero_zero(X_a))) <-> zero_zero(X_a) = ti(X_a,Xa) & ti(X_a,Ya) = zero_zero(X_a))))) # label(fact_311_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 249 (all X_a (linordered_semidom(X_a) -> (all N 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)),times_times(X_a,A_1,hAPP(nat,X_a,power_power(X_a,A_1),N)))))))) # label(fact_495_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 250 (all Xa all Ya (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),Xa)),hAPP(nat,int,semiring_1_of_nat(int),Ya))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Xa),Ya)))) # label(fact_677_Nat__Transfer_Otransfer__int__nat__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 251 (all P_4 all P_1 all Xa ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> (hBOOL(P_4) <-> hBOOL(P_1))) -> ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> hBOOL(P_4)) <-> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> hBOOL(P_1))))) # label(fact_906_imp__le__cong) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 252 (all K_1 all L bit0(times_times(int,K_1,L)) = times_times(int,bit0(K_1),L)) # label(fact_371_mult__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 253 (all Na all Ma (Ma = one_one(nat) & Na = one_one(nat) <-> one_one(nat) = times_times(nat,Na,Ma))) # label(fact_625_nat__mult__eq__one) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 254 (all X_a (ordere142940540dd_abs(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)),A_1)) -> ti(X_a,A_1) = abs_abs(X_a,A_1))))) # label(fact_947_abs__of__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 255 (all Ma all Na 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),Na)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),times_times(nat,K,Ma)),times_times(nat,K,Na)))))) # label(fact_982_nat__mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 256 (all M all N times_times(int,hAPP(nat,int,semiring_1_of_nat(int),M),hAPP(nat,int,semiring_1_of_nat(int),N)) = hAPP(nat,int,semiring_1_of_nat(int),times_times(nat,M,N))) # label(fact_664_zmult__int) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 257 (all B_1_1 ti(fun(int,bool),quadRes(B_1_1)) = quadRes(B_1_1)) # label(tsy_c_Residues_OQuadRes_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 258 (all Z all W (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),W)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),Z)) -> ti(int,Z) = ti(int,W)))) # label(fact_256_zle__antisym) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 259 (all V_1 all W number_number_of(int,plus_plus(int,V_1,W)) = plus_plus(int,number_number_of(int,V_1),number_number_of(int,W))) # label(fact_17_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 260 (all N all B_1 all A_1 all P_2 (hBOOL(hAPP(int,bool,zprime,P_2)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,P_2),N)),times_times(int,A_1,B_1))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,P_2),N)),B_1)))))) # label(fact_788_zprime__power__zdvd__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 261 (all Wa all Z_1 ((exists N_1 ti(int,Z_1) = plus_plus(int,Wa,hAPP(nat,int,semiring_1_of_nat(int),N_1))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Wa),Z_1)))) # label(fact_477_zle__iff__zadd) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 262 (all X_a (linordered_semidom(X_a) -> (all N 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))))))) # label(fact_454_one__le__power) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 263 (all K_1 all I_2 all J_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_2),J_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),times_times(int,hAPP(nat,int,semiring_1_of_nat(int),K_1),I_2)),times_times(int,hAPP(nat,int,semiring_1_of_nat(int),K_1),J_1)))))) # label(fact_531_zmult__zless__mono2__lemma) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 264 (all X all Y (Y = X -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)))) # label(fact_866_dvd_Oeq__refl) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 265 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),pls)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(K)),pls)))) # label(fact_504_rel__simps_I29_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 266 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K)),pls)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),pls)))) # label(fact_51_rel__simps_I12_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 267 (all X_a (ordere216010020id_add(X_a) -> (all B_1 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_1),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,A_1,B_1)),zero_zero(X_a)))))))) # label(fact_438_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 268 (all X_a (comm_semiring_1(X_a) -> (all A_1 zero_zero(X_a) = times_times(X_a,zero_zero(X_a),A_1)))) # label(fact_334_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 269 (all X_a (linord893533164strict(X_a) -> (all C all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_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),times_times(X_a,C,A_1)),times_times(X_a,C,B_1)))))))) # label(fact_423_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 270 (all X_a (number_ring(X_a) -> (all V_1 all W all C minus_minus(X_a,number_number_of(X_a,plus_plus(int,V_1,W)),C) = plus_plus(X_a,number_number_of(X_a,V_1),minus_minus(X_a,number_number_of(X_a,W),C))))) # label(fact_482_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 271 (all B_1_1 all B_2_1 all X_a (power(X_a) -> ti(X_a,hAPP(nat,X_a,B_1_1,B_2_1)) = hAPP(nat,X_a,B_1_1,B_2_1))) # label(tsy_c_hAPP_4_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 272 (all X_a (comm_monoid_mult(X_a) -> (all A_1 all B_1 all N hAPP(nat,X_a,power_power(X_a,times_times(X_a,A_1,B_1)),N) = times_times(X_a,hAPP(nat,X_a,power_power(X_a,A_1),N),hAPP(nat,X_a,power_power(X_a,B_1),N))))) # label(fact_363_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 273 (all X_a (ring(X_a) -> (all A_2 all E_1 all C_1 all B all D (plus_plus(X_a,times_times(X_a,B,E_1),D) = plus_plus(X_a,times_times(X_a,A_2,E_1),C_1) <-> ti(X_a,D) = plus_plus(X_a,times_times(X_a,minus_minus(X_a,A_2,B),E_1),C_1))))) # label(fact_286_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 274 (all K_1 all L minus_minus(int,bit1(K_1),bit0(L)) = bit1(minus_minus(int,K_1,L))) # label(fact_644_diff__bin__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 275 (all X_a (linord581940658strict(X_a) -> (all Xa all Ya (ti(X_a,Xa) != zero_zero(X_a) | ti(X_a,Ya) != zero_zero(X_a) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),plus_plus(X_a,times_times(X_a,Xa,Xa),times_times(X_a,Ya,Ya)))))))) # label(fact_486_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 276 (all X_a (number_semiring(X_a) -> plus_plus(X_a,one_one(X_a),one_one(X_a)) = number_number_of(X_a,bit0(bit1(pls))))) # label(fact_9_semiring__one__add__one__is__two) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 277 (all K_1 all M all N plus_plus(nat,times_times(nat,K_1,M),times_times(nat,K_1,N)) = times_times(nat,K_1,plus_plus(nat,M,N))) # label(fact_613_add__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 278 (all X all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)))) # label(fact_853_dvd_Oless__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 279 (all B_1_1 quadRes(ti(int,B_1_1)) = quadRes(B_1_1)) # label(tsy_c_Residues_OQuadRes_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 280 (all N times_times(nat,zero_zero(nat),N) = zero_zero(nat)) # label(fact_594_mult__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 281 (all C all A_1 all B_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B_1),A_1)) -> (C = B_1 -> -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_848_dvd_Oord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 282 (all X_a (ordere236663937imp_le(X_a) -> (all A_2 all C_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,A_2,C_1)),plus_plus(X_a,B,C_1))))))) # label(fact_226_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 283 (all B_2 all Q_1 all R_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),plus_plus(int,times_times(int,B_2,Q_1),R_2))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_2),B_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Q_1)))))) # label(fact_560_q__pos__lemma) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 284 (all X_a (ordere142940540dd_abs(X_a) -> (all A_2 (zero_zero(X_a) = ti(X_a,A_2) <-> zero_zero(X_a) = abs_abs(X_a,A_2))))) # label(fact_931_abs__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 285 (all N all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M)) -> minus_minus(int,hAPP(nat,int,semiring_1_of_nat(int),M),hAPP(nat,int,semiring_1_of_nat(int),N)) = hAPP(nat,int,semiring_1_of_nat(int),minus_minus(nat,M,N)))) # label(fact_589_zdiff__int) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 286 (all X_a (semiring_1(X_a) -> semiring_1_of_nat(X_a) = ti(fun(nat,X_a),semiring_1_of_nat(X_a)))) # label(tsy_c_Nat_Osemiring__1__class_Oof__nat_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 287 (all X_a (ordered_ring(X_a) -> (all A_2 all E_1 all C_1 all B all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),plus_plus(X_a,times_times(X_a,minus_minus(X_a,B,A_2),E_1),D))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,times_times(X_a,A_2,E_1),C_1)),plus_plus(X_a,times_times(X_a,B,E_1),D))))))) # label(fact_287_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 288 (all X_a (linord219039673up_add(X_a) -> (all A_2 (plus_plus(X_a,A_2,A_2) = zero_zero(X_a) <-> ti(X_a,A_2) = zero_zero(X_a))))) # label(fact_216_double__zero__sym) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 289 (all C all A_1 all B_1 all M (hBOOL(hAPP(int,bool,zcong(A_1,B_1),M)) -> (hBOOL(hAPP(int,bool,zcong(B_1,C),M)) -> hBOOL(hAPP(int,bool,zcong(A_1,C),M))))) # label(fact_881_zcong__trans) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 290 (all X all Y (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) -> X != Y)) # label(fact_855_dvd_Oless__imp__neq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 291 (all I_2 all J_1 -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,I_2,J_1)),I_2))) # label(fact_118_not__add__less1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 292 (all X_a (ab_semigroup_add(X_a) -> (all A_1 all B_1 all C plus_plus(X_a,plus_plus(X_a,A_1,B_1),C) = plus_plus(X_a,A_1,plus_plus(X_a,B_1,C))))) # label(fact_206_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 293 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M)))) # label(fact_576_nat__le__linear) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 294 (all X_a (linordered_semidom(X_a) -> (all A_1 all N all B_1 (hAPP(nat,X_a,power_power(X_a,B_1),N) = hAPP(nat,X_a,power_power(X_a,A_1),N) -> (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_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> ti(X_a,A_1) = ti(X_a,B_1)))))))) # label(fact_526_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 295 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),Ma)),hAPP(nat,int,semiring_1_of_nat(int),Na))))) # label(fact_678_zle__int) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 296 (all P_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_2)) -> minus_minus(nat,nat_1(P_2),number_number_of(nat,bit0(bit1(pls)))) = nat_1(minus_minus(int,P_2,number_number_of(int,bit0(bit1(pls))))))) # label(fact_709_Int2_Oaux__1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 297 (all Xa all Na (hAPP(nat,nat,power_power(nat,Xa),Na) = one_one(nat) <-> Na = zero_zero(nat) | one_one(nat) = Xa)) # label(fact_192_exp__eq__1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 298 (all X_a (linord219039673up_add(X_a) -> (all A_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),plus_plus(X_a,A_2,A_2))))))) # label(fact_194_zero__less__double__add__iff__zero__less__single__add) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 299 (all X_a (number_ring(X_a) -> one_one(X_a) = number_number_of(X_a,bit1(pls)))) # label(fact_12_semiring__norm_I110_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 300 (all X_a (ordered_semiring(X_a) -> (all C all D_1 all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_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),times_times(X_a,A_1,C)),times_times(X_a,B_1,D_1)))))))))) # label(fact_271_mult__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 301 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),min)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K)),min)))) # label(fact_751_rel__simps_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 302 (all X_a (ordere216010020id_add(X_a) -> (all B_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(X_a),zero_zero(X_a)),B_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),plus_plus(X_a,A_1,B_1)))))))) # label(fact_489_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 303 (all A_1 all P_2 (hBOOL(hAPP(int,bool,zprime,P_2)) -> (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_2)) -> (hBOOL(hAPP(int,bool,zcong(times_times(int,A_1,A_1),one_one(int)),P_2)) -> ti(int,A_1) = one_one(int) | minus_minus(int,P_2,one_one(int)) = ti(int,A_1)))))) # label(fact_880_zcong__square__zless) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 304 (all X all Y (Y != X -> (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),Y)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Y),X))))) # label(fact_128_linorder__neqE__nat) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 305 (all X_a (linord581940658strict(X_a) -> (all Xa all Ya (zero_zero(X_a) = ti(X_a,Ya) & zero_zero(X_a) = ti(X_a,Xa) <-> plus_plus(X_a,times_times(X_a,Xa,Xa),times_times(X_a,Ya,Ya)) = zero_zero(X_a))))) # label(fact_428_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 306 (all C all D_1 all A_1 all B_1 all M (hBOOL(hAPP(int,bool,zcong(A_1,B_1),M)) -> (hBOOL(hAPP(int,bool,zcong(C,D_1),M)) -> hBOOL(hAPP(int,bool,zcong(plus_plus(int,A_1,C),plus_plus(int,B_1,D_1)),M))))) # label(fact_891_zcong__zadd) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 307 (all X_a (number_semiring(X_a) -> (all N number_number_of(X_a,hAPP(nat,int,semiring_1_of_nat(int),N)) = hAPP(nat,X_a,semiring_1_of_nat(X_a),N)))) # label(fact_94_number__of__int) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 308 (all I_2 all J_1 all K_1 minus_minus(nat,minus_minus(nat,I_2,J_1),K_1) = minus_minus(nat,minus_minus(nat,I_2,K_1),J_1)) # label(fact_267_diff__commute) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 309 (all X_a (comm_semiring_1(X_a) -> (all B_1 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),times_times(X_a,B_1,C))))))) # label(fact_732_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 310 (all X_a (idom(X_a) -> (all C_1 all A_2 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),times_times(X_a,C_1,A_2)),times_times(X_a,C_1,B))) <-> zero_zero(X_a) = ti(X_a,C_1) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_2),B)))))) # label(fact_740_dvd__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 311 (all M all N hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),minus_minus(nat,M,N)),M))) # label(fact_639_Nat_Odiff__le__self) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 312 (all X_a (linordered_semiring(X_a) -> (all C all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,C,A_1)),times_times(X_a,C,B_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),A_1),B_1))))))) # label(fact_301_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 313 (all X_a (number_ring(X_a) -> (all N one_one(X_a) = hAPP(nat,X_a,power_power(X_a,number_number_of(X_a,min)),times_times(nat,number_number_of(nat,bit0(bit1(pls))),N))))) # label(fact_791_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 314 (all Z_1 all Wa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Wa)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),nat_1(Wa)),nat_1(Z_1))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Wa),Z_1))))) # label(fact_688_nat__le__eq__zle) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 315 (all X_a (linord581940658strict(X_a) -> (all A_2 all B 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_2),B)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,C_1,A_2)),times_times(X_a,C_1,B)))))))) # label(fact_309_mult__le__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 316 (all X_a (number_ring(X_a) -> (all V_1 plus_plus(X_a,number_number_of(X_a,V_1),one_one(X_a)) = number_number_of(X_a,plus_plus(int,V_1,bit1(pls)))))) # label(fact_7_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 317 (all X_a (linord219039673up_add(X_a) -> (all A_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),plus_plus(X_a,A_2,A_2))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_2)))))) # label(fact_432_zero__le__double__add__iff__zero__le__single__add) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 318 (all K_1 all L minus_minus(int,bit0(K_1),bit0(L)) = bit0(minus_minus(int,K_1,L))) # label(fact_591_diff__bin__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 319 (all Xa all Wa (zero_zero(nat) = number_number_of(nat,Wa) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Xa)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,power_power(nat,Xa),number_number_of(nat,Wa)))))) # label(fact_149_zero__less__power__nat__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 320 (all M all K_1 all N minus_minus(nat,plus_plus(nat,M,K_1),plus_plus(nat,N,K_1)) = minus_minus(nat,M,N)) # label(fact_369_diff__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 321 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 all B_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),abs_abs(X_a,minus_minus(X_a,abs_abs(X_a,A_1),abs_abs(X_a,B_1)))),abs_abs(X_a,minus_minus(X_a,A_1,B_1))))))) # label(fact_941_abs__triangle__ineq3) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 322 (all X all Y all N (zero_zero(nat) != N -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,X),N)),Y)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))))) # label(fact_809_divides__exp2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 323 (all X_a (linordered_semidom(X_a) -> (all A_1 all N all B_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)),hAPP(nat,X_a,power_power(X_a,B_1),N))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1))))))) # label(fact_494_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 324 (all C all A_1 all B_1 (A_1 = B_1 -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B_1),C)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C))))) # label(fact_862_dvd_Oord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 325 (all M all N (times_times(nat,M,N) = M -> one_one(nat) = N | M = zero_zero(nat))) # label(fact_657_mult__eq__self__implies__10) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 326 (all B_1_1 all B_2_1 all X_a (dvd(X_a) -> hAPP(X_a,bool,B_1_1,B_2_1) = hAPP(X_a,bool,B_1_1,ti(X_a,B_2_1)))) # label(tsy_c_hAPP_3_arg2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 327 (all U_1 all M all N all J_1 all I_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_1),I_2)) -> minus_minus(nat,plus_plus(nat,times_times(nat,I_2,U_1),M),plus_plus(nat,times_times(nat,J_1,U_1),N)) = minus_minus(nat,plus_plus(nat,times_times(nat,minus_minus(nat,I_2,J_1),U_1),M),N))) # label(fact_988_nat__diff__add__eq1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 328 (all X_a (linordered_idom(X_a) -> (all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),one_one(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,X,Y)),X)))))))) # label(fact_312_mult__right__le__one__le) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 329 (all X_a (linordered_semidom(X_a) -> (all N 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),times_times(X_a,A_1,hAPP(nat,X_a,power_power(X_a,A_1),N))),hAPP(nat,X_a,power_power(X_a,A_1),N)))))))) # label(fact_524_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 330 (all A_2 all B (abs_abs(int,A_2) = abs_abs(int,B) <-> hAPP(nat,int,power_power(int,B),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,int,power_power(int,A_2),number_number_of(nat,bit0(bit1(pls)))))) # label(fact_967_power2__eq__iff__abs__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 331 (all B_1_1 all B_2_1 all X_a (dvd(X_a) -> hAPP(X_a,fun(X_a,bool),B_1_1,ti(X_a,B_2_1)) = hAPP(X_a,fun(X_a,bool),B_1_1,B_2_1))) # label(tsy_c_hAPP_2_arg2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 332 (all B_1_1 all B_2_1 all X_a (monoid_add(X_a) -> plus_plus(X_a,B_1_1,B_2_1) = ti(X_a,plus_plus(X_a,B_1_1,B_2_1)))) # label(tsy_c_Groups_Oplus__class_Oplus_0_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 333 (all X_a (linordered_idom(X_a) -> (all A_1 all B_1 abs_abs(X_a,times_times(X_a,A_1,B_1)) = times_times(X_a,abs_abs(X_a,A_1),abs_abs(X_a,B_1))))) # label(fact_951_abs__mult) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 334 (all B_1_1 all B_2_1 all X_a (order(X_a) -> hAPP(X_a,bool,B_1_1,B_2_1) = hAPP(X_a,bool,B_1_1,ti(X_a,B_2_1)))) # label(tsy_c_hAPP_1_arg2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 335 (all K all I_1 all J (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J)) -> (K = minus_minus(nat,J,I_1) <-> plus_plus(nat,K,I_1) = J))) # label(fact_673_le__imp__diff__is__add) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 336 (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_756_rel__simps_I30_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 337 (all Va all V_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),number_number_of(nat,Va)),number_number_of(nat,V_3))) <-> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Va),V_3)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Va),pls))))) # label(fact_681_le__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 338 (all M all N hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,power_power(nat,M),N)) = hAPP(nat,int,power_power(int,hAPP(nat,int,semiring_1_of_nat(int),M)),N)) # label(fact_20_zpower__int) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 339 (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]. 1.52/1.80 340 (all L min != bit0(L)) # label(fact_765_rel__simps_I42_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 341 (all K_1 all M all N (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),times_times(int,K_1,M)),times_times(int,K_1,N))) -> (ti(int,K_1) != zero_zero(int) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M),N))))) # label(fact_746_zdvd__mult__cancel) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 342 (all M all K_1 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),plus_plus(nat,M,K_1)),N)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)))) # label(fact_615_add__leD1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 343 (all M all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),M)) -> (hBOOL(hAPP(int,bool,zcong(X,zero_zero(int)),M)) -> ti(int,X) = zero_zero(int))))) # label(fact_824_Int2_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 344 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) <-> zero_zero(nat) = minus_minus(nat,Ma,Na))) # label(fact_658_diff__is__0__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 345 (all X_a (comm_semiring_1(X_a) -> (all X all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) | ti(X_a,X) = one_one(X_a) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),hAPP(nat,X_a,power_power(X_a,X),N))))))) # label(fact_789_dvd__power) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 346 (all P_1 ((all X_1 hBOOL(hAPP(int,bool,P_1,hAPP(nat,int,semiring_1_of_nat(int),X_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,P_1,X_1)))))) # label(fact_474_transfer__int__nat__quantifiers_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 347 (all B_2 all Q_1 all R_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,times_times(int,B_2,Q_1),R_2)),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_1),zero_zero(int))))))) # label(fact_561_q__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 348 (all M hBOOL(hAPP(int,bool,zcong(M,zero_zero(int)),M))) # label(fact_816_zcong__id) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 349 (all Ma all K all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),times_times(nat,Ma,K)),times_times(nat,Na,K))) <-> K = zero_zero(nat) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),Na)))) # label(fact_805_nat__mult__dvd__cancel__disj_H) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 350 (all X_a (ordere223160158up_add(X_a) -> (all C all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,C,A_1)),plus_plus(X_a,C,B_1))))))) # label(fact_223_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 351 (all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z_1)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z_1)))) # label(fact_511_int__one__le__iff__zero__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 352 (all K all L_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L_1)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K)),bit0(L_1))))) # label(fact_375_rel__simps_I31_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 353 (all X_a (linord219039673up_add(X_a) -> (all A_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,A_2,A_2)),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),zero_zero(X_a))))))) # label(fact_195_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 354 (all X_a (comm_semiring_1(X_a) -> (all X all P_2 all Q hAPP(nat,X_a,power_power(X_a,X),times_times(nat,P_2,Q)) = hAPP(nat,X_a,power_power(X_a,hAPP(nat,X_a,power_power(X_a,X),P_2)),Q)))) # label(fact_599_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 355 (all Na all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Ma)) -> (one_one(nat) = Na <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),times_times(nat,Ma,Na)),Ma))))) # label(fact_831_dvd__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 356 (all W hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),W))) # label(fact_251_zle__refl) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 357 (all C all D_1 all A_1 all B_1 all M (hBOOL(hAPP(int,bool,zcong(A_1,B_1),M)) -> (hBOOL(hAPP(int,bool,zcong(C,D_1),M)) -> hBOOL(hAPP(int,bool,zcong(times_times(int,A_1,C),times_times(int,B_1,D_1)),M))))) # label(fact_890_zcong__zmult) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 358 (all C all A_1 all B_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),times_times(nat,A_1,C)),times_times(nat,B_1,C))))) # label(fact_798_divides__mul__r) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 359 (all X_b (ring(X_b) & number(X_b) -> (all V_1 all B_1 all C minus_minus(X_b,times_times(X_b,number_number_of(X_b,V_1),B_1),times_times(X_b,number_number_of(X_b,V_1),C)) = times_times(X_b,number_number_of(X_b,V_1),minus_minus(X_b,B_1,C))))) # label(fact_262_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 360 (all Na (zero_zero(nat) != Na <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)))) # label(fact_145_neq0__conv) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 361 (all X_a (comm_semiring_1(X_a) -> (all A_1 times_times(X_a,A_1,one_one(X_a)) = ti(X_a,A_1)))) # label(fact_348_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 362 (all X_a (linord581940658strict(X_a) -> (all A_2 all B 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),A_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,C_1,A_2)),times_times(X_a,C_1,B)))))))) # label(fact_308_mult__le__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 363 (all A_2 all B (hBOOL(hAPP(int,bool,zcong(A_2,B),zero_zero(int))) <-> ti(int,A_2) = ti(int,B))) # label(fact_885_IntPrimes_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 364 (all D_1 all C all A_1 all B_1 all M (hBOOL(hAPP(int,bool,zcong(A_1,B_1),M)) -> (ti(int,C) = ti(int,B_1) -> (hBOOL(hAPP(int,bool,zcong(C,D_1),M)) -> hBOOL(hAPP(int,bool,zcong(A_1,D_1),M)))))) # label(fact_818_zcong__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 365 (all X_a (linordered_semidom(X_a) -> (all N 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))))))) # label(fact_179_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 366 (all A_2 all B all C_1 (C_1 != zero_zero(real) -> (times_times(real,C_1,A_2) = times_times(real,C_1,B) <-> A_2 = B))) # label(fact_603_real__mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 367 (all X_a (comm_semiring_1(X_a) -> (all X ti(X_a,X) = hAPP(nat,X_a,power_power(X_a,X),one_one(nat))))) # label(fact_227_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 368 (all X_a (ordere236663937imp_le(X_a) -> (all A_1 all C all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,A_1,C)),plus_plus(X_a,B_1,C))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1)))))) # label(fact_221_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.80 369 (all M all N times_times(nat,M,N) = times_times(nat,N,M)) # label(fact_575_nat__mult__commute) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 370 (all I_2 all K_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J_1)) -> minus_minus(nat,plus_plus(nat,I_2,J_1),K_1) = plus_plus(nat,I_2,minus_minus(nat,J_1,K_1)))) # label(fact_670_add__diff__assoc) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 371 (all B_1_1 all B_2_1 zcong(B_1_1,B_2_1) = zcong(ti(int,B_1_1),B_2_1)) # label(tsy_c_IntPrimes_Ozcong_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 372 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B_1 all C all D_1 plus_plus(X_a,plus_plus(X_a,A_1,C),plus_plus(X_a,B_1,D_1)) = plus_plus(X_a,plus_plus(X_a,A_1,B_1),plus_plus(X_a,C,D_1))))) # label(fact_202_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 373 (all X all N hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,power_power(int,abs_abs(int,X)),N)))) # label(fact_955_zero__le__zpower__abs) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 374 (all Z all X all Y all M (hBOOL(hAPP(int,bool,zcong(X,Y),M)) -> hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,X),Z),hAPP(nat,int,power_power(int,Y),Z)),M)))) # label(fact_819_zcong__zpower) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 375 (all X_a (comm_ring_1(X_a) -> (all Z all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),Y)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),Z)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),minus_minus(X_a,Y,Z)))))))) # label(fact_736_dvd__diff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 376 (all X_a (semiri456707255roduct(X_a) -> (all Wa all Ya all Xa all Z_1 (plus_plus(X_a,times_times(X_a,Wa,Ya),times_times(X_a,Xa,Z_1)) = plus_plus(X_a,times_times(X_a,Wa,Z_1),times_times(X_a,Xa,Ya)) <-> ti(X_a,Wa) = ti(X_a,Xa) | ti(X_a,Ya) = ti(X_a,Z_1))))) # label(fact_341_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 377 (all X_b (number(X_b) & semiring(X_b) -> (all A_1 all B_1 all V_1 plus_plus(X_b,times_times(X_b,A_1,number_number_of(X_b,V_1)),times_times(X_b,B_1,number_number_of(X_b,V_1))) = times_times(X_b,plus_plus(X_b,A_1,B_1),number_number_of(X_b,V_1))))) # label(fact_444_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 378 (all B_1_1 all B_2_1 all X_a (power(X_a) -> ti(X_a,times_times(X_a,B_1_1,B_2_1)) = times_times(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_1_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 379 (all K all L_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K)),bit1(L_1))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L_1)))) # label(fact_505_rel__simps_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 380 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) <-> (exists K_2 plus_plus(nat,Ma,K_2) = Na))) # label(fact_622_le__iff__add) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 381 (all A_1 all B_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1)) -> (A_1 != B_1 -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B_1),A_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1))))) # label(fact_863_dvd_Ole__neq__trans) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 382 (all I_2 all J_1 all K_1 minus_minus(nat,I_2,plus_plus(nat,J_1,K_1)) = minus_minus(nat,minus_minus(nat,I_2,J_1),K_1)) # label(fact_367_diff__diff__left) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 383 (all X_a (comm_monoid_add(X_a) -> (all A_1 plus_plus(X_a,A_1,zero_zero(X_a)) = ti(X_a,A_1)))) # label(fact_213_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 384 (all X_a (number(X_a) -> (all Wa all Xa (ti(X_a,Xa) = number_number_of(X_a,Wa) <-> number_number_of(X_a,Wa) = ti(X_a,Xa))))) # label(fact_41_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 385 (all N 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)))) # label(fact_696_two__realpow__ge__one) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 386 (all X_a (dvd(X_a) -> ti(fun(X_a,fun(X_a,bool)),dvd_dvd(X_a)) = dvd_dvd(X_a))) # label(tsy_c_Rings_Odvd__class_Odvd_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 387 (all A_2 all B all Ma (hBOOL(hAPP(int,bool,zcong(A_2,B),Ma)) <-> hBOOL(hAPP(int,bool,zcong(B,A_2),Ma)))) # label(fact_883_zcong__sym) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 388 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 all B_1 all C all D_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),abs_abs(X_a,minus_minus(X_a,plus_plus(X_a,A_1,B_1),plus_plus(X_a,C,D_1)))),plus_plus(X_a,abs_abs(X_a,minus_minus(X_a,A_1,C)),abs_abs(X_a,minus_minus(X_a,B_1,D_1)))))))) # label(fact_927_abs__diff__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 389 (all C all A_1 all B_1 (A_1 = B_1 -> (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),C),B_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B_1),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_857_dvd_Oord__eq__less__trans) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 390 (all X_a (number_ring(X_a) -> (all V_1 all W number_number_of(X_a,plus_plus(int,V_1,W)) = plus_plus(X_a,number_number_of(X_a,V_1),number_number_of(X_a,W))))) # label(fact_89_number__of__add) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 391 (all A_1 abs_abs(int,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls))))) = hAPP(nat,int,power_power(int,abs_abs(int,A_1)),number_number_of(nat,bit0(bit1(pls))))) # label(fact_966_abs__power2__distrib) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 392 (all X_a (dvd(X_a) & comm_ring(X_a) -> (all Ta all D all D_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),D_2)) -> (all X_1 all K_2 (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),plus_plus(X_a,minus_minus(X_a,X_1,times_times(X_a,K_2,D_2)),Ta))) <-> -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),plus_plus(X_a,X_1,Ta))))))))) # label(fact_910_inf__period_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 393 (all Y all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y)) -> nat_1(times_times(int,X,Y)) = times_times(nat,nat_1(X),nat_1(Y))))) # label(fact_686_Nat__Transfer_Otransfer__nat__int__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 394 (all X_a (ordere216010020id_add(X_a) -> (all B_1 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_1),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,A_1,B_1)),zero_zero(X_a)))))))) # label(fact_492_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 395 (all N hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N))) # label(fact_601_less__eq__nat_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 396 (all X_a (number_ring(X_a) -> (all V_1 ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,V_1))) -> hAPP(nat,X_a,semiring_1_of_nat(X_a),number_number_of(nat,V_1)) = zero_zero(X_a)) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,V_1))) -> hAPP(nat,X_a,semiring_1_of_nat(X_a),number_number_of(nat,V_1)) = number_number_of(X_a,V_1)))))) # label(fact_549_of__nat__number__of__lemma) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 397 (all B_1_1 all B_2_1 all X_a (power(X_a) -> times_times(X_a,ti(X_a,B_1_1),B_2_1) = times_times(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_1_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 398 (all Ya all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Ya)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),nat_1(Xa)),nat_1(Ya))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Xa),Ya)))))) # label(fact_387_transfer__nat__int__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 399 (all X_a (ordere1490568538miring(X_a) -> (all C all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_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),times_times(X_a,C,A_1)),times_times(X_a,C,B_1)))))))) # label(fact_275_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 400 (all X_b (number(X_b) & semiring(X_b) -> (all V_1 all B_1 all C times_times(X_b,number_number_of(X_b,V_1),plus_plus(X_b,B_1,C)) = plus_plus(X_b,times_times(X_b,number_number_of(X_b,V_1),B_1),times_times(X_b,number_number_of(X_b,V_1),C))))) # label(fact_443_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 401 (all B_1_1 succ(ti(int,B_1_1)) = succ(B_1_1)) # label(tsy_c_Int_Osucc_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 402 (all X_a (mult_zero(X_a) -> (all A_1 times_times(X_a,zero_zero(X_a),A_1) = zero_zero(X_a)))) # label(fact_328_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 403 (all Z_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),nat_1(Z_1))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z_1)))) # label(fact_521_zero__less__nat__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 404 (all M all K_1 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),plus_plus(nat,M,K_1)),N)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N)))) # label(fact_616_add__leD2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 405 (all Ya all Xa (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) -> (Ya = Xa <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))))) # label(fact_865_dvd_Oantisym__conv) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 406 (all X_a (linordered_idom(X_a) -> (all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),one_one(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,Y,X)),X)))))))) # label(fact_313_mult__left__le__one__le) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 407 (all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),zero_zero(int))) -> nat_1(Z) = zero_zero(nat))) # label(fact_314_nat__le__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 408 (all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),number_number_of(int,bit0(bit1(pls))))) -> ti(int,X) = zero_zero(int) | ti(int,X) = one_one(int)))) # label(fact_554_int__pos__lt__two__imp__zero__or__one) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 409 (all Ma all Na (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)),Na)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),times_times(nat,Ma,Na))))) # label(fact_652_nat__0__less__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 410 (all X_a (ordere216010020id_add(X_a) -> (all B_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)),B_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),plus_plus(X_a,A_1,B_1)))))))) # label(fact_196_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 411 (all C_1 all Xa all Ta all A_2 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_2),D)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_2),plus_plus(int,Xa,Ta))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_2),plus_plus(int,plus_plus(int,Xa,times_times(int,C_1,D)),Ta)))))) # label(fact_749_zdvd__period) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 412 (all A_1 all B_1 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_1),number_number_of(nat,bit0(bit1(pls))))) = times_times(int,plus_plus(int,A_1,B_1),minus_minus(int,A_1,B_1))) # label(fact_698_zspecial__product) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 413 (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_506_less__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 414 (all L_1 (bit0(L_1) = pls <-> ti(int,L_1) = pls)) # label(fact_71_rel__simps_I38_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 415 (all X all Y all Z plus_plus(int,Y,plus_plus(int,X,Z)) = plus_plus(int,X,plus_plus(int,Y,Z))) # label(fact_46_zadd__left__commute) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 416 (all M all N plus_plus(nat,N,M) = plus_plus(nat,M,N)) # label(fact_122_nat__add__commute) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 417 (all L minus_minus(int,min,bit0(L)) = bit1(minus_minus(int,min,L))) # label(fact_784_diff__bin__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 418 (all K_1 all L all I_2 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),L)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),plus_plus(nat,I_2,K_1)),plus_plus(nat,J_1,L)))))) # label(fact_617_add__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 419 (all X_a (number_semiring(X_a) -> (all X all Y hAPP(nat,X_a,power_power(X_a,plus_plus(X_a,X,Y)),number_number_of(nat,bit0(bit1(pls)))) = plus_plus(X_a,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,X_a,power_power(X_a,Y),number_number_of(nat,bit0(bit1(pls))))),times_times(X_a,times_times(X_a,number_number_of(X_a,bit0(bit1(pls))),X),Y))))) # label(fact_550_power2__sum) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 420 (all X_a (linord581940658strict(X_a) -> (all C all B_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_1),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),times_times(X_a,C,A_1)),times_times(X_a,C,B_1)))))))) # label(fact_425_mult__strict__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 421 (all X_a (number_ring(X_a) -> (all K_1 plus_plus(X_a,one_one(X_a),number_number_of(X_a,K_1)) = number_number_of(X_a,succ(K_1))))) # label(fact_532_number__of__succ) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 422 (all K_1 minus_minus(int,K_1,min) = succ(K_1)) # label(fact_777_diff__bin__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 423 (all X_a (ordere779506340up_add(X_a) -> (all C all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,C,A_1)),plus_plus(X_a,C,B_1))))))) # label(fact_356_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 424 (all X_a (ordered_ring_abs(X_a) -> (all B_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),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_1)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),zero_zero(X_a)))) -> abs_abs(X_a,times_times(X_a,A_1,B_1)) = times_times(X_a,abs_abs(X_a,A_1),abs_abs(X_a,B_1)))))) # label(fact_925_abs__eq__mult) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 425 (all A_1 all B_1 hAPP(nat,int,power_power(int,minus_minus(int,A_1,B_1)),number_number_of(nat,bit1(bit1(pls)))) = minus_minus(int,plus_plus(int,minus_minus(int,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit1(bit1(pls)))),times_times(int,times_times(int,number_number_of(int,bit1(bit1(pls))),hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls))))),B_1)),times_times(int,times_times(int,number_number_of(int,bit1(bit1(pls))),A_1),hAPP(nat,int,power_power(int,B_1),number_number_of(nat,bit0(bit1(pls)))))),hAPP(nat,int,power_power(int,B_1),number_number_of(nat,bit1(bit1(pls)))))) # label(fact_702_zdiff__power3) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 426 (all I_2 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),N)) -> minus_minus(nat,N,minus_minus(nat,N,I_2)) = I_2)) # label(fact_636_diff__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 427 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) -> minus_minus(nat,M,N) = zero_zero(nat))) # label(fact_659_diff__is__0__eq_H) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 428 (all X_a (number_semiring(X_a) -> (all Z plus_plus(X_a,Z,Z) = times_times(X_a,Z,number_number_of(X_a,bit0(bit1(pls))))))) # label(fact_536_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 429 (all X_a (number_ring(X_a) -> (all W plus_plus(X_a,one_one(X_a),number_number_of(X_a,W)) = number_number_of(X_a,plus_plus(int,bit1(pls),W))))) # label(fact_6_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 430 (all X_a (linordered_semidom(X_a) -> (all Ma all Na (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Ma)),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)))))) # label(fact_167_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 431 (all X_a (dvd(X_a) & semiring_0(X_a) -> (all P_1 all L_1 ((exists X_1 hBOOL(hAPP(X_a,bool,P_1,times_times(X_a,L_1,X_1)))) <-> (exists X_1 (hBOOL(hAPP(X_a,bool,P_1,X_1)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),L_1),plus_plus(X_a,X_1,zero_zero(X_a)))))))))) # label(fact_908_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 432 (all A_1 all N all B_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,A_1),N)),hAPP(nat,nat,power_power(nat,B_1),N))) -> (zero_zero(nat) != N -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1))))) # label(fact_808_divides__rev) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 433 (all X_a (linordered_semidom(X_a) -> (all N all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_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),hAPP(nat,X_a,power_power(X_a,A_1),N)),hAPP(nat,X_a,power_power(X_a,B_1),N)))))))) # label(fact_450_power__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 434 (all Xa all P_3 (hBOOL(hAPP(int,bool,zcong(Xa,zero_zero(int)),P_3)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_3),Xa)))) # label(fact_801_zcong__eq__zdvd__prop) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 435 (all Wa all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),plus_plus(int,Z_1,one_one(int)))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Wa),Z_1)))) # label(fact_516_zle__add1__eq__le) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 436 (all B_1_1 all B_2_1 all X_a (ab_semigroup_add(X_a) -> ti(X_a,plus_plus(X_a,B_1_1,B_2_1)) = plus_plus(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Oplus__class_Oplus_1_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 437 (all Na all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),minus_minus(nat,Na,Ma))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)))) # label(fact_451_zero__less__diff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 438 (all X_a (linordered_idom(X_a) -> (all A_2 (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_2),number_number_of(nat,bit0(bit1(pls)))))) <-> ti(X_a,A_2) != zero_zero(X_a))))) # label(fact_95_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 439 (all B_1 all Q_1 all R_2 all Q all R_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,times_times(int,B_1,Q_1),R_2)),plus_plus(int,times_times(int,B_1,Q),R_1))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_2),B_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_1),B_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_1),Q))))))) # label(fact_707_unique__quotient__lemma) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 440 (all V_1 ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls)) -> plus_plus(nat,one_one(nat),number_number_of(nat,V_1)) = number_number_of(nat,succ(V_1))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls)) -> plus_plus(nat,one_one(nat),number_number_of(nat,V_1)) = one_one(nat)))) # label(fact_233_nat__1__add__number__of) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 441 (all Xa all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,power_power(nat,Xa),Na))) <-> zero_zero(nat) = Na | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Xa)))) # label(fact_143_nat__zero__less__power__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 442 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) | N = M -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)))) # label(fact_605_less__or__eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 443 (all X_a (ordered_semiring(X_a) -> (all C all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_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),times_times(X_a,A_1,C)),times_times(X_a,B_1,C)))))))) # label(fact_277_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 444 (all P_1 all K all I_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),I_1)) -> (hBOOL(hAPP(int,bool,P_1,plus_plus(int,K,one_one(int)))) -> ((all I (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),I)) -> (hBOOL(hAPP(int,bool,P_1,I)) -> hBOOL(hAPP(int,bool,P_1,plus_plus(int,I,one_one(int))))))) -> hBOOL(hAPP(int,bool,P_1,I_1)))))) # label(fact_112_int__gr__induct) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 445 (all M abs_abs(int,hAPP(nat,int,semiring_1_of_nat(int),M)) = hAPP(nat,int,semiring_1_of_nat(int),M)) # label(fact_936_abs__int__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 446 (all X_a (ordered_ring(X_a) -> (all C all B_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),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),times_times(X_a,A_1,C)),times_times(X_a,B_1,C)))))))) # label(fact_274_mult__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 447 (all X_a (linordered_idom(X_a) -> (all A_1 all N abs_abs(X_a,hAPP(nat,X_a,power_power(X_a,A_1),N)) = hAPP(nat,X_a,power_power(X_a,abs_abs(X_a,A_1)),N)))) # label(fact_934_power__abs) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 448 (all X hAPP(nat,real,power_power(real,times_times(real,number_number_of(real,bit0(bit1(pls))),X)),number_number_of(nat,bit0(bit1(pls)))) = times_times(real,number_number_of(real,bit0(bit0(bit1(pls)))),hAPP(nat,real,power_power(real,X),number_number_of(nat,bit0(bit1(pls)))))) # label(fact_695_four__x__squared) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 449 (all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),one_one(nat))) <-> Ma = one_one(nat))) # label(fact_813_nat__dvd__1__iff__1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 450 (all X_a (linord20386208strict(X_a) -> (all C all D_1 all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_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)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,A_1,C)),times_times(X_a,B_1,D_1)))))))))) # label(fact_306_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 451 (all X_a (ordere142940540dd_abs(X_a) -> (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)) -> abs_abs(X_a,A_1) = ti(X_a,A_1))))) # label(fact_944_abs__of__pos) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 452 (all X_a (number_semiring(X_a) -> (all V_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_1)) -> number_number_of(X_a,succ(V_1)) = plus_plus(X_a,one_one(X_a),number_number_of(X_a,V_1)))))) # label(fact_479_semiring__1__add__number__of) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 453 (all A_1 all R_1 all B_1 all M all C all D_1 all N plus_plus(int,times_times(int,minus_minus(int,A_1,times_times(int,R_1,B_1)),M),times_times(int,minus_minus(int,C,times_times(int,R_1,D_1)),N)) = minus_minus(int,plus_plus(int,times_times(int,A_1,M),times_times(int,C,N)),times_times(int,R_1,plus_plus(int,times_times(int,B_1,M),times_times(int,D_1,N))))) # label(fact_894_xzgcda__linear__aux1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 454 (all Xa all Ya (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)))) # label(fact_868_dvd_Oless__le__not__le) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 455 (all U all Ma all Na all J all I_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),I_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,times_times(nat,minus_minus(nat,I_1,J),U),Ma)),Na)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,times_times(nat,I_1,U),Ma)),plus_plus(nat,times_times(nat,J,U),Na)))))) # label(fact_978_nat__less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 456 (all Y all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y)) -> plus_plus(nat,nat_1(X),nat_1(Y)) = nat_1(plus_plus(int,X,Y))))) # label(fact_396_Nat__Transfer_Otransfer__nat__int__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 457 (all K_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K_1)) -> (exists N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) & hAPP(nat,int,semiring_1_of_nat(int),N_1) = ti(int,K_1))))) # label(fact_190_zero__less__imp__eq__int) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 458 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx times_times(X_a,times_times(X_a,Lx,Ly),Rx) = times_times(X_a,times_times(X_a,Lx,Rx),Ly)))) # label(fact_244_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 459 (all R_1 all Q all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (ti(int,A_1) = plus_plus(int,R_1,times_times(int,A_1,Q)) -> (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),one_one(int))))))) # label(fact_556_self__quotient__aux2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 460 -(all S all W_1 -(hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),times_times(int,number_number_of(int,bit0(bit1(pls))),abs_abs(int,W_1))),plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n)))) & minus_minus(int,y,times_times(int,S,plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n)))) = ti(int,W_1))) # label(fact_915__096_B_Bthesis_O_A_I_B_Bs_Aw_O_Aw_A_061_Ay_A_N_As_A_K_A_I1_A_L_Aint_An_) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 461 (all X_a (monoid_mult(X_a) -> (all A_1 times_times(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_527_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 462 (all X_a (linord20386208strict(X_a) -> (all A_1 all C all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,A_1,C)),times_times(X_a,B_1,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_1))))))) # label(fact_302_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 463 (all Z_1 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Z_1),hAPP(nat,int,semiring_1_of_nat(int),Ma))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),nat_1(abs_abs(int,Z_1))),Ma)))) # label(fact_963_dvd__int__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 464 (all Va (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),Va)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),number_number_of(nat,Va))))) # label(fact_161_less__0__number__of) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 465 (all K_1 bit1(K_1) = plus_plus(int,plus_plus(int,one_one(int),K_1),K_1)) # label(fact_92_Bit1__def) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 466 (all X_a (linordered_semidom(X_a) -> (all M all N 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))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))))))) # label(fact_393_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 467 (all X_a (number_semiring(X_a) -> (all Z times_times(X_a,number_number_of(X_a,bit0(bit1(pls))),Z) = plus_plus(X_a,Z,Z)))) # label(fact_538_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 468 (all W all Z1 all Z2 times_times(int,W,plus_plus(int,Z1,Z2)) = plus_plus(int,times_times(int,W,Z1),times_times(int,W,Z2))) # label(fact_381_zadd__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 469 (all X_a (comm_semiring_1(X_a) -> (all A_1 all C plus_plus(X_a,A_1,C) = plus_plus(X_a,C,A_1)))) # label(fact_210_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 470 (all X_a (zero(X_a) -> (all Xa (ti(X_a,Xa) = zero_zero(X_a) <-> zero_zero(X_a) = ti(X_a,Xa))))) # label(fact_198_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 471 (all B_1_1 all B_2_1 all X_a (order(X_a) -> hAPP(X_a,bool,ti(fun(X_a,bool),B_1_1),B_2_1) = hAPP(X_a,bool,B_1_1,B_2_1))) # label(tsy_c_hAPP_1_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 472 (all X_a (linord20386208strict(X_a) -> (all A_1 all C all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,A_1,C)),times_times(X_a,B_1,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_1))))))) # label(fact_299_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 473 (all B_1_1 all B_2_1 ti(fun(int,bool),zcong(B_1_1,B_2_1)) = zcong(B_1_1,B_2_1)) # label(tsy_c_IntPrimes_Ozcong_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 474 (all A_2 all P_3 (hBOOL(hAPP(int,bool,zcong(A_2,minus_minus(int,P_3,one_one(int))),P_3)) <-> hBOOL(hAPP(int,bool,zcong(times_times(int,A_2,minus_minus(int,P_3,one_one(int))),one_one(int)),P_3)))) # label(fact_912_inv__not__p__minus__1__aux) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 475 (all B_1_1 all B_2_1 all X_a (order(X_a) -> hAPP(X_a,fun(X_a,bool),B_1_1,B_2_1) = hAPP(X_a,fun(X_a,bool),B_1_1,ti(X_a,B_2_1)))) # label(tsy_c_hAPP_0_arg2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 476 (all X_a (number_ring(X_a) -> (all V_1 all W minus_minus(X_a,number_number_of(X_a,V_1),number_number_of(X_a,W)) = number_number_of(X_a,minus_minus(int,V_1,W))))) # label(fact_264_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 477 (all B_1_1 all B_2_1 all X_a (dvd(X_a) -> times_times(X_a,B_1_1,B_2_1) = ti(X_a,times_times(X_a,B_1_1,B_2_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_0_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 478 (all K_1 all I_2 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),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),times_times(nat,K_1,I_2)),times_times(nat,K_1,J_1)))))) # label(fact_648_mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 479 (all C all A_1 all B_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_1),B_1)) -> (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),minus_minus(nat,A_1,C)),minus_minus(nat,B_1,C)))))) # label(fact_661_diff__less__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 480 (all K all Ma all Na (times_times(nat,K,Ma) = times_times(nat,K,Na) <-> Na = Ma | zero_zero(nat) = K)) # label(fact_980_nat__mult__eq__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 481 (all X_a (comm_semiring_1(X_a) -> (all A_1 all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) -> 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))))))) # label(fact_742_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 482 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 all B_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),minus_minus(X_a,abs_abs(X_a,A_1),abs_abs(X_a,B_1))),abs_abs(X_a,minus_minus(X_a,B_1,A_1))))))) # label(fact_939_abs__triangle__ineq2__sym) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 483 (all Na all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Ma)) -> (Na = one_one(nat) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),times_times(nat,Na,Ma)),Ma))))) # label(fact_832_dvd__mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 484 (all W all Z1 all Z2 times_times(int,W,minus_minus(int,Z1,Z2)) = minus_minus(int,times_times(int,W,Z1),times_times(int,W,Z2))) # label(fact_593_zdiff__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 485 (all Z all W times_times(int,Z,W) = times_times(int,W,Z)) # label(fact_252_zmult__commute) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 486 (all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),Na))))) # label(fact_162_zero__less__int__conv) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 487 (all X_a (linord581940658strict(X_a) -> (all B_1 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_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)),times_times(X_a,A_1,B_1)))))))) # label(fact_420_mult__neg__neg) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 488 (all V_1 nat_1(number_number_of(int,V_1)) = number_number_of(nat,V_1)) # label(fact_427_nat__number__of__def) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 489 (all X_a (comm_semiring_1(X_a) -> (all N all M all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),Y)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),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),N)),hAPP(nat,X_a,power_power(X_a,Y),M)))))))) # label(fact_743_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 490 (all A_1 all B_1 hBOOL(hAPP(int,bool,zcong(A_1,B_1),one_one(int)))) # label(fact_886_zcong__1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 491 (all X_a (ordere223160158up_add(X_a) -> (all C all D_1 all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,A_1,C)),plus_plus(X_a,B_1,D_1)))))))) # label(fact_440_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 492 (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]. 1.52/1.81 493 (all K_1 all I_2 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),times_times(nat,K_1,I_2)),times_times(nat,K_1,J_1))))) # label(fact_580_mult__le__mono2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 494 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,Xa)),number_number_of(X_a,Ya))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),Ya)))))) # label(fact_257_le__number__of) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 495 (all X_a (linordered_semiring(X_a) -> (all A_1 all C all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,A_1,C)),times_times(X_a,B_1,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_1))))))) # label(fact_303_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 496 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Xa),bit1(pls))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),number_number_of(X_a,Xa)),one_one(X_a))))))) # label(fact_62_less__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 497 (all X all Y all Z hAPP(nat,int,power_power(int,X),times_times(nat,Y,Z)) = hAPP(nat,int,power_power(int,hAPP(nat,int,power_power(int,X),Y)),Z)) # label(fact_641_zpower__zpower) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 498 (all N all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) -> nat_1(hAPP(nat,int,power_power(int,Z),N)) = hAPP(nat,nat,power_power(nat,nat_1(Z)),N))) # label(fact_400_nat__power__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 499 (all X_a (comm_semiring_1(X_a) -> (all A_1 ti(X_a,A_1) = plus_plus(X_a,zero_zero(X_a),A_1)))) # label(fact_218_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 500 (all X_a (monoid_mult(X_a) -> (all A_1 ti(X_a,A_1) = times_times(X_a,one_one(X_a),A_1)))) # label(fact_352_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 501 (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_723_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 502 (all I_2 all U_1 all J_1 all K_1 plus_plus(nat,times_times(nat,I_2,U_1),plus_plus(nat,times_times(nat,J_1,U_1),K_1)) = plus_plus(nat,times_times(nat,plus_plus(nat,I_2,J_1),U_1),K_1)) # label(fact_981_left__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 503 (all X_a (number_ring(X_a) -> number_number_of(X_a,pls) = zero_zero(X_a))) # label(fact_82_number__of__Pls) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 504 (all X_a (semiring_1(X_a) -> one_one(X_a) = hAPP(nat,X_a,semiring_1_of_nat(X_a),one_one(nat)))) # label(fact_171_of__nat__1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 505 (all A_1 all B_1 all C (ti(int,C) = minus_minus(int,A_1,B_1) -> plus_plus(int,C,B_1) = ti(int,A_1))) # label(fact_719_Int2_Oaux1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 506 (all M all N all K_1 plus_plus(nat,times_times(nat,M,K_1),times_times(nat,N,K_1)) = times_times(nat,plus_plus(nat,M,N),K_1)) # label(fact_612_add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 507 (all X_a (linordered_ring(X_a) -> (all X all Y hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),plus_plus(X_a,times_times(X_a,X,X),times_times(X_a,Y,Y))))))) # label(fact_310_sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 508 (all X_a (ordered_ab_group_add(X_a) -> (all A_2 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),minus_minus(X_a,A_2,B)),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B)))))) # label(fact_406_less__iff__diff__less__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 509 (all X_a (comm_semiring_1(X_a) -> (all X hAPP(nat,X_a,power_power(X_a,X),number_number_of(nat,bit0(bit1(pls)))) = times_times(X_a,X,X)))) # label(fact_541_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 510 (all Xa all Ya 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_eq(real),times_times(real,Xa,Z_1)),times_times(real,Ya,Z_1))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya))))) # label(fact_587_real__mult__le__cancel__iff1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 511 (all X_a (ring_n68954251visors(X_a) -> (all A_2 all B (zero_zero(X_a) = ti(X_a,B) | ti(X_a,A_2) = zero_zero(X_a) <-> times_times(X_a,A_2,B) = zero_zero(X_a))))) # label(fact_330_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 512 (all K_1 all L bit0(L) != bit1(K_1)) # label(fact_68_rel__simps_I50_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 513 (all N all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M)) -> plus_plus(nat,N,minus_minus(nat,M,N)) = M)) # label(fact_669_le__add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 514 (all X_a (cancel146912293up_add(X_a) -> (all A_1 all B_1 all C (plus_plus(X_a,A_1,B_1) = plus_plus(X_a,A_1,C) -> ti(X_a,C) = ti(X_a,B_1))))) # label(fact_200_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 515 (all X_a (linordered_semidom(X_a) -> (all N all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_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(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> 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)),hAPP(nat,X_a,power_power(X_a,B_1),N))))))))) # label(fact_539_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 516 (all P_1 ((all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) -> hBOOL(hAPP(nat,bool,P_1,nat_1(X_1))))) <-> (all X1 hBOOL(hAPP(nat,bool,P_1,X1))))) # label(fact_292_all__nat) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 517 (all X_a (number_ring(X_a) -> (all X all Y hAPP(nat,X_a,power_power(X_a,minus_minus(X_a,X,Y)),number_number_of(nat,bit0(bit1(pls)))) = minus_minus(X_a,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,X_a,power_power(X_a,Y),number_number_of(nat,bit0(bit1(pls))))),times_times(X_a,times_times(X_a,number_number_of(X_a,bit0(bit1(pls))),X),Y))))) # label(fact_520_power2__diff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 518 (all N (N != zero_zero(nat) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)))) # label(fact_139_gr0I) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 519 (all X_a (one(X_a) -> ti(X_a,one_one(X_a)) = one_one(X_a))) # label(tsy_c_Groups_Oone__class_Oone_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 520 (all B_1_1 all B_2_1 all X_a (dvd(X_a) -> hAPP(X_a,bool,ti(fun(X_a,bool),B_1_1),B_2_1) = hAPP(X_a,bool,B_1_1,B_2_1))) # label(tsy_c_hAPP_3_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 521 (all Xa all Ya (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) <-> Xa = Ya)) # label(fact_871_dvd_Oeq__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 522 (all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,plus_plus(int,one_one(int),Z_1),Z_1)),zero_zero(int))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_1),zero_zero(int))))) # label(fact_64_odd__less__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 523 (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_760_rel__simps_I28_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 524 (all K all L_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K)),bit1(L_1))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L_1)))) # label(fact_31_rel__simps_I17_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 525 (all X_a (ordered_ab_group_add(X_a) -> (all A_2 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),minus_minus(X_a,A_2,B)),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B)))))) # label(fact_238_le__iff__diff__le__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 526 (all V_2 all K_1 all V_1 ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls)) -> times_times(nat,number_number_of(nat,times_times(int,V_1,V_2)),K_1) = times_times(nat,number_number_of(nat,V_1),times_times(nat,number_number_of(nat,V_2),K_1))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls)) -> zero_zero(nat) = times_times(nat,number_number_of(nat,V_1),times_times(nat,number_number_of(nat,V_2),K_1))))) # label(fact_693_nat__number__of__mult__left) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 527 (all X_a (ordere453448008miring(X_a) -> (all B_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)),B_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),times_times(X_a,A_1,B_1)))))))) # label(fact_284_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 528 (all K all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),Na)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),plus_plus(nat,Na,K))))) # label(fact_814_dvd__reduce) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 529 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 all B_1 plus_plus(X_a,abs_abs(X_a,A_1),abs_abs(X_a,B_1)) = abs_abs(X_a,plus_plus(X_a,abs_abs(X_a,A_1),abs_abs(X_a,B_1)))))) # label(fact_932_abs__add__abs) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 530 (all Ma all Na 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),times_times(nat,K,Ma)),times_times(nat,K,Na))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),Na))))) # label(fact_985_nat__mult__dvd__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 531 (all X_a (group_add(X_a) -> (all A_1 ti(X_a,A_1) = minus_minus(X_a,A_1,zero_zero(X_a))))) # label(fact_323_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 532 (all A_1 all N all P_2 (hBOOL(hAPP(int,bool,zprime,P_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),hAPP(nat,int,power_power(int,A_1),N))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),A_1))))) # label(fact_739_zprime__zdvd__power) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 533 (all X_a (power(X_a) & mult_zero(X_a) & zero_neq_one(X_a) & no_zero_divisors(X_a) -> (all A_2 all Wa (zero_zero(nat) != number_number_of(nat,Wa) & ti(X_a,A_2) = zero_zero(X_a) <-> zero_zero(X_a) = hAPP(nat,X_a,power_power(X_a,A_2),number_number_of(nat,Wa)))))) # label(fact_86_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 534 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all N one_one(X_a) = abs_abs(X_a,hAPP(nat,X_a,power_power(X_a,number_number_of(X_a,min)),N))))) # label(fact_958_abs__power__minus__one) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 535 (all K all L_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L_1)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),minus_minus(int,K,L_1)),zero_zero(int))))) # label(fact_647_less__bin__lemma) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 536 (all X_a (ordere453448008miring(X_a) -> (all B_1 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_1)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),zero_zero(X_a))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,A_1,B_1)),zero_zero(X_a))))))) # label(fact_269_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 537 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),times_times(X_a,A_1,B_1)))))) # label(fact_735_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 538 (all B_1_1 all B_2_1 all X_a (dvd(X_a) -> hAPP(X_a,fun(X_a,bool),B_1_1,B_2_1) = hAPP(X_a,fun(X_a,bool),ti(fun(X_a,fun(X_a,bool)),B_1_1),B_2_1))) # label(tsy_c_hAPP_2_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 539 (all X_a (monoid_mult(X_a) -> (all A_1 times_times(X_a,A_1,A_1) = hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_540_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 540 (all B_1_1 all B_2_1 all X_a (dvd(X_a) -> ti(fun(X_a,bool),hAPP(X_a,fun(X_a,bool),B_1_1,B_2_1)) = hAPP(X_a,fun(X_a,bool),B_1_1,B_2_1))) # label(tsy_c_hAPP_2_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 541 (all X_a (number_ring(X_a) -> (all A_1 times_times(X_a,number_number_of(X_a,bit1(pls)),A_1) = ti(X_a,A_1)))) # label(fact_498_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 542 (all Ma all Na (one_one(int) = times_times(int,Ma,Na) <-> ti(int,Na) = one_one(int) & ti(int,Ma) = one_one(int) | ti(int,Na) = number_number_of(int,min) & number_number_of(int,min) = ti(int,Ma))) # label(fact_785_zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 543 (all Z ti(int,Z) = times_times(int,Z,one_one(int))) # label(fact_379_zmult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 544 (all X hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),X))) # label(fact_842_dvd_Oorder__refl) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 545 (all K all L_1 (bit1(L_1) = bit1(K) <-> ti(int,L_1) = ti(int,K))) # label(fact_42_rel__simps_I51_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 546 (all K_1 all M hBOOL(hAPP(int,bool,zcong(K_1,K_1),M))) # label(fact_882_zcong__refl) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 547 (all Wa all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z_1)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,Wa,one_one(int))),Z_1)))) # label(fact_517_add1__zle__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 548 (all K_1 all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),minus_minus(nat,M,N))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),N)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),M)))))) # label(fact_806_dvd__diffD) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 549 (all X_a (ordered_ring(X_a) -> (all A_2 all E_1 all C_1 all B all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,times_times(X_a,A_2,E_1),C_1)),plus_plus(X_a,times_times(X_a,B,E_1),D))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),plus_plus(X_a,times_times(X_a,minus_minus(X_a,B,A_2),E_1),D))))))) # label(fact_296_less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 550 (all X_a (ordered_ring(X_a) -> (all B_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)),B_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_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)),times_times(X_a,A_1,B_1))))))) # label(fact_270_split__mult__pos__le) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 551 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all Xa all Ya (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Xa),Ya)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),number_number_of(X_a,Xa)),number_number_of(X_a,Ya))))))) # label(fact_18_less__number__of) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 552 (all A_2 all B all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Ma),minus_minus(int,A_2,B))) <-> hBOOL(hAPP(int,bool,zcong(A_2,B),Ma)))) # label(fact_896_zcong__def) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 553 (all Z ti(int,Z) = times_times(int,one_one(int),Z)) # label(fact_378_zmult__1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 554 (all X_a (number_ring(X_a) -> (all A_1 plus_plus(X_a,number_number_of(X_a,pls),A_1) = ti(X_a,A_1)))) # label(fact_84_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 555 (all X_a (linordered_idom(X_a) -> (all A_1 all N 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),times_times(nat,number_number_of(nat,bit0(bit1(pls))),N))))))) # label(fact_700_zero__le__even__power_H) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 556 (all Z all X all Y all P_2 (hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,X),Y),one_one(int)),P_2)) -> hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,X),times_times(nat,Y,Z)),one_one(int)),P_2)))) # label(fact_913_zcong__zpower__zmult) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 557 (all M all I_2 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),J_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),plus_plus(nat,J_1,M))))) # label(fact_132_trans__less__add1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 558 (all K all Ma all Na (plus_plus(nat,K,Na) = plus_plus(nat,K,Ma) <-> Ma = Na)) # label(fact_125_nat__add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 559 (all P_2 all Y all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),Y)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),hAPP(nat,int,power_power(int,Y),N)))))) # label(fact_779_zpower__zdvd__prop1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 560 (all Xa all Ya (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,semiring_1_of_nat(int),Xa)),hAPP(nat,int,semiring_1_of_nat(int),Ya))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)))) # label(fact_811_zdvd__int) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 561 (all N all M zero_zero(nat) = minus_minus(nat,N,plus_plus(nat,N,M))) # label(fact_453_diff__add__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 562 (all Xa all Na (ti(int,Xa) != zero_zero(int) | Na = zero_zero(nat) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(nat,int,power_power(int,abs_abs(int,Xa)),Na))))) # label(fact_960_zero__less__zpower__abs__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 563 (all K all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),Na)) | zero_zero(nat) = K <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),times_times(nat,K,Ma)),times_times(nat,K,Na))))) # label(fact_984_nat__mult__dvd__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 564 (all X_a (linord581940658strict(X_a) -> (all A_2 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),zero_zero(X_a))) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),times_times(X_a,A_2,B))))))) # label(fact_291_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 565 (all B_1_1 all B_2_1 all X_a (power(X_a) -> hAPP(nat,X_a,B_1_1,B_2_1) = hAPP(nat,X_a,ti(fun(nat,X_a),B_1_1),B_2_1))) # label(tsy_c_hAPP_4_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 566 (all Y all X all P_2 (-hBOOL(hAPP(int,bool,zcong(X,zero_zero(int)),P_2)) -> (hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,Y),number_number_of(nat,bit0(bit1(pls)))),X),P_2)) -> -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),Y))))) # label(fact_837_Euler_Oaux____1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 567 (all X_a (linordered_idom(X_a) -> (all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X)) -> times_times(X_a,abs_abs(X_a,Y),X) = abs_abs(X_a,times_times(X_a,Y,X)))))) # label(fact_924_abs__mult__pos) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 568 (all X_a (ring(X_a) -> (all X all Y all A_1 all B_1 minus_minus(X_a,times_times(X_a,X,Y),times_times(X_a,A_1,B_1)) = plus_plus(X_a,times_times(X_a,X,minus_minus(X_a,Y,B_1)),times_times(X_a,minus_minus(X_a,X,A_1),B_1))))) # label(fact_289_mult__diff__mult) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 569 (all X_a (linordered_semidom(X_a) -> (all A_1 all N all N_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),N_2)) -> (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)),hAPP(nat,X_a,power_power(X_a,A_1),N_2)))))))) # label(fact_316_power__increasing) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 570 (all Z all X all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) -> (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z),Y)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),Z)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z),X)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Z))))) # label(fact_845_dvd_Oless__trans) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 571 (all B_1_1 bit1(B_1_1) = bit1(ti(int,B_1_1))) # label(tsy_c_Int_OBit1_arg1) # label(hypothesis) # label(non_clause). [assumption]. 1.52/1.81 572 (all Y all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),plus_plus(int,X,Y)))))) # label(fact_469_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 573 (all I_2 all K_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J_1)) -> plus_plus(nat,minus_minus(nat,J_1,K_1),I_2) = minus_minus(nat,plus_plus(nat,J_1,I_2),K_1))) # label(fact_676_diff__add__assoc2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 574 (all X_a (ring(X_a) -> (all A_2 all E_1 all C_1 all B all D (plus_plus(X_a,times_times(X_a,B,E_1),D) = plus_plus(X_a,times_times(X_a,A_2,E_1),C_1) <-> ti(X_a,C_1) = plus_plus(X_a,times_times(X_a,minus_minus(X_a,B,A_2),E_1),D))))) # label(fact_288_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 575 (all X_a (number_semiring(X_a) -> zero_zero(X_a) = number_number_of(X_a,pls))) # label(fact_81_semiring__numeral__0__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 576 (all I_2 all K_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J_1)) -> minus_minus(nat,I_2,minus_minus(nat,J_1,K_1)) = minus_minus(nat,plus_plus(nat,I_2,K_1),J_1))) # label(fact_666_diff__diff__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 577 (all Z1 all Z2 all Z3 times_times(real,Z1,times_times(real,Z2,Z3)) = times_times(real,times_times(real,Z1,Z2),Z3)) # label(fact_715_real__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 578 (all Ma all Wa ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa)) -> zero_zero(nat) = Ma) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa)) -> ti(int,Wa) = hAPP(nat,int,semiring_1_of_nat(int),Ma)) <-> nat_1(Wa) = Ma)) # label(fact_394_nat__eq__iff2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 579 (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_465_rel__simps_I27_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 580 (all M all I_2 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),J_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),plus_plus(nat,M,J_1))))) # label(fact_133_trans__less__add2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 581 (all Wa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),zero_zero(int))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(Wa)),zero_zero(int))))) # label(fact_58_bin__less__0__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 582 (all K_1 all L bit1(plus_plus(int,K_1,L)) = plus_plus(int,bit0(K_1),bit1(L))) # label(fact_91_add__Bit0__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 583 (all X_a (linord20386208strict(X_a) -> (all B_1 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_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,A_1,B_1)),zero_zero(X_a)))))))) # label(fact_419_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 584 (all K all L_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,K)),number_number_of(int,L_1))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L_1)))) # label(fact_16_less__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 585 (all Ma all Va (number_number_of(int,Va) = hAPP(nat,int,semiring_1_of_nat(int),Ma) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,Va))) & Ma = nat_1(number_number_of(int,Va)))) # label(fact_398_int__eq__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 586 (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_30_less__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 587 (all B_1_1 all B_2_1 all X_a (no_zero_divisors(X_a) -> times_times(X_a,B_1_1,ti(X_a,B_2_1)) = times_times(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_3_arg2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.81 588 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),abs_abs(real,X)),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,power_power(real,X),number_number_of(nat,bit0(bit1(pls))))),one_one(real))))) # label(fact_975_less__one__imp__sqr__less__one) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 589 (all X_a (linordered_idom(X_a) -> (all A_1 times_times(X_a,abs_abs(X_a,A_1),abs_abs(X_a,A_1)) = times_times(X_a,A_1,A_1)))) # label(fact_952_abs__mult__self) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 590 (all V_2 all V_1 ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls)) -> times_times(nat,number_number_of(nat,V_1),number_number_of(nat,V_2)) = number_number_of(nat,times_times(int,V_1,V_2))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls)) -> zero_zero(nat) = times_times(nat,number_number_of(nat,V_1),number_number_of(nat,V_2))))) # label(fact_694_mult__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 591 (all W pls = times_times(int,pls,W)) # label(fact_370_mult__Pls) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 592 (all X_a (one(X_a) -> (all Xa (one_one(X_a) = ti(X_a,Xa) <-> one_one(X_a) = ti(X_a,Xa))))) # label(fact_211_one__reorient) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 593 (all L bit1(minus_minus(int,min,L)) = minus_minus(int,pls,bit1(L))) # label(fact_782_diff__bin__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 594 (all K all P_1 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D)) -> ((all X_1 (hBOOL(hAPP(int,bool,P_1,X_1)) -> hBOOL(hAPP(int,bool,P_1,plus_plus(int,X_1,D))))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) -> (all X_1 (hBOOL(hAPP(int,bool,P_1,X_1)) -> hBOOL(hAPP(int,bool,P_1,plus_plus(int,X_1,times_times(int,K,D)))))))))) # label(fact_977_incr__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 595 (all X all Y all Z plus_plus(nat,X,plus_plus(nat,Y,Z)) = plus_plus(nat,Y,plus_plus(nat,X,Z))) # label(fact_123_nat__add__left__commute) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 596 (all B_1_1 all B_2_1 all X_a (group_add(X_a) -> minus_minus(X_a,B_1_1,ti(X_a,B_2_1)) = minus_minus(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Ominus__class_Ominus_0_arg2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 597 (all M times_times(nat,M,zero_zero(nat)) = zero_zero(nat)) # label(fact_595_mult__0__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 598 (all R_1 all Q all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (plus_plus(int,R_1,times_times(int,A_1,Q)) = 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)))))) # label(fact_557_self__quotient__aux1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 599 (all Z_3 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_eq(int),zero_zero(int)),Z_3)) -> (ti(int,Z_1) = ti(int,Z_3) <-> nat_1(Z_1) = nat_1(Z_3))))) # label(fact_261_eq__nat__nat__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 600 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),K)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),bit0(K))))) # label(fact_754_rel__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 601 (all K_1 all I_2 all J_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),I_2),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_2),K_1))))) # label(fact_716_real__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 602 (all Xa all Ya (Ya = Xa <-> hAPP(nat,int,semiring_1_of_nat(int),Ya) = hAPP(nat,int,semiring_1_of_nat(int),Xa))) # label(fact_159_Nat__Transfer_Otransfer__int__nat__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 603 (all K_1 all I_2 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),plus_plus(nat,I_2,K_1)),plus_plus(nat,J_1,K_1))))) # label(fact_618_add__le__mono1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 604 (all A_1 all B_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A_1),B_1)) | B_1 = zero_zero(nat))) # label(fact_804_divides__ge) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 605 (all X_a (ordere216010020id_add(X_a) -> (all B_1 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_1),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),plus_plus(X_a,A_1,C)))))))) # label(fact_436_add__increasing) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 606 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx all Ry times_times(X_a,Lx,times_times(X_a,Ly,times_times(X_a,Rx,Ry))) = times_times(X_a,times_times(X_a,Lx,Ly),times_times(X_a,Rx,Ry))))) # label(fact_245_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 607 (all Xa (-hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),times_times(real,Xa,Xa))) <-> Xa = zero_zero(real))) # label(fact_653_not__real__square__gt__zero) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 608 (all Z ti(int,Z) = plus_plus(int,Z,zero_zero(int))) # label(fact_79_zadd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 609 (all X all Y all Z (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Z)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),times_times(real,Z,X)),times_times(real,Z,Y)))))) # label(fact_654_real__mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 610 (all Na 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),Na)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),minus_minus(nat,Ma,K)),minus_minus(nat,Na,K))))))) # label(fact_633_le__diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 611 (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_373_less__eq__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 612 (all N (hAPP(nat,int,power_power(int,number_number_of(int,min)),N) = number_number_of(int,min) | hAPP(nat,int,power_power(int,number_number_of(int,min)),N) = one_one(int))) # label(fact_876_neg__one__power) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 613 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all Ya (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),Ya)) <-> 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,Ya))))))) # label(fact_389_le__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 614 (all X_a (linord20386208strict(X_a) -> (all C all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_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),times_times(X_a,A_1,C)),times_times(X_a,B_1,C)))))))) # label(fact_421_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 615 (all K_1 all L times_times(int,bit1(K_1),L) = plus_plus(int,bit0(times_times(int,K_1,L)),L)) # label(fact_509_mult__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 616 (all K_1 all L bit0(K_1) != bit1(L)) # label(fact_69_rel__simps_I49_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 617 (all X all Y times_times(int,hAPP(nat,int,semiring_1_of_nat(int),X),hAPP(nat,int,semiring_1_of_nat(int),Y)) = hAPP(nat,int,semiring_1_of_nat(int),times_times(nat,X,Y))) # label(fact_663_Nat__Transfer_Otransfer__int__nat__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 618 (all X_a (linordered_idom(X_a) -> (all N abs_abs(X_a,hAPP(nat,X_a,semiring_1_of_nat(X_a),N)) = hAPP(nat,X_a,semiring_1_of_nat(X_a),N)))) # label(fact_935_abs__of__nat) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 619 (all B_1 all Q_1 all R_2 all Q all R_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,times_times(int,B_1,Q_1),R_2)),plus_plus(int,times_times(int,B_1,Q),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_1),R_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_1),R_2)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q),Q_1))))))) # label(fact_705_unique__quotient__lemma__neg) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 620 (all Ma all Na (Ma = Na <-> hAPP(nat,int,semiring_1_of_nat(int),Na) = hAPP(nat,int,semiring_1_of_nat(int),Ma))) # label(fact_48_int__int__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 621 (all B_1 all Q all R_1 all B_2 all Q_1 all R_2 (plus_plus(int,times_times(int,B_2,Q_1),R_2) = plus_plus(int,times_times(int,B_1,Q),R_1) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),plus_plus(int,times_times(int,B_2,Q_1),R_2))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_2),B_2)) -> (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_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),B_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q),Q_1))))))))) # label(fact_706_zdiv__mono2__lemma) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 622 (all U all Ma all Na all J all I_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),I_1)) -> (Na = plus_plus(nat,times_times(nat,minus_minus(nat,I_1,J),U),Ma) <-> plus_plus(nat,times_times(nat,I_1,U),Ma) = plus_plus(nat,times_times(nat,J,U),Na)))) # label(fact_989_nat__eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 623 (all X_a (order(X_a) -> (all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) -> (ti(X_a,Y) != ti(X_a,X) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))))))) # label(fact_884_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 624 (all B_1 all D_1 all A_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_1),A_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_1),plus_plus(nat,A_1,B_1))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_1),B_1))))) # label(fact_815_divides__add__revr) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 625 (all N nat_1(hAPP(nat,int,semiring_1_of_nat(int),N)) = N) # label(fact_346_nat__int) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 626 (all Z all W (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),Z)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),W)))) # label(fact_253_zle__linear) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 627 (all X_a (semiring_1(X_a) -> (all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) -> plus_plus(X_a,hAPP(nat,X_a,semiring_1_of_nat(X_a),nat_1(X)),hAPP(nat,X_a,semiring_1_of_nat(X_a),nat_1(X))) = hAPP(nat,X_a,semiring_1_of_nat(X_a),nat_1(times_times(int,number_number_of(int,bit0(bit1(pls))),X))))))) # label(fact_470_of__nat__double) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 628 (all A_1 (one_one(int) = hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls)))) -> one_one(int) = abs_abs(int,A_1))) # label(fact_968_power2__eq1__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 629 (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_773_one__dvd) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 630 (all X_a (linordered_idom(X_a) -> (all X all Y hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,X_a,power_power(X_a,Y),number_number_of(nat,bit0(bit1(pls)))))))))) # label(fact_552_sum__power2__ge__zero) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 631 (all T all A ti(T,ti(T,A)) = ti(T,A)) # label(help_ti_idem) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 632 (all B_1_1 bit0(B_1_1) = ti(int,bit0(B_1_1))) # label(tsy_c_Int_OBit0_res) # label(hypothesis) # label(non_clause). [assumption]. 1.52/1.82 633 (all X_a (ordere236663937imp_le(X_a) -> (all A_2 all C_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,A_2,C_1)),plus_plus(X_a,B,C_1))))))) # label(fact_359_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 634 (all Xa all Ya (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,semiring_1_of_nat(int),Xa)),hAPP(nat,int,semiring_1_of_nat(int),Ya))))) # label(fact_810_Nat__Transfer_Otransfer__int__nat__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 635 (all A_1 all B_1 hAPP(nat,int,power_power(int,minus_minus(int,A_1,B_1)),number_number_of(nat,bit0(bit1(pls)))) = plus_plus(int,minus_minus(int,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls)))),times_times(int,times_times(int,number_number_of(int,bit0(bit1(pls))),A_1),B_1)),hAPP(nat,int,power_power(int,B_1),number_number_of(nat,bit0(bit1(pls)))))) # label(fact_703_zdiff__power2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 636 (all X all Y plus_plus(int,hAPP(nat,int,semiring_1_of_nat(int),X),hAPP(nat,int,semiring_1_of_nat(int),Y)) = hAPP(nat,int,semiring_1_of_nat(int),plus_plus(nat,X,Y))) # label(fact_174_Nat__Transfer_Otransfer__int__nat__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 637 (all X_a (linord20386208strict(X_a) -> (all C all D_1 all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B_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),times_times(X_a,A_1,C)),times_times(X_a,B_1,D_1)))))))))) # label(fact_307_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 638 (all X_a (comm_semiring_1(X_a) -> (all A_1 all C all D_1 plus_plus(X_a,C,plus_plus(X_a,A_1,D_1)) = plus_plus(X_a,A_1,plus_plus(X_a,C,D_1))))) # label(fact_209_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 639 (all Ma all Na 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),times_times(nat,K,Ma)),times_times(nat,K,Na))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na))))) # label(fact_986_nat__mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 640 (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_eq(int),bit1(K1)),bit0(K2))))) # label(fact_508_less__eq__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 641 (all N all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N),M)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),M),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),N),M)))) # label(fact_738_zdvd__bounds) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 642 (all I_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_1),zero_zero(int))) <-> nat_1(I_1) = zero_zero(nat))) # label(fact_315_nat__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 643 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 all B_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),minus_minus(X_a,abs_abs(X_a,A_1),abs_abs(X_a,B_1))),abs_abs(X_a,minus_minus(X_a,A_1,B_1))))))) # label(fact_940_abs__triangle__ineq2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 644 (all K_1 succ(bit1(K_1)) = bit0(succ(K_1))) # label(fact_515_succ__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 645 (all P_1 ((exists X_1 hBOOL(hAPP(int,bool,P_1,hAPP(nat,int,semiring_1_of_nat(int),X_1)))) <-> (exists X_1 (hBOOL(hAPP(int,bool,P_1,X_1)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)))))) # label(fact_473_transfer__int__nat__quantifiers_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 646 (all U all Ma all Na all I_1 all J (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J)) -> (Ma = plus_plus(nat,times_times(nat,minus_minus(nat,J,I_1),U),Na) <-> plus_plus(nat,times_times(nat,I_1,U),Ma) = plus_plus(nat,times_times(nat,J,U),Na)))) # label(fact_992_nat__eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 647 (all K_1 plus_plus(int,K_1,K_1) = bit0(K_1)) # label(fact_78_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 648 (all K_1 all I_2 all J_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_2),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_2),K_1))))) # label(fact_255_zle__trans) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 649 (all K all Na all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),plus_plus(int,Na,times_times(int,K,Ma)))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),Na)))) # label(fact_748_zdvd__reduce) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 650 (all X_a (ordere142940540dd_abs(X_a) -> (all A_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),abs_abs(X_a,A_2)),zero_zero(X_a))) <-> ti(X_a,A_2) = zero_zero(X_a))))) # label(fact_948_abs__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 651 (all Xa all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya)) <-> Xa = Ya | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya)))) # label(fact_611_less__eq__real__def) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 652 (all X_a (ordere236663937imp_le(X_a) -> (all C_1 all A_2 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,C_1,A_2)),plus_plus(X_a,C_1,B))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B)))))) # label(fact_225_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 653 (all L_1 (min = bit1(L_1) <-> min = ti(int,L_1))) # label(fact_770_rel__simps_I43_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 654 (all N all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) -> hAPP(nat,nat,power_power(nat,nat_1(X)),N) = nat_1(hAPP(nat,int,power_power(int,X),N)))) # label(fact_399_Nat__Transfer_Otransfer__nat__int__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 655 (all X_a (no_zero_divisors(X_a) -> (all B_1 all A_1 (ti(X_a,A_1) != zero_zero(X_a) -> (zero_zero(X_a) != ti(X_a,B_1) -> zero_zero(X_a) != times_times(X_a,A_1,B_1)))))) # label(fact_331_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 656 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all Ya (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),Ya)) <-> 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,Ya))))))) # label(fact_50_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 657 (all B_1_1 all B_2_1 all X_a (power(X_a) -> hAPP(nat,X_a,B_1_1,B_2_1) = hAPP(nat,X_a,B_1_1,ti(nat,B_2_1)))) # label(tsy_c_hAPP_4_arg2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 658 (all X_a (semiri456707255roduct(X_a) -> (all B all A_2 (zero_zero(X_a) = ti(X_a,A_2) <-> ti(X_a,B) = plus_plus(X_a,B,A_2))))) # label(fact_212_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 659 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K)),pls)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),pls)))) # label(fact_54_rel__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 660 (all N zero_zero(nat) = minus_minus(nat,zero_zero(nat),N)) # label(fact_338_diff__0__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 661 (all X_a (number_ring(X_a) -> (all A_1 plus_plus(X_a,A_1,number_number_of(X_a,pls)) = ti(X_a,A_1)))) # label(fact_85_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 662 (all B_1_1 all B_2_1 minus_minus(nat,B_1_1,B_2_1) = ti(nat,minus_minus(nat,B_1_1,B_2_1))) # label(tsy_c_Groups_Ominus__class_Ominus_1_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 663 (all Ma all Na (zero_zero(nat) = Na & Ma = zero_zero(nat) <-> plus_plus(nat,Ma,Na) = zero_zero(nat))) # label(fact_154_add__is__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 664 (all Y all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),times_times(int,X,Y)))))) # label(fact_283_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 665 (all N 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),minus_minus(nat,J_1,N)),K_1)))) # label(fact_360_less__imp__diff__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 666 (all X_a (zero_neq_one(X_a) -> one_one(X_a) != zero_zero(X_a))) # label(fact_164_zero__neq__one) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 667 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),K)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),bit1(K))))) # label(fact_755_rel__simps_I26_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 668 (all A_1 all B_1 (A_1 != B_1 -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B_1),A_1))))) # label(fact_867_dvd_Oneq__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 669 (all W all Z nat_1(abs_abs(int,times_times(int,W,Z))) = times_times(nat,nat_1(abs_abs(int,W)),nat_1(abs_abs(int,Z)))) # label(fact_918_nat__abs__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 670 (all Y all N all P_2 (hBOOL(hAPP(int,bool,zprime,P_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),hAPP(nat,int,power_power(int,Y),N))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),Y)))))) # label(fact_790_zpower__zdvd__prop2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 671 (all X_a (monoid_mult(X_a) -> (all A_1 all N times_times(X_a,hAPP(nat,X_a,power_power(X_a,A_1),N),A_1) = times_times(X_a,A_1,hAPP(nat,X_a,power_power(X_a,A_1),N))))) # label(fact_362_power__commutes) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 672 (all X_a (linord20386208strict(X_a) -> (all C all D_1 all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_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)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,A_1,C)),times_times(X_a,B_1,D_1)))))))))) # label(fact_305_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 673 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> abs_abs(X_a,number_number_of(X_a,min)) = one_one(X_a))) # label(fact_929_abs__minus__one) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 674 (all B_1_1 all B_2_1 legendre(B_1_1,B_2_1) = legendre(ti(int,B_1_1),B_2_1)) # label(tsy_c_Residues_OLegendre_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 675 (all V_2 all V_1 ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls)) -> number_number_of(nat,V_2) = plus_plus(nat,number_number_of(nat,V_1),number_number_of(nat,V_2))) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_2),pls)) -> number_number_of(nat,plus_plus(int,V_1,V_2)) = plus_plus(nat,number_number_of(nat,V_1),number_number_of(nat,V_2))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_2),pls)) -> number_number_of(nat,V_1) = plus_plus(nat,number_number_of(nat,V_1),number_number_of(nat,V_2)))))) # label(fact_36_add__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 676 (all X_a (comm_semiring_1(X_a) -> (all X all Y all Z times_times(X_a,X,plus_plus(X_a,Y,Z)) = plus_plus(X_a,times_times(X_a,X,Y),times_times(X_a,X,Z))))) # label(fact_345_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 677 (all Ma all K all Na (times_times(nat,Na,K) = times_times(nat,Ma,K) <-> K = zero_zero(nat) | Na = Ma)) # label(fact_598_mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 678 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 all B_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),abs_abs(X_a,plus_plus(X_a,A_1,B_1))),plus_plus(X_a,abs_abs(X_a,A_1),abs_abs(X_a,B_1))))))) # label(fact_942_abs__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 679 (all K all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),plus_plus(nat,K,Ma)),plus_plus(nat,K,Na))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)))) # label(fact_621_nat__add__left__cancel__le) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 680 (all X_a (comm_semiring_1(X_a) -> (all C all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B_1)) -> (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),plus_plus(X_a,B_1,C)))))))) # label(fact_774_dvd__add) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 681 (all B_1_1 all X_a (number(X_a) -> number_number_of(X_a,B_1_1) = ti(X_a,number_number_of(X_a,B_1_1)))) # label(tsy_c_Int_Onumber__class_Onumber__of_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 682 (all X all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) -> X = Y))) # label(fact_859_dvd_Oantisym) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 683 (all X_a (ordered_ring(X_a) -> (all A_2 all E_1 all C_1 all B all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,times_times(X_a,A_2,E_1),C_1)),plus_plus(X_a,times_times(X_a,B,E_1),D))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,times_times(X_a,minus_minus(X_a,A_2,B),E_1),C_1)),D)))))) # label(fact_295_less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 684 (all K all L_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K)),bit0(L_1))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L_1)))) # label(fact_34_rel__simps_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 685 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx all Ry times_times(X_a,times_times(X_a,Lx,Ly),times_times(X_a,Rx,Ry)) = times_times(X_a,times_times(X_a,Lx,Rx),times_times(X_a,Ly,Ry))))) # label(fact_247_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 686 (all Z all W (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Z),W)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),W),Z)) -> W = Z))) # label(fact_717_real__le__antisym) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 687 (all Z all W plus_plus(int,W,Z) = plus_plus(int,Z,W)) # label(fact_47_zadd__commute) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 688 (all N plus_plus(nat,zero_zero(nat),N) = N) # label(fact_156_plus__nat_Oadd__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 689 (all N hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),N)))) # label(fact_475_zero__zle__int) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 690 (all B_1_1 all B_2_1 all X_a (ab_semigroup_add(X_a) -> plus_plus(X_a,ti(X_a,B_1_1),B_2_1) = plus_plus(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Oplus__class_Oplus_1_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 691 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),Ma)),hAPP(nat,int,semiring_1_of_nat(int),Na))))) # label(fact_150_zless__int) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 692 (all B_1 all A_1 all P_2 (hBOOL(hAPP(int,bool,zprime,P_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,zcong(times_times(int,A_1,B_1),zero_zero(int)),P_2)) -> hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_2)) | hBOOL(hAPP(int,bool,zcong(B_1,zero_zero(int)),P_2)))))) # label(fact_828_zcong__zprime__prod__zero) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 693 (all X_a (ordere236663937imp_le(X_a) -> (all C all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,C,A_1)),plus_plus(X_a,C,B_1))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1)))))) # label(fact_220_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 694 (all X_a (linord626643107strict(X_a) -> (all V_1 all U_1 all Y all X all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),U_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),V_1)) -> (one_one(X_a) = plus_plus(X_a,U_1,V_1) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,times_times(X_a,U_1,X),times_times(X_a,V_1,Y))),A_1)))))))))) # label(fact_401_convex__bound__lt) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 695 (all Xa all Na (Na = zero_zero(nat) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Xa)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,power_power(nat,Xa),Na))))) # label(fact_147_zero__less__power__nat__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 696 (all X_a all B_1_1 all B_2_1 all B_3 if(X_a,B_1_1,B_2_1,B_3) = if(X_a,ti(bool,B_1_1),B_2_1,B_3)) # label(tsy_c_If_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 697 (all X_a (number_ring(X_a) -> (all Z plus_plus(X_a,Z,Z) = times_times(X_a,Z,number_number_of(X_a,bit0(bit1(pls))))))) # label(fact_535_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 698 (all X_a (cancel_semigroup_add(X_a) -> (all B all A_2 all C_1 (ti(X_a,B) = ti(X_a,C_1) <-> plus_plus(X_a,C_1,A_2) = plus_plus(X_a,B,A_2))))) # label(fact_203_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 699 (all K_1 all M all N (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_1),minus_minus(int,M,N))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_1),N)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_1),M))))) # label(fact_726_zdvd__zdiffD) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 700 (all X all Z all D_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),minus_minus(int,X,times_times(int,plus_plus(int,abs_abs(int,minus_minus(int,X,Z)),one_one(int)),D_1))),Z)))) # label(fact_970_decr__lemma) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 701 (all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) -> ti(int,Z) = hAPP(nat,int,semiring_1_of_nat(int),nat_1(Z)))) # label(fact_319_nat__0__le) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 702 (all A_1 all B_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),minus_minus(int,B_1,one_one(int)))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B_1)))) # label(fact_897_norR__mem__unique__aux) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 703 (all N all M hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),plus_plus(nat,N,M)))) # label(fact_623_le__add1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 704 (all K_1 all M all N minus_minus(nat,plus_plus(nat,K_1,M),plus_plus(nat,K_1,N)) = minus_minus(nat,M,N)) # label(fact_368_Nat_Odiff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 705 (all X_a (linordered_semidom(X_a) -> (all A_1 all N all N_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N_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(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_2)),hAPP(nat,X_a,power_power(X_a,A_1),N))))))))) # label(fact_110_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 706 (all X_a (ordered_ring(X_a) -> (all C all B_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),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),times_times(X_a,C,A_1)),times_times(X_a,C,B_1)))))))) # label(fact_273_mult__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 707 (all B_1_1 all B_2_1 all X_a (group_add(X_a) -> minus_minus(X_a,B_1_1,B_2_1) = ti(X_a,minus_minus(X_a,B_1_1,B_2_1)))) # label(tsy_c_Groups_Ominus__class_Ominus_0_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 708 (all M all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),X)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),M)) -> -hBOOL(hAPP(int,bool,zcong(X,zero_zero(int)),M))))) # label(fact_821_zcong__not__zero) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 709 (all Y all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),times_times(real,X,Y)))))) # label(fact_655_real__mult__order) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 710 (all K_1 min != bit0(K_1)) # label(fact_766_rel__simps_I45_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 711 (all X_a (linord20386208strict(X_a) -> (all C all D_1 all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_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)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,A_1,C)),times_times(X_a,B_1,D_1)))))))))) # label(fact_304_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 712 (all K_1 all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),minus_minus(nat,M,N))) -> (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),M)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),N)))))) # label(fact_807_dvd__diffD1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 713 (all M all I_2 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),plus_plus(nat,J_1,M))))) # label(fact_620_trans__le__add1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 714 (all X_a (ordere216010020id_add(X_a) -> (all B_1 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_1),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_1),plus_plus(X_a,A_1,C)))))))) # label(fact_490_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 715 (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_441_not__one__le__zero) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 716 (all Z1 all Z2 all Z3 plus_plus(int,plus_plus(int,Z1,Z2),Z3) = plus_plus(int,Z1,plus_plus(int,Z2,Z3))) # label(fact_45_zadd__assoc) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 717 (all P_1 all I_1 (hBOOL(hAPP(nat,bool,P_1,nat_1(I_1))) <-> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),zero_zero(int))) -> hBOOL(hAPP(nat,bool,P_1,zero_zero(nat)))) & (all N_1 (ti(int,I_1) = hAPP(nat,int,semiring_1_of_nat(int),N_1) -> hBOOL(hAPP(nat,bool,P_1,N_1)))))) # label(fact_523_split__nat) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 718 (all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Xa),one_one(int))) <-> abs_abs(int,Xa) = one_one(int))) # label(fact_923_zdvd1__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 719 (all B_1 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_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_1),M)) -> (hBOOL(hAPP(int,bool,zcong(A_1,B_1),M)) -> ti(int,B_1) = ti(int,A_1))))))) # label(fact_899_zcong__zless__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 720 (all X_a (comm_semiring_1(X_a) -> (all A_1 plus_plus(X_a,A_1,zero_zero(X_a)) = ti(X_a,A_1)))) # label(fact_214_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 721 (all B_1_1 all B_2_1 all X_a (semiring(X_a) -> times_times(X_a,B_1_1,B_2_1) = ti(X_a,times_times(X_a,B_1_1,B_2_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_2_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 722 (all B_1_1 ti(nat,nat_1(B_1_1)) = nat_1(B_1_1)) # label(tsy_c_Int_Onat_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 723 (all X_a (number_semiring(X_a) -> (all V_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_1)) -> number_number_of(X_a,succ(V_1)) = plus_plus(X_a,number_number_of(X_a,V_1),one_one(X_a)))))) # label(fact_480_semiring__number__of__add__1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 724 (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_546_zero__le__power2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 725 (all W all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W),Z)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,W,one_one(int))),Z)))) # label(fact_518_zless__imp__add1__zle) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 726 (all X_a (number_ring(X_a) -> (all V_1 all W all Z times_times(X_a,number_number_of(X_a,times_times(int,V_1,W)),Z) = times_times(X_a,number_number_of(X_a,V_1),times_times(X_a,number_number_of(X_a,W),Z))))) # label(fact_260_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 727 (all P_1 all Ma all Na ((hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,Na),Ma))) -> ((Na = Ma -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,Na),Ma))) -> ((hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),Ma)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,Na),Ma))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,Na),Ma)))))) # label(fact_138_nat__less__cases) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 728 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),abs_abs(X_a,A_1)),zero_zero(X_a)))))) # label(fact_946_abs__not__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 729 (all Ya all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Ya)) -> (ti(int,Xa) = ti(int,Ya) <-> nat_1(Xa) = nat_1(Ya))))) # label(fact_282_transfer__nat__int__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 730 (all K_1 number_number_of(int,K_1) = ti(int,K_1)) # label(fact_120_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 731 (all X_a (semiring_1(X_a) -> (all M all N plus_plus(X_a,hAPP(nat,X_a,semiring_1_of_nat(X_a),M),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)) = hAPP(nat,X_a,semiring_1_of_nat(X_a),plus_plus(nat,M,N))))) # label(fact_170_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 732 (all N 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)) -> -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N),M))))) # label(fact_745_zdvd__not__zless) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 733 -(hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n))),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))) -> -hBOOL(hAPP(int,bool,twoSqu658283162sum2sq,times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n)))))) # label(fact_559_smaller_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 734 (all Ma all Na (Na != zero_zero(nat) & zero_zero(nat) = Ma <-> hAPP(nat,nat,power_power(nat,Ma),Na) = zero_zero(nat))) # label(fact_228_nat__power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 735 (all Xa all Ya (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) <-> Ya = Xa)) # label(fact_873_divides__antisym) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 736 (all Z all Z_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_2),Z)) -> nat_1(minus_minus(int,Z,Z_2)) = minus_minus(nat,nat_1(Z),nat_1(Z_2))))) # label(fact_684_nat__diff__distrib) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 737 (all K_1 succ(bit0(K_1)) = bit1(K_1)) # label(fact_514_succ__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 738 (all K all L_1 (ti(int,K) = ti(int,L_1) <-> bit0(K) = bit0(L_1))) # label(fact_43_rel__simps_I48_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 739 (all B_1_1 all X_a (ordere142940540dd_abs(X_a) -> abs_abs(X_a,B_1_1) = abs_abs(X_a,ti(X_a,B_1_1)))) # label(tsy_c_Groups_Oabs__class_Oabs_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 740 (all K_1 all I_2 all J_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_2),J_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,K_1,I_2)),plus_plus(int,K_1,J_1))))) # label(fact_383_zadd__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 741 (all B_1_1 all B_2_1 all X_a (mult_zero(X_a) -> times_times(X_a,ti(X_a,B_1_1),B_2_1) = times_times(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_4_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 742 (all Na all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma)) -> (one_one(int) = ti(int,Na) & ti(int,Ma) = one_one(int) <-> one_one(int) = times_times(int,Ma,Na)))) # label(fact_510_pos__zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 743 (all Z_2 all Z 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_2),Z)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,W_2,Z_2)),plus_plus(int,W,Z)))))) # label(fact_471_zadd__zless__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 744 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 abs_abs(X_a,A_1) = abs_abs(X_a,abs_abs(X_a,A_1))))) # label(fact_937_abs__idempotent) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 745 (all X_a (linorder(X_a) & number(X_a) -> (all Va all Wa (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),number_number_of(X_a,Wa)),number_number_of(X_a,Va))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,Va)),number_number_of(X_a,Wa))))))) # label(fact_448_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 746 (all X all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) -> -(-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X))))) # label(fact_844_dvd_Oless__asym) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 747 (all Xa all Ya 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_eq(real),times_times(real,Z_1,Xa)),times_times(real,Z_1,Ya))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya))))) # label(fact_588_real__mult__le__cancel__iff2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 748 (all U all Ma all Na all I_1 all J (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,times_times(nat,I_1,U),Ma)),plus_plus(nat,times_times(nat,J,U),Na))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),plus_plus(nat,times_times(nat,minus_minus(nat,J,I_1),U),Na)))))) # label(fact_979_nat__less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 749 (all X_a (number_ring(X_a) -> number_number_of(X_a,pls) = zero_zero(X_a))) # label(fact_83_semiring__norm_I112_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 750 (all X_a (linord20386208strict(X_a) -> (all C all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,C,A_1)),times_times(X_a,C,B_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_eq(X_a),A_1),B_1))))))) # label(fact_298_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 751 (all Z all X all Y (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),Z)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z),X)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Z))))) # label(fact_847_dvd_Oless__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 752 (all Xa all Ya (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),Xa)),hAPP(nat,int,semiring_1_of_nat(int),Ya))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Xa),Ya)))) # label(fact_148_Nat__Transfer_Otransfer__int__nat__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 753 (all M all N all I_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),I_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,power_power(nat,I_2),M)),hAPP(nat,nat,power_power(nat,I_2),N))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))))) # label(fact_141_nat__power__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 754 (all X_a (ring_char_0(X_a) & number_ring(X_a) -> (all Xa all Ya (number_number_of(X_a,Ya) = number_number_of(X_a,Xa) <-> ti(int,Ya) = ti(int,Xa))))) # label(fact_40_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 755 (all M all Y all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),X)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Y)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M)) -> (hBOOL(hAPP(int,bool,zcong(X,Y),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y),M)) -> ti(int,X) = ti(int,Y)))))))) # label(fact_820_zcong__less__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 756 (all Ma all Na (Ma != Na & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)))) # label(fact_609_nat__less__le) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 757 (all J all K all I_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),minus_minus(nat,J,K)),I_1)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),plus_plus(nat,I_1,K))))) # label(fact_667_le__diff__conv) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 758 (all M all N plus_plus(int,hAPP(nat,int,semiring_1_of_nat(int),M),hAPP(nat,int,semiring_1_of_nat(int),N)) = hAPP(nat,int,semiring_1_of_nat(int),plus_plus(nat,M,N))) # label(fact_23_zadd__int) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 759 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B_1 times_times(X_a,B_1,A_1) = times_times(X_a,A_1,B_1)))) # label(fact_239_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 760 (all X_a (linord20386208strict(X_a) -> (all B_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),B_1),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,B_1,A_1)),zero_zero(X_a)))))))) # label(fact_415_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 761 (all A_1 all P_2 ((-hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_2)) -> (-hBOOL(hAPP(int,bool,quadRes(P_2),A_1)) -> legendre(A_1,P_2) = number_number_of(int,min)) & (hBOOL(hAPP(int,bool,quadRes(P_2),A_1)) -> one_one(int) = legendre(A_1,P_2))) & (hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_2)) -> zero_zero(int) = legendre(A_1,P_2)))) # label(fact_877_Legendre__def) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 762 (all X_a (linord20386208strict(X_a) -> (all B_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),times_times(X_a,B_1,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_1))))))) # label(fact_417_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 763 (all X_a (comm_semiring_1(X_a) -> (all A_1 all M plus_plus(X_a,times_times(X_a,A_1,M),M) = times_times(X_a,plus_plus(X_a,A_1,one_one(X_a)),M)))) # label(fact_447_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 764 (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)),plus_plus(X_a,one_one(X_a),one_one(X_a)))))) # label(fact_111_zero__less__two) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 765 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),abs_abs(X_a,A_1)))))) # label(fact_953_abs__ge__self) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 766 (all Ma all Ta all K (zero_zero(int) != ti(int,K) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Ma),Ta)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),times_times(int,K,Ma)),times_times(int,K,Ta)))))) # label(fact_907_zdvd__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 767 (all A_2 all Ma (hBOOL(hAPP(int,bool,zcong(A_2,zero_zero(int)),Ma)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Ma),A_2)))) # label(fact_802_zcong__zero__equiv__div) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 768 (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_874_neg__one__power__eq__mod__m) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 769 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K1)),bit1(K2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)))) # label(fact_463_less__eq__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 770 (all Z all W times_times(real,W,Z) = times_times(real,Z,W)) # label(fact_713_real__mult__commute) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 771 (all X_a (linordered_idom(X_a) -> one_one(X_a) = abs_abs(X_a,one_one(X_a)))) # label(fact_933_abs__one) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 772 (all K_1 all L all I_2 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),J_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_1),L)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,I_2,K_1)),plus_plus(nat,J_1,L)))))) # label(fact_135_add__less__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 773 (all X_a (ring_1(X_a) -> (all X minus_minus(X_a,times_times(X_a,X,X),one_one(X_a)) = times_times(X_a,plus_plus(X_a,X,one_one(X_a)),minus_minus(X_a,X,one_one(X_a)))))) # label(fact_297_real__squared__diff__one__factored) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 774 (all X_a (comm_semiring_1(X_a) -> (all Lx all Rx all Ry times_times(X_a,times_times(X_a,Lx,Rx),Ry) = times_times(X_a,Lx,times_times(X_a,Rx,Ry))))) # label(fact_241_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 775 -(all X_1 all Y_1 times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n))) != plus_plus(int,hAPP(nat,int,power_power(int,X_1),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,int,power_power(int,Y_1),number_number_of(nat,bit0(bit1(pls)))))) # label(fact_568__096_B_Bthesis_O_A_I_B_Bx_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_061_A_I4_A) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 776 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 all B_1 abs_abs(X_a,minus_minus(X_a,A_1,B_1)) = abs_abs(X_a,minus_minus(X_a,B_1,A_1))))) # label(fact_950_abs__minus__commute) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 777 (all X_a (comm_semiring_1(X_a) -> (all A_1 zero_zero(X_a) = times_times(X_a,A_1,zero_zero(X_a))))) # label(fact_333_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 778 (all K all L_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(K)),bit0(L_1))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L_1)))) # label(fact_507_rel__simps_I33_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 779 (all X all Y all Z times_times(int,hAPP(nat,int,power_power(int,X),Y),hAPP(nat,int,power_power(int,X),Z)) = hAPP(nat,int,power_power(int,X),plus_plus(nat,Y,Z))) # label(fact_478_zpower__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 780 (all B_1_1 all B_2_1 all X_a (no_zero_divisors(X_a) -> ti(X_a,times_times(X_a,B_1_1,B_2_1)) = times_times(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_3_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 781 (all D_1 all I_2 (zero_zero(int) != ti(int,I_2) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),I_2)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),abs_abs(int,D_1)),abs_abs(int,I_2)))))) # label(fact_917_dvd__imp__le__int) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 782 (all X all P_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_2)) -> (hBOOL(hAPP(int,bool,zcong(X,number_number_of(int,min)),P_2)) -> -hBOOL(hAPP(int,bool,zcong(X,one_one(int)),P_2))))) # label(fact_836_zcong__neg__1__impl__ne__1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 783 (all V_1 all W times_times(int,number_number_of(int,V_1),number_number_of(int,W)) = number_number_of(int,times_times(int,V_1,W))) # label(fact_384_times__numeral__code_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 784 (all M all N all Z plus_plus(int,hAPP(nat,int,semiring_1_of_nat(int),plus_plus(nat,M,N)),Z) = plus_plus(int,hAPP(nat,int,semiring_1_of_nat(int),M),plus_plus(int,hAPP(nat,int,semiring_1_of_nat(int),N),Z))) # label(fact_22_zadd__int__left) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 785 (all Na (Na = zero_zero(nat) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),zero_zero(nat))))) # label(fact_602_le__0__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 786 (all Va (zero_zero(nat) = number_number_of(nat,Va) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Va),pls)))) # label(fact_501_eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 787 (all X_a (semiring(X_a) -> (all A_1 all E all B_1 all C plus_plus(X_a,times_times(X_a,plus_plus(X_a,A_1,B_1),E),C) = plus_plus(X_a,times_times(X_a,A_1,E),plus_plus(X_a,times_times(X_a,B_1,E),C))))) # label(fact_340_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 788 (all A_1 all B_1 plus_plus(int,plus_plus(int,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls)))),times_times(int,times_times(int,number_number_of(int,bit0(bit1(pls))),A_1),B_1)),hAPP(nat,int,power_power(int,B_1),number_number_of(nat,bit0(bit1(pls))))) = hAPP(nat,int,power_power(int,plus_plus(int,A_1,B_1)),number_number_of(nat,bit0(bit1(pls))))) # label(fact_235_zadd__power2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 789 (all N all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N),M)) -> N = M | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),times_times(nat,number_number_of(nat,bit0(bit1(pls))),N)),M)) | zero_zero(nat) = M)) # label(fact_838_divides__cases) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 790 (all X_a (linordered_idom(X_a) -> (all Ma all K (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Ma),K)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),abs_abs(X_a,Ma)),K)))))) # label(fact_920_abs__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 791 (all M hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),times_times(nat,M,M)))) # label(fact_573_le__square) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 792 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all Ya (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,Ya))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),Ya)))))) # label(fact_403_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 793 (all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),abs_abs(int,Z_1)),one_one(int))) <-> ti(int,Z_1) = zero_zero(int))) # label(fact_956_zabs__less__one__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 794 (all X_a (ordere216010020id_add(X_a) -> (all B_1 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_1),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,A_1,B_1)),zero_zero(X_a)))))))) # label(fact_493_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 795 (all Ma all K all Na (plus_plus(nat,Na,K) = plus_plus(nat,Ma,K) <-> Na = Ma)) # label(fact_126_nat__add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 796 (all X_a (linordered_semidom(X_a) -> (all A_1 all N all N_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N_2)) -> (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)),hAPP(nat,X_a,power_power(X_a,A_1),N_2)))))))) # label(fact_184_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 797 (all N all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,power_power(int,X),N))))) # label(fact_476_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 798 (all X_a (monoid_add(X_a) -> (all A_1 plus_plus(X_a,zero_zero(X_a),A_1) = ti(X_a,A_1)))) # label(fact_219_add__0__left) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 799 (all A_1 all P_2 (hBOOL(hAPP(int,bool,zprime,P_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,zcong(times_times(int,A_1,A_1),one_one(int)),P_2)) -> hBOOL(hAPP(int,bool,zcong(A_1,minus_minus(int,P_2,one_one(int))),P_2)) | hBOOL(hAPP(int,bool,zcong(A_1,one_one(int)),P_2)))))) # label(fact_902_zcong__square) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 800 (all X_a ti(X_a,undefined(X_a)) = undefined(X_a)) # label(tsy_c_HOL_Oundefined_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 801 (all X all Y (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) -> -(-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X))))) # label(fact_854_dvd_Oless__not__sym) # label(axiom) # label(non_clause). [assumption]. 1.52/1.82 802 (all K_1 all I_2 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),times_times(nat,I_2,K_1)),times_times(nat,J_1,K_1))))) # label(fact_579_mult__le__mono1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 803 (all J_1 all I_2 -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,J_1,I_2)),I_2))) # label(fact_119_not__add__less2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 804 (all X_a (linord20386208strict(X_a) -> (all B_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),B_1),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,A_1,B_1)),zero_zero(X_a)))))))) # label(fact_414_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 805 (all K_1 all L plus_plus(int,bit1(K_1),bit0(L)) = bit1(plus_plus(int,K_1,L))) # label(fact_90_add__Bit1__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 806 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all Ya (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(pls)),Ya)) <-> 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,Ya))))))) # label(fact_63_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 807 (all K all L_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L_1)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,K)),number_number_of(int,L_1))))) # label(fact_385_less__eq__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 808 (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)) -> zero_zero(X_a) = ti(X_a,A_1))))) # label(fact_775_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 809 (all Ma all Na (one_one(nat) = Na & one_one(nat) = Ma <-> one_one(nat) = times_times(nat,Ma,Na))) # label(fact_626_nat__mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 810 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),times_times(X_a,B_1,A_1)))))) # label(fact_734_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 811 (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_177_not__one__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 812 (all Ma all K all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) & 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),times_times(nat,Ma,K)),times_times(nat,Na,K))))) # label(fact_650_mult__less__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 813 (all U all Ma all Na all I_1 all J (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),plus_plus(nat,times_times(nat,I_1,U),Ma)),plus_plus(nat,times_times(nat,J,U),Na))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),plus_plus(nat,times_times(nat,minus_minus(nat,J,I_1),U),Na)))))) # label(fact_990_nat__le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 814 (all X_a (ordere453448008miring(X_a) -> (all B_1 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_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,A_1,B_1)),zero_zero(X_a)))))))) # label(fact_279_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 815 (all P_1 all Xa all Ya ((hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Xa),Ya)) -> hBOOL(hAPP(int,bool,P_1,zero_zero(int)))) & (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ya),Xa)) -> hBOOL(hAPP(int,bool,P_1,minus_minus(int,hAPP(nat,int,semiring_1_of_nat(int),Xa),hAPP(nat,int,semiring_1_of_nat(int),Ya))))) <-> hBOOL(hAPP(int,bool,P_1,hAPP(nat,int,semiring_1_of_nat(int),minus_minus(nat,Xa,Ya)))))) # label(fact_903_zdiff__int__split) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 816 (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_376_less__eq__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 817 (all K_1 all A_1 all B_1 all M (hBOOL(hAPP(int,bool,zcong(A_1,B_1),M)) -> hBOOL(hAPP(int,bool,zcong(times_times(int,K_1,A_1),times_times(int,K_1,B_1)),M)))) # label(fact_889_zcong__scalar2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 818 (all X_a (linord20386208strict(X_a) -> (all B_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)),B_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),times_times(X_a,A_1,B_1)))))))) # label(fact_413_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 819 (all N hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N))) # label(fact_563_le0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 820 (all X_a (monoid_mult(X_a) -> (all A_1 times_times(X_a,A_1,one_one(X_a)) = ti(X_a,A_1)))) # label(fact_349_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 821 (all X_a (ordere223160158up_add(X_a) -> (all C all D_1 all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,A_1,C)),plus_plus(X_a,B_1,D_1)))))))) # label(fact_439_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 822 (all K (pls = ti(int,K) <-> pls = bit0(K))) # label(fact_70_rel__simps_I44_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 823 (all X all P_2 (hBOOL(hAPP(int,bool,zprime,P_2)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),X)) -> hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,X),nat_1(minus_minus(int,P_2,one_one(int)))),one_one(int)),P_2))))) # label(fact_879_Little__Fermat) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 824 (all M all N minus_minus(nat,plus_plus(nat,M,N),N) = M) # label(fact_365_diff__add__inverse2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 825 (all Z_1 all Wa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_1),Wa)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),Wa)) & ti(int,Z_1) != ti(int,Wa))) # label(fact_380_zless__le) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 826 (all Z1 all Z2 all W times_times(real,plus_plus(real,Z1,Z2),W) = plus_plus(real,times_times(real,Z1,W),times_times(real,Z2,W))) # label(fact_640_real__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 827 (all X_a (linordered_semidom(X_a) -> (all M all N (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)))))) # label(fact_169_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 828 (all X_a (power(X_a) -> (all M all N ((zero_zero(nat) = N -> one_one(X_a) = hAPP(nat,X_a,power_power(X_a,M),N)) & (N != zero_zero(nat) -> times_times(X_a,M,hAPP(nat,X_a,power_power(X_a,M),minus_minus(nat,N,one_one(nat)))) = hAPP(nat,X_a,power_power(X_a,M),N)))))) # label(fact_558_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 829 (all A_1 hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit1(bit1(pls)))) = times_times(int,A_1,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls)))))) # label(fact_543_cube__square) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 830 (all P_1 all A_2 all B (hBOOL(hAPP(nat,bool,P_1,minus_minus(nat,A_2,B))) <-> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_2),B)) -> hBOOL(hAPP(nat,bool,P_1,zero_zero(nat)))) & (all D_3 (A_2 = plus_plus(nat,B,D_3) -> hBOOL(hAPP(nat,bool,P_1,D_3)))))) # label(fact_500_nat__diff__split) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 831 (all Na 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),Na)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),minus_minus(nat,Ma,K)),minus_minus(nat,Na,K))))))) # label(fact_662_less__diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 832 (all B_1_1 all B_2_1 all X_a (order(X_a) -> ti(fun(X_a,bool),hAPP(X_a,fun(X_a,bool),B_1_1,B_2_1)) = hAPP(X_a,fun(X_a,bool),B_1_1,B_2_1))) # label(tsy_c_hAPP_0_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 833 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all N hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)),hAPP(nat,X_a,power_power(X_a,number_number_of(X_a,bit0(bit1(pls)))),N)))))) # label(fact_103_of__nat__less__two__power) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 834 (all N 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)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M),N)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N),M)) -> ti(int,M) = ti(int,N)))))) # label(fact_747_zdvd__antisym__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 835 (all X_a (ordered_ring(X_a) -> (all B_1 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_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)),times_times(X_a,A_1,B_1)))))))) # label(fact_278_mult__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 836 (all N all X all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,X),N)),hAPP(nat,nat,power_power(nat,Y),N))))) # label(fact_812_divides__exp) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 837 (all X_a (comm_semiring_1(X_a) -> (all M all A_1 all N all B_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),N)),B_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) -> 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_1))))))) # label(fact_744_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 838 (all X_a (ab_group_add(X_a) -> (all A_1 all C all B_1 all D_1 plus_plus(X_a,minus_minus(X_a,A_1,B_1),minus_minus(X_a,C,D_1)) = minus_minus(X_a,plus_plus(X_a,A_1,C),plus_plus(X_a,B_1,D_1))))) # label(fact_325_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 839 (all X_a (number_ring(X_a) -> (all V_1 all W times_times(X_a,number_number_of(X_a,V_1),number_number_of(X_a,W)) = number_number_of(X_a,times_times(int,V_1,W))))) # label(fact_258_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 840 (all B_1_1 all B_2_1 all X_a (monoid_add(X_a) -> plus_plus(X_a,B_1_1,B_2_1) = plus_plus(X_a,B_1_1,ti(X_a,B_2_1)))) # label(tsy_c_Groups_Oplus__class_Oplus_0_arg2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 841 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B_1 all C plus_plus(X_a,plus_plus(X_a,A_1,B_1),C) = plus_plus(X_a,plus_plus(X_a,A_1,C),B_1)))) # label(fact_205_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 842 (all X_a (ordere236663937imp_le(X_a) -> (all C_1 all A_2 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,C_1,A_2)),plus_plus(X_a,C_1,B))))))) # label(fact_358_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 843 (all K_1 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),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_eq(nat),K_1),N))))) # label(fact_826_dvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 844 (all X_a (ring_1(X_a) -> (all N all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M)) -> hAPP(nat,X_a,semiring_1_of_nat(X_a),minus_minus(nat,M,N)) = minus_minus(X_a,hAPP(nat,X_a,semiring_1_of_nat(X_a),M),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)))))) # label(fact_682_of__nat__diff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 845 (all C_1 all D all A_2 all B all Ma (hBOOL(hAPP(int,bool,zcong(A_2,B),Ma)) -> (hBOOL(hAPP(int,bool,zcong(C_1,times_times(int,D,B)),Ma)) <-> hBOOL(hAPP(int,bool,zcong(C_1,times_times(int,D,A_2)),Ma))))) # label(fact_823_zcong__zmult__prop2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 846 (all Xa all Ya (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) | Ya = Xa <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)))) # label(fact_870_dvd_Ole__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 847 (all X_a (semiring_char_0(X_a) -> (all Ma all Na (hAPP(nat,X_a,semiring_1_of_nat(X_a),Ma) = hAPP(nat,X_a,semiring_1_of_nat(X_a),Na) <-> Na = Ma)))) # label(fact_158_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 848 one_one(int) = t -> (exists X_1 exists Y_1 plus_plus(int,hAPP(nat,int,power_power(int,X_1),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,int,power_power(int,Y_1),number_number_of(nat,bit0(bit1(pls))))) = plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) # label(fact_571__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 849 (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_442_zero__le__one) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 850 (all Z all X all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),Z)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z),Y)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z),X)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Z))))) # label(fact_856_dvd_Ole__less__trans) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 851 (all A_1 all M all B_1 hBOOL(hAPP(int,bool,zcong(times_times(int,A_1,M),times_times(int,B_1,M)),M))) # label(fact_887_zcong__zmult__self) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 852 (all B_1_1 all B_2_1 all X_a (semiring(X_a) -> times_times(X_a,ti(X_a,B_1_1),B_2_1) = times_times(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_2_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 853 (all M all N (one_one(int) = abs_abs(int,times_times(int,M,N)) -> abs_abs(int,M) = one_one(int))) # label(fact_938_abs__zmult__eq__1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 854 (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),times_times(nat,number_number_of(nat,bit0(bit1(pls))),K_1))),zero_zero(X_a))) -> ti(X_a,A_1) = zero_zero(X_a))))) # label(fact_699_even__power__le__0__imp__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 855 (all X_a (semiring_1(X_a) -> (all M all N hAPP(nat,X_a,semiring_1_of_nat(X_a),times_times(nat,M,N)) = times_times(X_a,hAPP(nat,X_a,semiring_1_of_nat(X_a),M),hAPP(nat,X_a,semiring_1_of_nat(X_a),N))))) # label(fact_265_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 856 (all K all P_1 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D)) -> ((all X_1 (hBOOL(hAPP(int,bool,P_1,X_1)) -> hBOOL(hAPP(int,bool,P_1,minus_minus(int,X_1,D))))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) -> (all X_1 (hBOOL(hAPP(int,bool,P_1,X_1)) -> hBOOL(hAPP(int,bool,P_1,minus_minus(int,X_1,times_times(int,K,D)))))))))) # label(fact_972_decr__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 857 (all B_1_1 all X_a (power(X_a) -> ti(fun(nat,X_a),power_power(X_a,B_1_1)) = power_power(X_a,B_1_1))) # label(tsy_c_Power_Opower__class_Opower_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 858 (all N N = times_times(nat,N,one_one(nat))) # label(fact_627_nat__mult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 859 (all M plus_plus(nat,M,zero_zero(nat)) = M) # label(fact_155_Nat_Oadd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 860 (all Wa all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z_1)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Wa),minus_minus(int,Z_1,one_one(int)))))) # label(fact_680_zle__diff1__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 861 (all X_a (linordered_semidom(X_a) -> (all M 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,semiring_1_of_nat(X_a),M)))))) # label(fact_457_zero__le__imp__of__nat) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 862 (all Z all X all D_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z),plus_plus(int,X,times_times(int,plus_plus(int,abs_abs(int,minus_minus(int,X,Z)),one_one(int)),D_1)))))) # label(fact_969_incr__lemma) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 863 (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_763_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 864 (all X hAPP(nat,int,power_power(int,X),number_number_of(nat,bit0(bit0(bit1(pls))))) = hAPP(nat,int,power_power(int,hAPP(nat,int,power_power(int,X),number_number_of(nat,bit0(bit1(pls))))),number_number_of(nat,bit0(bit1(pls))))) # label(fact_10_quartic__square__square) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 865 (all Wa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(Wa)),zero_zero(int))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),zero_zero(int))))) # label(fact_56_bin__less__0__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 866 (all X_a (linordered_idom(X_a) -> (all L all K_1 (abs_abs(X_a,K_1) = abs_abs(X_a,L) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),L),K_1)))))) # label(fact_919_dvd__if__abs__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 867 (all X_a (linord581940658strict(X_a) -> (all A_2 all B 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),times_times(X_a,C_1,A_2)),times_times(X_a,C_1,B))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_2))))))) # label(fact_418_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 868 (all X_a (number_ring(X_a) -> (all Z plus_plus(X_a,Z,Z) = times_times(X_a,number_number_of(X_a,bit0(bit1(pls))),Z)))) # label(fact_537_mult__2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 869 (all M all N (M = plus_plus(nat,M,N) -> zero_zero(nat) = N)) # label(fact_153_add__eq__self__zero) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 870 (all S_1 all T_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),S_1),T_2)) -> S_1 != T_2)) # label(fact_131_less__not__refl3) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 871 (all M M = minus_minus(nat,M,zero_zero(nat))) # label(fact_337_minus__nat_Odiff__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 872 (all X_a (linordered_semidom(X_a) -> (all A_1 all N all N_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),N_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),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_2)),hAPP(nat,X_a,power_power(X_a,A_1),N))))))))) # label(fact_391_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 873 (all K all Ma all Na (Na = Ma | zero_zero(nat) = K <-> times_times(nat,K,Ma) = times_times(nat,K,Na))) # label(fact_597_mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 874 (all X_a (linordered_semidom(X_a) -> (all N 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)),times_times(X_a,A_1,hAPP(nat,X_a,power_power(X_a,A_1),N)))))))) # label(fact_496_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 875 (all Na 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),Na)) -> (Ma = Na <-> minus_minus(nat,Ma,K) = minus_minus(nat,Na,K))))) # label(fact_635_eq__diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 876 (all X_a (linord581940658strict(X_a) -> (all C_1 all A_2 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B)) & 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),B),A_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),times_times(X_a,C_1,A_2)),times_times(X_a,C_1,B))))))) # label(fact_411_mult__less__cancel__left__disj) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 877 (all X_a (linord1278240602ring_1(X_a) -> (all V_1 all U_1 all Y all X all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),U_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),V_1)) -> (one_one(X_a) = plus_plus(X_a,U_1,V_1) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,times_times(X_a,U_1,X),times_times(X_a,V_1,Y))),A_1)))))))))) # label(fact_386_convex__bound__le) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 878 (all Wa all Z_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),nat_1(Wa)),nat_1(Z_1))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z_1)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z_1)))) # label(fact_484_zless__nat__conj) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 879 (all A_1 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1)) -> (exists X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1)) & (all Y_1 (hAPP(nat,real,power_power(real,Y_1),N) = A_1 & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y_1)) -> X_1 = Y_1)) & hAPP(nat,real,power_power(real,X_1),N) = A_1))))) # label(fact_229_realpow__pos__nth__unique) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 880 (all X_a (ordered_semiring(X_a) -> (all C all D_1 all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_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)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),times_times(X_a,A_1,C)),times_times(X_a,B_1,D_1)))))))))) # label(fact_272_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 881 (all Z times_times(nat,number_number_of(nat,bit0(bit1(pls))),Z) = plus_plus(nat,Z,Z)) # label(fact_691_nat__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 882 (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_750_rel__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 883 (all I_2 all K_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J_1)) -> minus_minus(nat,plus_plus(nat,J_1,I_2),K_1) = plus_plus(nat,minus_minus(nat,J_1,K_1),I_2))) # label(fact_675_add__diff__assoc2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 884 (all B_1_1 all B_2_1 all X_a (mult_zero(X_a) -> times_times(X_a,B_1_1,ti(X_a,B_2_1)) = times_times(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_4_arg2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 885 (all X_a (comm_semiring_1(X_a) -> (all A_1 all M all B_1 plus_plus(X_a,times_times(X_a,A_1,M),times_times(X_a,B_1,M)) = times_times(X_a,plus_plus(X_a,A_1,B_1),M)))) # label(fact_342_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 886 (all X_a all X all Y ti(X_a,Y) = if(X_a,fFalse,X,Y)) # label(help_If_2_1_T) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 887 (all K_1 all A_1 all B_1 all M (hBOOL(hAPP(int,bool,zcong(A_1,B_1),M)) -> hBOOL(hAPP(int,bool,zcong(times_times(int,A_1,K_1),times_times(int,B_1,K_1)),M)))) # label(fact_888_zcong__scalar) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 888 (all X_a (linordered_semidom(X_a) -> (all B_1 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_1),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_1),plus_plus(X_a,A_1,C)))))))) # label(fact_176_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 889 (all L pls != bit1(L)) # label(fact_67_rel__simps_I39_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 890 (all X_a (comm_semiring_1(X_a) -> (all N all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),Y)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,power_power(X_a,X),N)),hAPP(nat,X_a,power_power(X_a,Y),N))))))) # label(fact_772_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 891 (all L minus_minus(int,pls,bit0(L)) = bit0(minus_minus(int,pls,L))) # label(fact_645_diff__bin__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 892 (all Z_1 all Wa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z_1)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),nat_1(Wa)),nat_1(Z_1)))))) # label(fact_388_nat__less__eq__zless) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 893 (all B_1_1 all B_2_1 zcong(B_1_1,ti(int,B_2_1)) = zcong(B_1_1,B_2_1)) # label(tsy_c_IntPrimes_Ozcong_arg2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 894 -(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)),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))))) # label(fact_797__096_B_Bthesis_O_A_I_B_Bs1_O_A_091s1_A_094_A2_A_061_A_N1_093_A_Imod_A4_) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 895 (all X_a (monoid_mult(X_a) -> (all A_1 all M all N hAPP(nat,X_a,power_power(X_a,A_1),plus_plus(nat,M,N)) = 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))))) # label(fact_459_power__add) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 896 (all X_a (linordered_ring(X_a) -> (all X all Y -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,times_times(X_a,X,X),times_times(X_a,Y,Y))),zero_zero(X_a)))))) # label(fact_485_not__sum__squares__lt__zero) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 897 (all I_1 all J all K (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),minus_minus(nat,J,K))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,I_1,K)),J)))) # label(fact_455_less__diff__conv) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 898 (all K_1 plus_plus(int,K_1,one_one(int)) = succ(K_1)) # label(fact_519_succ__def) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 899 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B_1 all C times_times(X_a,plus_plus(X_a,A_1,B_1),C) = plus_plus(X_a,times_times(X_a,A_1,C),times_times(X_a,B_1,C))))) # label(fact_343_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 900 (all B_1 all A_1 all P_2 (hBOOL(hAPP(int,bool,zprime,P_2)) -> (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_2)) & -hBOOL(hAPP(int,bool,zcong(B_1,zero_zero(int)),P_2)) -> -hBOOL(hAPP(int,bool,zcong(times_times(int,A_1,B_1),zero_zero(int)),P_2)))))) # label(fact_829_zcong__zprime__prod__zero__contra) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 901 (all Z ti(int,Z) = plus_plus(int,zero_zero(int),Z)) # label(fact_80_zadd__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 902 (all X_b (monoid_mult(X_b) & number(X_b) -> (all W hAPP(nat,X_b,power_power(X_b,number_number_of(X_b,W)),number_number_of(nat,bit0(bit1(pls)))) = times_times(X_b,number_number_of(X_b,W),number_number_of(X_b,W))))) # label(fact_545_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 903 (all M all K_1 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),plus_plus(nat,M,K_1)),N)) -> -(hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N))))) # label(fact_614_add__leE) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 904 (all Xa all Ya (zero_zero(real) = plus_plus(real,hAPP(nat,real,power_power(real,Xa),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,real,power_power(real,Ya),number_number_of(nat,bit0(bit1(pls))))) <-> zero_zero(real) = Xa & Ya = zero_zero(real))) # label(fact_106_realpow__two__sum__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 905 (all L minus_minus(int,min,bit1(L)) = bit0(minus_minus(int,min,L))) # label(fact_783_diff__bin__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 906 (all X_a (linord20386208strict(X_a) -> (all C all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,C,A_1)),times_times(X_a,C,B_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),A_1),B_1))))))) # label(fact_300_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 907 (all U_1 all M all N all I_2 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_1)) -> minus_minus(nat,M,plus_plus(nat,times_times(nat,minus_minus(nat,J_1,I_2),U_1),N)) = minus_minus(nat,plus_plus(nat,times_times(nat,I_2,U_1),M),plus_plus(nat,times_times(nat,J_1,U_1),N)))) # label(fact_991_nat__diff__add__eq2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 908 (all X_a (ordere236663937imp_le(X_a) -> (all A_1 all C all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,A_1,C)),plus_plus(X_a,B_1,C))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1)))))) # label(fact_354_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 909 (all A_1 all B_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_1),B_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B_1),A_1)) -> abs_abs(int,A_1) = abs_abs(int,B_1)))) # label(fact_922_zdvd__antisym__abs) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 910 (all I_1 all K all J (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),J)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),plus_plus(nat,I_1,K)),J)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),minus_minus(nat,J,K)))))) # label(fact_671_le__diff__conv2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 911 (all X_a (monoid_mult(X_a) -> (all A_1 hAPP(nat,X_a,power_power(X_a,A_1),one_one(nat)) = ti(X_a,A_1)))) # label(fact_157_power__one__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 912 (all B_1_1 all B_2_1 all X_a (ab_semigroup_mult(X_a) -> times_times(X_a,B_1_1,ti(X_a,B_2_1)) = times_times(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_5_arg2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 913 (all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),plus_plus(int,one_one(int),Z))))) # label(fact_530_le__imp__0__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 914 (all X_a (comm_semiring_1(X_a) -> (all Lx all Rx all Ry times_times(X_a,Lx,times_times(X_a,Rx,Ry)) = times_times(X_a,Rx,times_times(X_a,Lx,Ry))))) # label(fact_240_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 915 (all X_a (number_ring(X_a) -> number_number_of(X_a,bit1(pls)) = one_one(X_a))) # label(fact_13_numeral__1__eq__1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 916 (all B_1_1 (hBOOL(B_1_1) <-> hBOOL(ti(bool,B_1_1)))) # label(tsy_c_hBOOL_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 917 (all X_a all B_1_1 all B_2_1 all B_3 if(X_a,B_1_1,ti(X_a,B_2_1),B_3) = if(X_a,B_1_1,B_2_1,B_3)) # label(tsy_c_If_arg2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 918 (all X_a (cancel_semigroup_add(X_a) -> (all A_2 all B all C_1 (ti(X_a,C_1) = ti(X_a,B) <-> plus_plus(X_a,A_2,B) = plus_plus(X_a,A_2,C_1))))) # label(fact_204_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 919 (all X all N hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,power_power(nat,X),N)) = hAPP(nat,int,power_power(int,hAPP(nat,int,semiring_1_of_nat(int),X)),N)) # label(fact_175_Nat__Transfer_Otransfer__int__nat__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 920 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) -> (M != N -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))))) # label(fact_606_le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 921 (all B_1_1 all B_2_1 all X_a (order(X_a) -> ti(bool,hAPP(X_a,bool,B_1_1,B_2_1)) = hAPP(X_a,bool,B_1_1,B_2_1))) # label(tsy_c_hAPP_1_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 922 (all X_a (ordered_ring(X_a) -> (all A_2 all E_1 all C_1 all B all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,times_times(X_a,A_2,E_1),C_1)),plus_plus(X_a,times_times(X_a,B,E_1),D))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,times_times(X_a,minus_minus(X_a,A_2,B),E_1),C_1)),D)))))) # label(fact_285_le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 923 (all B_1 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)),times_times(int,A_1,B_1))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_1))))) # label(fact_466_pos__zmult__pos) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 924 (all P (fTrue = ti(bool,P) | ti(bool,P) = fFalse)) # label(help_If_3_1_T) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 925 (all X_a (ordere216010020id_add(X_a) -> (all Ya all Xa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Xa)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Ya)) -> (plus_plus(X_a,Xa,Ya) = zero_zero(X_a) <-> zero_zero(X_a) = ti(X_a,Xa) & zero_zero(X_a) = ti(X_a,Ya))))))) # label(fact_435_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 926 -(all S -(hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),S)) & hBOOL(hAPP(int,bool,zcong(s1,S),plus_plus(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),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))))) # label(fact_843__096_B_Bthesis_O_A_I_B_Bs_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 927 (all X_a (ring_11004092258visors(X_a) -> (all A_2 (zero_zero(X_a) = ti(X_a,A_2) <-> zero_zero(X_a) = hAPP(nat,X_a,power_power(X_a,A_2),number_number_of(nat,bit0(bit1(pls)))))))) # label(fact_5_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 928 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx all Ry times_times(X_a,Rx,times_times(X_a,times_times(X_a,Lx,Ly),Ry)) = times_times(X_a,times_times(X_a,Lx,Ly),times_times(X_a,Rx,Ry))))) # label(fact_246_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 929 (all Na (zero_zero(nat) = Na <-> zero_zero(int) = hAPP(nat,int,semiring_1_of_nat(int),Na))) # label(fact_27_int__eq__0__conv) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 930 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),bit1(pls))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,Xa)),one_one(X_a))))))) # label(fact_404_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 931 (all Y all X (hBOOL(hAPP(int,bool,twoSqu658283162sum2sq,X)) -> (hBOOL(hAPP(int,bool,twoSqu658283162sum2sq,Y)) -> hBOOL(hAPP(int,bool,twoSqu658283162sum2sq,times_times(int,X,Y)))))) # label(fact_679_is__mult__sum2sq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 932 (all X_a (linord20386208strict(X_a) -> (all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),times_times(X_a,A_1,B_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_1))))))) # label(fact_416_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 933 (all X_a (ordered_semiring(X_a) -> (all C all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_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),times_times(X_a,C,A_1)),times_times(X_a,C,B_1)))))))) # label(fact_276_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 934 (all K_1 -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),K_1)),zero_zero(int)))) # label(fact_61_int__less__0__conv) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 935 (all X_a (ordered_ab_group_add(X_a) -> (all A_2 all B all C_1 all D (minus_minus(X_a,A_2,B) = minus_minus(X_a,C_1,D) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B))))))) # label(fact_324_diff__eq__diff__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 936 (all X_a (semiring_1(X_a) -> (all M all N hAPP(nat,X_a,power_power(X_a,hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),N) = hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,power_power(nat,M),N))))) # label(fact_172_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 937 (all X_b (number(X_b) & ring(X_b) -> (all A_1 all B_1 all V_1 times_times(X_b,minus_minus(X_b,A_1,B_1),number_number_of(X_b,V_1)) = minus_minus(X_b,times_times(X_b,A_1,number_number_of(X_b,V_1)),times_times(X_b,B_1,number_number_of(X_b,V_1)))))) # label(fact_263_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 938 (all X_a (comm_semiring(X_a) -> (all A_1 all B_1 all C times_times(X_a,plus_plus(X_a,A_1,B_1),C) = plus_plus(X_a,times_times(X_a,A_1,C),times_times(X_a,B_1,C))))) # label(fact_339_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 939 (all B_1_1 all B_2_1 all X_a (cancel_semigroup_add(X_a) -> plus_plus(X_a,B_1_1,B_2_1) = ti(X_a,plus_plus(X_a,B_1_1,B_2_1)))) # label(tsy_c_Groups_Oplus__class_Oplus_2_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 940 (all X_a (monoid_mult(X_a) -> (all N hAPP(nat,X_a,power_power(X_a,one_one(X_a)),N) = one_one(X_a)))) # label(fact_166_power__one) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 941 (all K_1 all L minus_minus(int,bit1(K_1),bit1(L)) = bit0(minus_minus(int,K_1,L))) # label(fact_643_diff__bin__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 942 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),abs_abs(X_a,A_1)),B_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1)))))) # label(fact_954_abs__le__D1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 943 (all N 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)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N),M))))) # label(fact_803_nat__dvd__not__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 944 (all I_2 all K_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J_1)) -> minus_minus(nat,plus_plus(nat,I_2,J_1),K_1) = plus_plus(nat,I_2,minus_minus(nat,J_1,K_1)))) # label(fact_674_diff__add__assoc) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 945 (all X_a (semiring_0(X_a) & power(X_a) -> (all N ((zero_zero(nat) != N -> hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),N) = zero_zero(X_a)) & (zero_zero(nat) = N -> hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),N) = one_one(X_a)))))) # label(fact_109_power__0__left) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 946 (all X_a (power(X_a) & no_zero_divisors(X_a) & zero_neq_one(X_a) & mult_zero(X_a) -> (all A_2 all Na (zero_zero(X_a) = ti(X_a,A_2) & Na != zero_zero(nat) <-> zero_zero(X_a) = hAPP(nat,X_a,power_power(X_a,A_2),Na))))) # label(fact_185_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 947 (all X_a (comm_semiring_1(X_a) -> (all X all P_2 all Q times_times(X_a,hAPP(nat,X_a,power_power(X_a,X),P_2),hAPP(nat,X_a,power_power(X_a,X),Q)) = hAPP(nat,X_a,power_power(X_a,X),plus_plus(nat,P_2,Q))))) # label(fact_460_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 948 (all X_a (ordere142940540dd_abs(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)),abs_abs(X_a,A_1)))))) # label(fact_949_abs__ge__zero) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 949 (all X_a (mult_zero(X_a) -> (all A_1 times_times(X_a,A_1,zero_zero(X_a)) = zero_zero(X_a)))) # label(fact_329_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 950 (all X -(-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),X)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),X)))) # label(fact_872_dvd_Oless__irrefl) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 951 (all Z all W (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),W),Z)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Z),W)))) # label(fact_714_real__le__linear) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 952 (all X_a (monoid_add(X_a) -> (all A_1 ti(X_a,A_1) = plus_plus(X_a,A_1,zero_zero(X_a))))) # label(fact_215_add__0__right) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 953 (all B_1_1 all B_2_1 all X_a (ab_semigroup_add(X_a) -> plus_plus(X_a,B_1_1,B_2_1) = plus_plus(X_a,B_1_1,ti(X_a,B_2_1)))) # label(tsy_c_Groups_Oplus__class_Oplus_1_arg2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 954 (all X_a (zero_neq_one(X_a) -> zero_zero(X_a) != one_one(X_a))) # label(fact_163_one__neq__zero) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 955 (all Z all X all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),Z)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Z))))) # label(fact_858_dvd_Oorder__trans) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 956 (all X_a (linordered_idom(X_a) -> (all X all Y -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,X_a,power_power(X_a,Y),number_number_of(nat,bit0(bit1(pls)))))),zero_zero(X_a)))))) # label(fact_98_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 957 (all X_a (dvd(X_a) -> (all A_1 all B_1 all K_1 (times_times(X_a,B_1,K_1) = A_1 -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B_1),A_1)))))) # label(fact_730_dvdI) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 958 (all X_a (dvd(X_a) & comm_ring(X_a) -> (all Ta all D all D_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),D_2)) -> (all X_1 all K_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),plus_plus(X_a,X_1,Ta))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),plus_plus(X_a,minus_minus(X_a,X_1,times_times(X_a,K_2,D_2)),Ta))))))))) # label(fact_909_inf__period_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 959 (all Na (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),Na)),zero_zero(int))) <-> Na = zero_zero(nat))) # label(fact_512_int__le__0__conv) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 960 (all Wa all Ma ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa)) -> hAPP(nat,int,semiring_1_of_nat(int),Ma) = ti(int,Wa)) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa)) -> zero_zero(nat) = Ma) <-> Ma = nat_1(Wa))) # label(fact_395_nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 961 (all I_2 all J_1 all K_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,I_2,J_1)),K_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),K_1)))) # label(fact_137_add__lessD1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 962 (all M all N hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,power_power(nat,M),N)) = hAPP(nat,int,power_power(int,hAPP(nat,int,semiring_1_of_nat(int),M)),N)) # label(fact_21_int__power) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 963 (all K_1 all L plus_plus(int,bit1(K_1),bit1(L)) = bit0(plus_plus(int,K_1,succ(L)))) # label(fact_533_add__Bit1__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 964 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) | 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)),plus_plus(nat,Ma,Na))))) # label(fact_144_add__gr__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 965 (all N 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)) -> minus_minus(nat,minus_minus(nat,M,K_1),minus_minus(nat,N,K_1)) = minus_minus(nat,M,N)))) # label(fact_634_Nat_Odiff__diff__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 966 (all B_1_1 bit1(B_1_1) = ti(int,bit1(B_1_1))) # label(tsy_c_Int_OBit1_res) # label(hypothesis) # label(non_clause). [assumption]. 1.52/1.83 967 (all X_a (linordered_idom(X_a) -> (all A_1 hAPP(nat,X_a,power_power(X_a,abs_abs(X_a,A_1)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_965_power2__abs) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 968 (all X_a (number_semiring(X_a) -> number_number_of(X_a,bit1(pls)) = one_one(X_a))) # label(fact_101_semiring__numeral__1__eq__1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 969 (all C all A_1 all B_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),times_times(nat,C,A_1)),times_times(nat,C,B_1))))) # label(fact_799_divides__mul__l) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 970 (all M 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)),M)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),minus_minus(nat,M,N)),M))))) # label(fact_452_diff__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 971 (all L all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),minus_minus(nat,M,L)),minus_minus(nat,N,L))))) # label(fact_637_diff__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 972 (all B_1_1 all B_2_1 all X_a (cancel_semigroup_add(X_a) -> plus_plus(X_a,B_1_1,B_2_1) = plus_plus(X_a,B_1_1,ti(X_a,B_2_1)))) # label(tsy_c_Groups_Oplus__class_Oplus_2_arg2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 973 (all K all F all Na ((all I (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),Na)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),abs_abs(int,minus_minus(int,hAPP(nat,int,F,plus_plus(nat,I,one_one(nat))),hAPP(nat,int,F,I)))),one_one(int))))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,F,zero_zero(nat))),K)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),hAPP(nat,int,F,Na))) -> (exists I (ti(int,K) = hAPP(nat,int,F,I) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),Na)))))))) # label(fact_993_nat0__intermed__int__val) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 974 (all K_1 all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),times_times(nat,K_1,M)),times_times(nat,K_1,N))) -> (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))))) # label(fact_827_dvd__mult__cancel) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 975 (all B_1_1 all B_2_1 all X_a (no_zero_divisors(X_a) -> times_times(X_a,B_1_1,B_2_1) = times_times(X_a,ti(X_a,B_1_1),B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_3_arg1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 976 (all X_a (comm_semiring_1(X_a) -> (all X all Y all Q times_times(X_a,hAPP(nat,X_a,power_power(X_a,X),Q),hAPP(nat,X_a,power_power(X_a,Y),Q)) = hAPP(nat,X_a,power_power(X_a,times_times(X_a,X,Y)),Q)))) # label(fact_364_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 977 (all X_a (linord581940658strict(X_a) -> (all C all B_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_1),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),times_times(X_a,A_1,C)),times_times(X_a,B_1,C)))))))) # label(fact_424_mult__strict__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 978 (all X_a (comm_semiring_1(X_a) -> (all C all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B_1),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C))))))) # label(fact_762_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 979 (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_187_power__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 980 (all N all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M)) -> plus_plus(nat,minus_minus(nat,M,N),N) = M)) # label(fact_672_le__add__diff__inverse2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 981 (all X_a (ordere216010020id_add(X_a) -> (all B_1 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_1),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_1),plus_plus(X_a,A_1,C)))))))) # label(fact_437_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 982 (all B_1_1 bit0(B_1_1) = bit0(ti(int,B_1_1))) # label(tsy_c_Int_OBit0_arg1) # label(hypothesis) # label(non_clause). [assumption]. 1.52/1.83 983 (all A_1 all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),X)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),A_1)) -> (ti(int,X) != minus_minus(int,A_1,one_one(int)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),minus_minus(int,A_1,one_one(int)))))))) # label(fact_646_Euler_Oaux1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 984 (all X_a (linordered_semidom(X_a) -> (all Ma all Na all A_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_2)) -> (hAPP(nat,X_a,power_power(X_a,A_2),Na) = hAPP(nat,X_a,power_power(X_a,A_2),Ma) <-> Na = Ma))))) # label(fact_181_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 985 (all A_1 all B_1 hAPP(nat,int,power_power(int,plus_plus(int,A_1,B_1)),number_number_of(nat,bit1(bit1(pls)))) = plus_plus(int,plus_plus(int,plus_plus(int,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit1(bit1(pls)))),times_times(int,times_times(int,number_number_of(int,bit1(bit1(pls))),hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls))))),B_1)),times_times(int,times_times(int,number_number_of(int,bit1(bit1(pls))),A_1),hAPP(nat,int,power_power(int,B_1),number_number_of(nat,bit0(bit1(pls)))))),hAPP(nat,int,power_power(int,B_1),number_number_of(nat,bit1(bit1(pls)))))) # label(fact_234_zadd__power3) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 986 (all M all N all K_1 times_times(nat,times_times(nat,M,N),K_1) = times_times(nat,M,times_times(nat,N,K_1))) # label(fact_577_nat__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 987 (all B_1_1 all B_2_1 all X_a (power(X_a) -> times_times(X_a,B_1_1,B_2_1) = times_times(X_a,B_1_1,ti(X_a,B_2_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_1_arg2) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 988 (all X_a (linordered_semidom(X_a) -> (all N 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,semiring_1_of_nat(X_a),N)))))) # label(fact_458_of__nat__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 989 (all X_a (linordered_semidom(X_a) -> (all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,X),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,X_a,power_power(X_a,Y),number_number_of(nat,bit0(bit1(pls)))))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))))))) # label(fact_547_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 990 (all X_a (linordered_semidom(X_a) -> (all N 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)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),times_times(X_a,M,N)))))))) # label(fact_431_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 991 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all Xa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,Xa)),zero_zero(X_a))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),pls)))))) # label(fact_390_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 992 (all Z Z = times_times(real,one_one(real),Z)) # label(fact_632_real__mult__1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 993 (all X all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) -> X != Y)) # label(fact_851_dvd_Oless__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 994 (all X_a (linordered_semidom(X_a) -> (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N))))))) # label(fact_168_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 995 (all X_a (ordered_ab_group_add(X_a) -> (all A_2 all B all C_1 all D (minus_minus(X_a,A_2,B) = minus_minus(X_a,C_1,D) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),D))))))) # label(fact_249_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 996 (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),times_times(X_a,A_1,A_1)),zero_zero(X_a)))))) # label(fact_409_not__square__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 997 (all N -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat)))) # label(fact_115_less__zeroE) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 998 (all X_a (comm_monoid_mult(X_a) -> (all A_1 times_times(X_a,one_one(X_a),A_1) = ti(X_a,A_1)))) # label(fact_350_mult__1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 999 (all B_1_1 all B_2_1 all X_a (ab_semigroup_mult(X_a) -> times_times(X_a,B_1_1,B_2_1) = ti(X_a,times_times(X_a,B_1_1,B_2_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_5_res) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 1000 (all X_a (linordered_idom(X_a) -> (all A_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,A_2,A_2)),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),zero_zero(X_a))))))) # label(fact_44_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 1001 (all Xa all Ya all P_1 ((hBOOL(P_1) -> hAPP(nat,int,semiring_1_of_nat(int),if(nat,P_1,Xa,Ya)) = hAPP(nat,int,semiring_1_of_nat(int),Xa)) & (-hBOOL(P_1) -> hAPP(nat,int,semiring_1_of_nat(int),Ya) = hAPP(nat,int,semiring_1_of_nat(int),if(nat,P_1,Xa,Ya))))) # label(fact_160_int__if__cong) # label(axiom) # label(non_clause). [assumption]. 1.52/1.83 1002 (all X_a (linordered_idom(X_a) -> (all B_1 all D_1 all A_1 all C (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),abs_abs(X_a,A_1)),C)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),abs_abs(X_a,B_1)),D_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),times_times(X_a,abs_abs(X_a,A_1),abs_abs(X_a,B_1))),times_times(X_a,C,D_1)))))))) # label(fact_943_abs__mult__less) # label(axiom) # label(non_clause). [assumption]. 1.52/1.84 1003 (all K all Ma all Na (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),Na)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),times_times(nat,K,Ma)),times_times(nat,K,Na))))) # label(fact_651_mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.84 1004 (all K all Ma all Na ((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),Na))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),times_times(nat,K,Ma)),times_times(nat,K,Na))))) # label(fact_586_mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.84 1005 (all X_a (ordere779506340up_add(X_a) -> (all C all A_1 all B_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,A_1,C)),plus_plus(X_a,B_1,C))))))) # label(fact_357_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.52/1.84 1006 (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_96_power2__less__0) # label(axiom) # label(non_clause). [assumption]. 1.52/1.84 1007 (all U all Ma all Na all J all I_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),I_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),plus_plus(nat,times_times(nat,minus_minus(nat,I_1,J),U),Ma)),Na)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),plus_plus(nat,times_times(nat,I_1,U),Ma)),plus_plus(nat,times_times(nat,J,U),Na)))))) # label(fact_987_nat__le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.52/1.84 1008 (all X_a (comm_semiring_1(X_a) -> (all X one_one(X_a) = hAPP(nat,X_a,power_power(X_a,X),zero_zero(nat))))) # label(fact_193_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.52/1.84 1009 (all X_a (group_add(X_a) -> (all A_1 all B_1 plus_plus(X_a,minus_minus(X_a,A_1,B_1),B_1) = ti(X_a,A_1)))) # label(fact_327_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 1.52/1.84 1010 (all P_2 all M ((M != zero_zero(nat) -> times_times(nat,P_2,hAPP(nat,nat,power_power(nat,P_2),minus_minus(nat,M,one_one(nat)))) = hAPP(nat,nat,power_power(nat,P_2),M)) & (zero_zero(nat) = M -> one_one(nat) = hAPP(nat,nat,power_power(nat,P_2),M)))) # label(fact_689_power__eq__if) # label(axiom) # label(non_clause). [assumption]. 1.52/1.84 1011 -(zero_zero(int) != hAPP(nat,int,power_power(int,plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n))),number_number_of(nat,bit0(bit1(pls))))) # label(conj_0) # label(negated_conjecture) # label(non_clause). [assumption]. 1.52/1.84 1.52/1.84 ============================== end of process non-clausal formulas === 1.52/1.84 1.52/1.84 ============================== PROCESS INITIAL CLAUSES =============== 1.52/1.84 1.52/1.84 ============================== PREDICATE ELIMINATION ================= 1.52/1.84 1012 dvd(int) # label(arity_Int_Oint___Rings_Odvd) # label(axiom). [assumption]. 1.52/1.84 1013 -dvd(A) | ti(bool,hAPP(A,bool,B,C)) = hAPP(A,bool,B,C) # label(tsy_c_hAPP_3_res) # label(axiom). [clausify(2)]. 1.52/1.84 1014 -dvd(A) | times_times(A,ti(A,B),C) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_0_arg1) # label(axiom). [clausify(8)]. 1.52/1.84 Derived: ti(bool,hAPP(int,bool,A,B)) = hAPP(int,bool,A,B). [resolve(1012,a,1013,a)]. 1.52/1.84 Derived: times_times(int,ti(int,A),B) = times_times(int,A,B). [resolve(1012,a,1014,a)]. 1.52/1.84 1015 dvd(real) # label(arity_RealDef_Oreal___Rings_Odvd) # label(axiom). [assumption]. 1.52/1.84 Derived: ti(bool,hAPP(real,bool,A,B)) = hAPP(real,bool,A,B). [resolve(1015,a,1013,a)]. 1.52/1.84 Derived: times_times(real,ti(real,A),B) = times_times(real,A,B). [resolve(1015,a,1014,a)]. 1.52/1.86 1016 -dvd(A) | times_times(A,B,ti(A,C)) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_0_arg2) # label(axiom). [clausify(246)]. 1.52/1.86 Derived: times_times(int,A,ti(int,B)) = times_times(int,A,B). [resolve(1016,a,1012,a)]. 1.52/1.86 Derived: times_times(real,A,ti(real,B)) = times_times(real,A,B). [resolve(1016,a,1015,a)]. 1.52/1.86 1017 -dvd(A) | hAPP(A,bool,B,ti(A,C)) = hAPP(A,bool,B,C) # label(tsy_c_hAPP_3_arg2) # label(axiom). [clausify(326)]. 1.52/1.86 Derived: hAPP(int,bool,A,ti(int,B)) = hAPP(int,bool,A,B). [resolve(1017,a,1012,a)]. 1.52/1.86 Derived: hAPP(real,bool,A,ti(real,B)) = hAPP(real,bool,A,B). [resolve(1017,a,1015,a)]. 1.52/1.86 1018 -dvd(A) | hAPP(A,fun(A,bool),B,ti(A,C)) = hAPP(A,fun(A,bool),B,C) # label(tsy_c_hAPP_2_arg2) # label(axiom). [clausify(331)]. 1.52/1.86 Derived: hAPP(int,fun(int,bool),A,ti(int,B)) = hAPP(int,fun(int,bool),A,B). [resolve(1018,a,1012,a)]. 1.52/1.86 Derived: hAPP(real,fun(real,bool),A,ti(real,B)) = hAPP(real,fun(real,bool),A,B). [resolve(1018,a,1015,a)]. 1.52/1.86 1019 -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(386)]. 1.52/1.86 Derived: dvd_dvd(int) = ti(fun(int,fun(int,bool)),dvd_dvd(int)). [resolve(1019,a,1012,a)]. 1.52/1.86 Derived: dvd_dvd(real) = ti(fun(real,fun(real,bool)),dvd_dvd(real)). [resolve(1019,a,1015,a)]. 1.52/1.86 1020 -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),plus_plus(A,minus_minus(A,D,times_times(A,E,C)),F))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),plus_plus(A,D,F))) # label(fact_910_inf__period_I4_J) # label(axiom). [clausify(392)]. 1.52/1.86 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),plus_plus(int,minus_minus(int,C,times_times(int,D,B)),E))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),plus_plus(int,C,E))). [resolve(1020,a,1012,a)]. 1.52/1.86 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),plus_plus(real,minus_minus(real,C,times_times(real,D,B)),E))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),plus_plus(real,C,E))). [resolve(1020,a,1015,a)]. 1.52/1.86 1021 -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),plus_plus(A,minus_minus(A,D,times_times(A,E,C)),F))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),plus_plus(A,D,F))) # label(fact_910_inf__period_I4_J) # label(axiom). [clausify(392)]. 1.52/1.86 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),plus_plus(int,minus_minus(int,C,times_times(int,D,B)),E))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),plus_plus(int,C,E))). [resolve(1021,a,1012,a)]. 1.52/1.86 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),plus_plus(real,minus_minus(real,C,times_times(real,D,B)),E))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),plus_plus(real,C,E))). [resolve(1021,a,1015,a)]. 1.52/1.86 1022 -dvd(A) | -semiring_0(A) | -hBOOL(hAPP(A,bool,B,times_times(A,C,D))) | hBOOL(hAPP(A,bool,B,f17(A,B,C))) # label(fact_908_unity__coeff__ex) # label(axiom). [clausify(431)]. 1.52/1.86 Derived: -semiring_0(int) | -hBOOL(hAPP(int,bool,A,times_times(int,B,C))) | hBOOL(hAPP(int,bool,A,f17(int,A,B))). [resolve(1022,a,1012,a)]. 1.52/1.86 Derived: -semiring_0(real) | -hBOOL(hAPP(real,bool,A,times_times(real,B,C))) | hBOOL(hAPP(real,bool,A,f17(real,A,B))). [resolve(1022,a,1015,a)]. 1.52/1.86 1023 -dvd(A) | -semiring_0(A) | -hBOOL(hAPP(A,bool,B,times_times(A,C,D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),plus_plus(A,f17(A,B,C),zero_zero(A)))) # label(fact_908_unity__coeff__ex) # label(axiom). [clausify(431)]. 1.52/1.86 Derived: -semiring_0(int) | -hBOOL(hAPP(int,bool,A,times_times(int,B,C))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),plus_plus(int,f17(int,A,B),zero_zero(int)))). [resolve(1023,a,1012,a)]. 1.52/1.87 Derived: -semiring_0(real) | -hBOOL(hAPP(real,bool,A,times_times(real,B,C))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),B),plus_plus(real,f17(real,A,B),zero_zero(real)))). [resolve(1023,a,1015,a)]. 1.52/1.87 1024 -dvd(A) | -semiring_0(A) | hBOOL(hAPP(A,bool,B,times_times(A,C,f18(A,B,C)))) | -hBOOL(hAPP(A,bool,B,D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),plus_plus(A,D,zero_zero(A)))) # label(fact_908_unity__coeff__ex) # label(axiom). [clausify(431)]. 1.52/1.87 Derived: -semiring_0(int) | hBOOL(hAPP(int,bool,A,times_times(int,B,f18(int,A,B)))) | -hBOOL(hAPP(int,bool,A,C)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),plus_plus(int,C,zero_zero(int)))). [resolve(1024,a,1012,a)]. 1.52/1.87 Derived: -semiring_0(real) | hBOOL(hAPP(real,bool,A,times_times(real,B,f18(real,A,B)))) | -hBOOL(hAPP(real,bool,A,C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),B),plus_plus(real,C,zero_zero(real)))). [resolve(1024,a,1015,a)]. 1.52/1.87 1025 -dvd(A) | times_times(A,B,C) = ti(A,times_times(A,B,C)) # label(tsy_c_Groups_Otimes__class_Otimes_0_res) # label(axiom). [clausify(477)]. 1.52/1.87 Derived: times_times(int,A,B) = ti(int,times_times(int,A,B)). [resolve(1025,a,1012,a)]. 1.52/1.87 Derived: times_times(real,A,B) = ti(real,times_times(real,A,B)). [resolve(1025,a,1015,a)]. 1.52/1.87 1026 dvd(nat) # label(arity_Nat_Onat___Rings_Odvd) # label(axiom). [assumption]. 1.52/1.87 Derived: ti(bool,hAPP(nat,bool,A,B)) = hAPP(nat,bool,A,B). [resolve(1026,a,1013,a)]. 1.52/1.87 Derived: times_times(nat,ti(nat,A),B) = times_times(nat,A,B). [resolve(1026,a,1014,a)]. 1.52/1.87 Derived: times_times(nat,A,ti(nat,B)) = times_times(nat,A,B). [resolve(1026,a,1016,a)]. 1.52/1.87 Derived: hAPP(nat,bool,A,ti(nat,B)) = hAPP(nat,bool,A,B). [resolve(1026,a,1017,a)]. 1.52/1.87 Derived: hAPP(nat,fun(nat,bool),A,ti(nat,B)) = hAPP(nat,fun(nat,bool),A,B). [resolve(1026,a,1018,a)]. 1.52/1.87 Derived: dvd_dvd(nat) = ti(fun(nat,fun(nat,bool)),dvd_dvd(nat)). [resolve(1026,a,1019,a)]. 1.52/1.87 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),plus_plus(nat,minus_minus(nat,C,times_times(nat,D,B)),E))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),plus_plus(nat,C,E))). [resolve(1026,a,1020,a)]. 1.52/1.87 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),plus_plus(nat,minus_minus(nat,C,times_times(nat,D,B)),E))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),plus_plus(nat,C,E))). [resolve(1026,a,1021,a)]. 1.52/1.87 Derived: -semiring_0(nat) | -hBOOL(hAPP(nat,bool,A,times_times(nat,B,C))) | hBOOL(hAPP(nat,bool,A,f17(nat,A,B))). [resolve(1026,a,1022,a)]. 1.52/1.87 Derived: -semiring_0(nat) | -hBOOL(hAPP(nat,bool,A,times_times(nat,B,C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),plus_plus(nat,f17(nat,A,B),zero_zero(nat)))). [resolve(1026,a,1023,a)]. 1.52/1.87 Derived: -semiring_0(nat) | hBOOL(hAPP(nat,bool,A,times_times(nat,B,f18(nat,A,B)))) | -hBOOL(hAPP(nat,bool,A,C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),plus_plus(nat,C,zero_zero(nat)))). [resolve(1026,a,1024,a)]. 1.52/1.87 Derived: times_times(nat,A,B) = ti(nat,times_times(nat,A,B)). [resolve(1026,a,1025,a)]. 1.52/1.87 1027 -dvd(A) | hAPP(A,bool,ti(fun(A,bool),B),C) = hAPP(A,bool,B,C) # label(tsy_c_hAPP_3_arg1) # label(axiom). [clausify(520)]. 1.52/1.87 Derived: hAPP(int,bool,ti(fun(int,bool),A),B) = hAPP(int,bool,A,B). [resolve(1027,a,1012,a)]. 1.52/1.87 Derived: hAPP(real,bool,ti(fun(real,bool),A),B) = hAPP(real,bool,A,B). [resolve(1027,a,1015,a)]. 1.52/1.87 Derived: hAPP(nat,bool,ti(fun(nat,bool),A),B) = hAPP(nat,bool,A,B). [resolve(1027,a,1026,a)]. 1.52/1.87 1028 -dvd(A) | hAPP(A,fun(A,bool),ti(fun(A,fun(A,bool)),B),C) = hAPP(A,fun(A,bool),B,C) # label(tsy_c_hAPP_2_arg1) # label(axiom). [clausify(538)]. 1.52/1.87 Derived: hAPP(int,fun(int,bool),ti(fun(int,fun(int,bool)),A),B) = hAPP(int,fun(int,bool),A,B). [resolve(1028,a,1012,a)]. 1.52/1.87 Derived: hAPP(real,fun(real,bool),ti(fun(real,fun(real,bool)),A),B) = hAPP(real,fun(real,bool),A,B). [resolve(1028,a,1015,a)]. 1.52/2.00 Derived: hAPP(nat,fun(nat,bool),ti(fun(nat,fun(nat,bool)),A),B) = hAPP(nat,fun(nat,bool),A,B). [resolve(1028,a,1026,a)]. 1.52/2.00 1029 -dvd(A) | ti(fun(A,bool),hAPP(A,fun(A,bool),B,C)) = hAPP(A,fun(A,bool),B,C) # label(tsy_c_hAPP_2_res) # label(axiom). [clausify(540)]. 1.52/2.00 Derived: ti(fun(int,bool),hAPP(int,fun(int,bool),A,B)) = hAPP(int,fun(int,bool),A,B). [resolve(1029,a,1012,a)]. 1.52/2.00 Derived: ti(fun(real,bool),hAPP(real,fun(real,bool),A,B)) = hAPP(real,fun(real,bool),A,B). [resolve(1029,a,1015,a)]. 1.52/2.00 Derived: ti(fun(nat,bool),hAPP(nat,fun(nat,bool),A,B)) = hAPP(nat,fun(nat,bool),A,B). [resolve(1029,a,1026,a)]. 1.52/2.00 1030 -dvd(A) | times_times(A,B,C) != D | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) # label(fact_730_dvdI) # label(axiom). [clausify(957)]. 1.52/2.00 Derived: times_times(int,A,B) != C | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),C)). [resolve(1030,a,1012,a)]. 1.52/2.00 Derived: times_times(real,A,B) != C | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),C)). [resolve(1030,a,1015,a)]. 1.52/2.00 Derived: times_times(nat,A,B) != C | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),C)). [resolve(1030,a,1026,a)]. 1.52/2.00 1031 -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),plus_plus(A,D,E))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),plus_plus(A,minus_minus(A,D,times_times(A,F,C)),E))) # label(fact_909_inf__period_I3_J) # label(axiom). [clausify(958)]. 1.52/2.00 1032 -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),plus_plus(A,D,E))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),plus_plus(A,minus_minus(A,D,times_times(A,F,C)),E))) # label(fact_909_inf__period_I3_J) # label(axiom). [clausify(958)]. 1.52/2.00 1033 number(int) # label(arity_Int_Oint___Int_Onumber) # label(axiom). [assumption]. 1.52/2.00 1034 -number(A) | number_number_of(A,ti(int,B)) = number_number_of(A,B) # label(tsy_c_Int_Onumber__class_Onumber__of_arg1) # label(axiom). [clausify(4)]. 1.52/2.00 Derived: number_number_of(int,ti(int,A)) = number_number_of(int,A). [resolve(1033,a,1034,a)]. 1.52/2.00 1035 -ring(A) | -number(A) | times_times(A,number_number_of(A,B),minus_minus(A,C,D)) = minus_minus(A,times_times(A,number_number_of(A,B),C),times_times(A,number_number_of(A,B),D)) # label(fact_262_right__diff__distrib__number__of) # label(axiom). [clausify(359)]. 1.52/2.00 1036 -number(A) | -semiring(A) | times_times(A,plus_plus(A,B,C),number_number_of(A,D)) = plus_plus(A,times_times(A,B,number_number_of(A,D)),times_times(A,C,number_number_of(A,D))) # label(fact_444_left__distrib__number__of) # label(axiom). [clausify(377)]. 1.52/2.00 Derived: -semiring(int) | times_times(int,plus_plus(int,A,B),number_number_of(int,C)) = plus_plus(int,times_times(int,A,number_number_of(int,C)),times_times(int,B,number_number_of(int,C))). [resolve(1036,a,1033,a)]. 1.52/2.00 1037 -number(A) | -semiring(A) | times_times(A,number_number_of(A,B),plus_plus(A,C,D)) = plus_plus(A,times_times(A,number_number_of(A,B),C),times_times(A,number_number_of(A,B),D)) # label(fact_443_right__distrib__number__of) # label(axiom). [clausify(400)]. 1.52/2.00 1038 -number(A) | number_number_of(A,B) = ti(A,number_number_of(A,B)) # label(tsy_c_Int_Onumber__class_Onumber__of_res) # label(axiom). [clausify(681)]. 1.52/2.00 Derived: number_number_of(int,A) = ti(int,number_number_of(int,A)). [resolve(1038,a,1033,a)]. 1.52/2.00 1039 number(real) # label(arity_RealDef_Oreal___Int_Onumber) # label(axiom). [assumption]. 1.52/2.00 Derived: number_number_of(real,ti(int,A)) = number_number_of(real,A). [resolve(1039,a,1034,a)]. 1.52/2.00 Derived: -ring(real) | times_times(real,number_number_of(real,A),minus_minus(real,B,C)) = minus_minus(real,times_times(real,number_number_of(real,A),B),times_times(real,number_number_of(real,A),C)). [resolve(1039,a,1035,b)]. 1.52/2.00 Derived: -semiring(real) | times_times(real,number_number_of(real,A),plus_plus(real,B,C)) = plus_plus(real,times_times(real,number_number_of(real,A),B),times_times(real,number_number_of(real,A),C)). [resolve(1039,a,1037,a)]. 1.52/2.02 Derived: number_number_of(real,A) = ti(real,number_number_of(real,A)). [resolve(1039,a,1038,a)]. 1.52/2.02 1040 -linorder(A) | -number(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,B)),number_number_of(A,C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,C)),number_number_of(A,B))) # label(fact_448_le__number__of__eq__not__less) # label(axiom). [clausify(745)]. 1.52/2.02 Derived: -linorder(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,A)),number_number_of(int,B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,B)),number_number_of(int,A))). [resolve(1040,b,1033,a)]. 1.52/2.02 Derived: -linorder(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,A)),number_number_of(real,B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,B)),number_number_of(real,A))). [resolve(1040,b,1039,a)]. 1.52/2.02 1041 -linorder(A) | -number(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,B)),number_number_of(A,C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,C)),number_number_of(A,B))) # label(fact_448_le__number__of__eq__not__less) # label(axiom). [clausify(745)]. 1.52/2.02 Derived: -linorder(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,A)),number_number_of(int,B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,B)),number_number_of(int,A))). [resolve(1041,b,1033,a)]. 1.52/2.02 Derived: -linorder(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,A)),number_number_of(real,B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,B)),number_number_of(real,A))). [resolve(1041,b,1039,a)]. 1.52/2.02 1042 number(nat) # label(arity_Nat_Onat___Int_Onumber) # label(axiom). [assumption]. 1.52/2.02 Derived: number_number_of(nat,ti(int,A)) = number_number_of(nat,A). [resolve(1042,a,1034,a)]. 1.52/2.02 Derived: -ring(nat) | times_times(nat,number_number_of(nat,A),minus_minus(nat,B,C)) = minus_minus(nat,times_times(nat,number_number_of(nat,A),B),times_times(nat,number_number_of(nat,A),C)). [resolve(1042,a,1035,b)]. 1.52/2.02 Derived: -semiring(nat) | times_times(nat,plus_plus(nat,A,B),number_number_of(nat,C)) = plus_plus(nat,times_times(nat,A,number_number_of(nat,C)),times_times(nat,B,number_number_of(nat,C))). [resolve(1042,a,1036,a)]. 1.52/2.02 Derived: -semiring(nat) | times_times(nat,number_number_of(nat,A),plus_plus(nat,B,C)) = plus_plus(nat,times_times(nat,number_number_of(nat,A),B),times_times(nat,number_number_of(nat,A),C)). [resolve(1042,a,1037,a)]. 1.52/2.02 Derived: number_number_of(nat,A) = ti(nat,number_number_of(nat,A)). [resolve(1042,a,1038,a)]. 1.52/2.02 Derived: -linorder(nat) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),number_number_of(nat,A)),number_number_of(nat,B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),number_number_of(nat,B)),number_number_of(nat,A))). [resolve(1042,a,1040,b)]. 1.52/2.02 Derived: -linorder(nat) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),number_number_of(nat,A)),number_number_of(nat,B))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),number_number_of(nat,B)),number_number_of(nat,A))). [resolve(1042,a,1041,b)]. 1.52/2.02 1043 -monoid_mult(A) | -number(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_545_power2__eq__square__number__of) # label(axiom). [clausify(902)]. 1.52/2.02 Derived: -monoid_mult(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(1043,b,1033,a)]. 1.52/2.02 Derived: -monoid_mult(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(1043,b,1039,a)]. 1.52/2.02 Derived: -monoid_mult(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(1043,b,1042,a)]. 1.52/2.07 1044 -number(A) | -ring(A) | times_times(A,minus_minus(A,B,C),number_number_of(A,D)) = minus_minus(A,times_times(A,B,number_number_of(A,D)),times_times(A,C,number_number_of(A,D))) # label(fact_263_left__diff__distrib__number__of) # label(axiom). [clausify(937)]. 1.52/2.07 Derived: -ring(int) | times_times(int,minus_minus(int,A,B),number_number_of(int,C)) = minus_minus(int,times_times(int,A,number_number_of(int,C)),times_times(int,B,number_number_of(int,C))). [resolve(1044,a,1033,a)]. 1.52/2.07 Derived: -ring(real) | times_times(real,minus_minus(real,A,B),number_number_of(real,C)) = minus_minus(real,times_times(real,A,number_number_of(real,C)),times_times(real,B,number_number_of(real,C))). [resolve(1044,a,1039,a)]. 1.52/2.07 Derived: -ring(nat) | times_times(nat,minus_minus(nat,A,B),number_number_of(nat,C)) = minus_minus(nat,times_times(nat,A,number_number_of(nat,C)),times_times(nat,B,number_number_of(nat,C))). [resolve(1044,a,1042,a)]. 1.52/2.07 1045 linordered_semidom(nat) # label(arity_Nat_Onat___Rings_Olinordered__semidom) # label(axiom). [assumption]. 1.52/2.07 1046 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),one_one(A))) # label(fact_178_zero__less__one) # label(axiom). [clausify(6)]. 1.52/2.07 1047 -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_108_one__less__power) # label(axiom). [clausify(10)]. 1.52/2.07 1048 -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_449_zero__le__power) # label(axiom). [clausify(18)]. 1.52/2.07 1049 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(nat,A,semiring_1_of_nat(A),B)),hAPP(nat,A,semiring_1_of_nat(A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) # label(fact_266_of__nat__le__iff) # label(axiom). [clausify(52)]. 1.52/2.07 1050 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(nat,A,semiring_1_of_nat(A),B)),hAPP(nat,A,semiring_1_of_nat(A),C))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) # label(fact_266_of__nat__le__iff) # label(axiom). [clausify(52)]. 1.52/2.07 1051 -linordered_semidom(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),ord_less(A),zero_zero(A)),hAPP(nat,A,semiring_1_of_nat(A),B))) # label(fact_107_of__nat__0__less__iff) # label(axiom). [clausify(115)]. 1.52/2.07 1052 -linordered_semidom(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),ord_less(A),zero_zero(A)),hAPP(nat,A,semiring_1_of_nat(A),B))) # label(fact_107_of__nat__0__less__iff) # label(axiom). [clausify(115)]. 1.52/2.07 1053 -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_548_power2__eq__imp__eq) # label(axiom). [clausify(148)]. 1.52/2.07 1054 -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_551_power2__less__imp__less) # label(axiom). [clausify(208)]. 1.52/2.07 1055 -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_183_power__less__imp__less__exp) # label(axiom). [clausify(226)]. 1.52/2.08 1056 -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_392_power__increasing__iff) # label(axiom). [clausify(228)]. 1.52/2.08 1057 -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_392_power__increasing__iff) # label(axiom). [clausify(228)]. 1.52/2.08 1058 -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_182_power__strict__increasing__iff) # label(axiom). [clausify(231)]. 1.52/2.08 1059 -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_182_power__strict__increasing__iff) # label(axiom). [clausify(231)]. 1.52/2.08 1060 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),plus_plus(A,B,one_one(A)))) # label(fact_180_less__add__one) # label(axiom). [clausify(241)]. 1.52/2.08 1061 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,semiring_1_of_nat(A),B)),zero_zero(A))) # label(fact_186_of__nat__less__0__iff) # label(axiom). [clausify(245)]. 1.52/2.08 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),one_one(nat))). [resolve(1045,a,1046,a)]. 1.52/2.08 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(1045,a,1047,a)]. 1.52/2.08 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,semiring_1_of_nat(nat),A)),hAPP(nat,nat,semiring_1_of_nat(nat),B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)). [resolve(1045,a,1049,a)]. 1.52/2.08 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,semiring_1_of_nat(nat),A)),hAPP(nat,nat,semiring_1_of_nat(nat),B))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)). [resolve(1045,a,1050,a)]. 1.52/2.08 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,semiring_1_of_nat(nat),A))). [resolve(1045,a,1051,a)]. 1.52/2.08 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,semiring_1_of_nat(nat),A))). [resolve(1045,a,1052,a)]. 1.52/2.08 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(1045,a,1053,a)]. 1.52/2.08 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(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(nat),A),B)). [resolve(1045,a,1054,a)]. 1.52/2.08 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(1045,a,1055,a)]. 1.52/2.08 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(1045,a,1056,a)]. 1.52/2.08 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(1045,a,1057,a)]. 1.52/2.08 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(1045,a,1059,a)]. 1.52/2.08 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),plus_plus(nat,A,one_one(nat)))). [resolve(1045,a,1060,a)]. 1.52/2.08 1062 -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)),times_times(A,B,hAPP(nat,A,power_power(A,B),C)))) # label(fact_495_power__gt1__lemma) # label(axiom). [clausify(249)]. 1.52/2.08 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)),times_times(nat,A,hAPP(nat,nat,power_power(nat,A),B)))). [resolve(1062,a,1045,a)]. 1.52/2.08 1063 -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_454_one__le__power) # label(axiom). [clausify(262)]. 1.52/2.08 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(1063,a,1045,a)]. 1.52/2.08 1064 -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)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C)) | ti(A,B) = ti(A,D) # label(fact_526_power__eq__imp__eq__base) # label(axiom). [clausify(294)]. 1.52/2.08 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)),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),zero_zero(nat)),B)) | ti(nat,A) = ti(nat,C). [resolve(1064,a,1045,a)]. 1.52/2.08 1065 -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_494_power__less__imp__less__base) # label(axiom). [clausify(323)]. 1.52/2.08 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(1065,a,1045,a)]. 1.52/2.08 1066 -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),times_times(A,B,hAPP(nat,A,power_power(A,B),C))),hAPP(nat,A,power_power(A,B),C))) # label(fact_524_power__Suc__less) # label(axiom). [clausify(329)]. 1.52/2.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),one_one(nat))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),times_times(nat,A,hAPP(nat,nat,power_power(nat,A),B))),hAPP(nat,nat,power_power(nat,A),B))). [resolve(1066,a,1045,a)]. 1.52/2.09 1067 -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_179_zero__less__power) # label(axiom). [clausify(365)]. 1.52/2.09 1068 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,semiring_1_of_nat(A),B)),hAPP(nat,A,semiring_1_of_nat(A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)) # label(fact_167_of__nat__less__iff) # label(axiom). [clausify(430)]. 1.52/2.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,semiring_1_of_nat(nat),A)),hAPP(nat,nat,semiring_1_of_nat(nat),B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)). [resolve(1068,a,1045,a)]. 1.52/2.09 1069 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,semiring_1_of_nat(A),B)),hAPP(nat,A,semiring_1_of_nat(A),C))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)) # label(fact_167_of__nat__less__iff) # label(axiom). [clausify(430)]. 1.52/2.09 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,semiring_1_of_nat(nat),A)),hAPP(nat,nat,semiring_1_of_nat(nat),B))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)). [resolve(1069,a,1045,a)]. 1.52/2.09 1070 -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_450_power__mono) # label(axiom). [clausify(433)]. 1.52/2.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,power_power(nat,A),C)),hAPP(nat,nat,power_power(nat,B),C))). [resolve(1070,a,1045,a)]. 1.52/2.09 1071 -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_393_power__le__imp__le__exp) # label(axiom). [clausify(466)]. 1.52/2.09 1072 linordered_semidom(int) # label(arity_Int_Oint___Rings_Olinordered__semidom) # label(axiom). [assumption]. 1.52/2.09 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(1072,a,1047,a)]. 1.52/2.09 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),A)),hAPP(nat,int,semiring_1_of_nat(int),B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)). [resolve(1072,a,1049,a)]. 1.52/2.09 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),A)),hAPP(nat,int,semiring_1_of_nat(int),B))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)). [resolve(1072,a,1050,a)]. 1.52/2.09 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(1072,a,1053,a)]. 1.52/2.10 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))))),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(int),A),B)). [resolve(1072,a,1054,a)]. 1.52/2.10 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(1072,a,1055,a)]. 1.52/2.10 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(1072,a,1056,a)]. 1.52/2.10 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(1072,a,1057,a)]. 1.52/2.10 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(1072,a,1059,a)]. 1.52/2.10 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),plus_plus(int,A,one_one(int)))). [resolve(1072,a,1060,a)]. 1.52/2.10 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)),times_times(int,A,hAPP(nat,int,power_power(int,A),B)))). [resolve(1072,a,1062,a)]. 1.52/2.10 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(1072,a,1063,a)]. 1.52/2.10 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)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | ti(int,A) = ti(int,C). [resolve(1072,a,1064,a)]. 1.52/2.10 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(1072,a,1065,a)]. 1.52/2.10 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),times_times(int,A,hAPP(nat,int,power_power(int,A),B))),hAPP(nat,int,power_power(int,A),B))). [resolve(1072,a,1066,a)]. 1.52/2.10 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(1072,a,1067,a)]. 1.52/2.10 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(1072,a,1070,a)]. 1.52/2.10 1073 -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_539_power__strict__mono) # label(axiom). [clausify(515)]. 1.52/2.13 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(1073,a,1045,a)]. 1.52/2.13 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(1073,a,1072,a)]. 1.52/2.13 1074 -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_316_power__increasing) # label(axiom). [clausify(569)]. 1.52/2.13 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(1074,a,1045,a)]. 1.52/2.13 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(1074,a,1072,a)]. 1.52/2.13 1075 -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_110_power__strict__decreasing) # label(axiom). [clausify(705)]. 1.52/2.13 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(1075,a,1045,a)]. 1.52/2.13 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(1075,a,1072,a)]. 1.52/2.13 1076 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),zero_zero(A))) # label(fact_441_not__one__le__zero) # label(axiom). [clausify(715)]. 1.52/2.13 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),zero_zero(nat))). [resolve(1076,a,1045,a)]. 1.52/2.13 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),zero_zero(int))). [resolve(1076,a,1072,a)]. 1.52/2.13 1077 linordered_semidom(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semidom) # label(axiom). [assumption]. 1.52/2.13 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),one_one(real))). [resolve(1077,a,1046,a)]. 1.52/2.13 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(1077,a,1047,a)]. 1.52/2.13 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,power_power(real,A),B))). [resolve(1077,a,1048,a)]. 1.52/2.13 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,semiring_1_of_nat(real),A)),hAPP(nat,real,semiring_1_of_nat(real),B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)). [resolve(1077,a,1049,a)]. 1.52/2.13 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,semiring_1_of_nat(real),A)),hAPP(nat,real,semiring_1_of_nat(real),B))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)). [resolve(1077,a,1050,a)]. 1.52/2.13 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),ord_less(real),zero_zero(real)),hAPP(nat,real,semiring_1_of_nat(real),A))). [resolve(1077,a,1051,a)]. 1.52/2.13 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),ord_less(real),zero_zero(real)),hAPP(nat,real,semiring_1_of_nat(real),A))). [resolve(1077,a,1052,a)]. 1.52/2.13 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(1077,a,1053,a)]. 1.52/2.13 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))))),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(real),A),B)). [resolve(1077,a,1054,a)]. 1.52/2.13 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(1077,a,1055,a)]. 1.52/2.13 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(1077,a,1056,a)]. 1.52/2.13 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(1077,a,1057,a)]. 1.52/2.13 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(1077,a,1059,a)]. 1.52/2.13 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),plus_plus(real,A,one_one(real)))). [resolve(1077,a,1060,a)]. 1.52/2.13 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,semiring_1_of_nat(real),A)),zero_zero(real))). [resolve(1077,a,1061,a)]. 1.52/2.13 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)),times_times(real,A,hAPP(nat,real,power_power(real,A),B)))). [resolve(1077,a,1062,a)]. 1.52/2.14 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(1077,a,1063,a)]. 1.52/2.14 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)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | ti(real,A) = ti(real,C). [resolve(1077,a,1064,a)]. 1.52/2.14 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(1077,a,1065,a)]. 1.52/2.14 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),times_times(real,A,hAPP(nat,real,power_power(real,A),B))),hAPP(nat,real,power_power(real,A),B))). [resolve(1077,a,1066,a)]. 1.52/2.14 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(1077,a,1067,a)]. 1.52/2.14 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,semiring_1_of_nat(real),A)),hAPP(nat,real,semiring_1_of_nat(real),B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)). [resolve(1077,a,1068,a)]. 1.52/2.14 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,semiring_1_of_nat(real),A)),hAPP(nat,real,semiring_1_of_nat(real),B))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)). [resolve(1077,a,1069,a)]. 1.52/2.14 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(1077,a,1070,a)]. 1.52/2.14 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(1077,a,1073,a)]. 1.52/2.14 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(1077,a,1074,a)]. 1.52/2.14 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(1077,a,1075,a)]. 1.52/2.14 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),zero_zero(real))). [resolve(1077,a,1076,a)]. 1.52/2.14 1078 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),plus_plus(A,one_one(A),one_one(A)))) # label(fact_111_zero__less__two) # label(axiom). [clausify(764)]. 1.52/2.14 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),plus_plus(nat,one_one(nat),one_one(nat)))). [resolve(1078,a,1045,a)]. 1.52/2.15 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),plus_plus(int,one_one(int),one_one(int)))). [resolve(1078,a,1072,a)]. 1.52/2.15 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),plus_plus(real,one_one(real),one_one(real)))). [resolve(1078,a,1077,a)]. 1.52/2.15 1079 -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_184_power__strict__increasing) # label(axiom). [clausify(796)]. 1.52/2.15 1080 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),zero_zero(A))) # label(fact_177_not__one__less__zero) # label(axiom). [clausify(811)]. 1.52/2.15 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),zero_zero(int))). [resolve(1080,a,1072,a)]. 1.52/2.15 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),zero_zero(real))). [resolve(1080,a,1077,a)]. 1.52/2.15 1081 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,semiring_1_of_nat(A),B)),hAPP(nat,A,semiring_1_of_nat(A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)) # label(fact_169_of__nat__less__imp__less) # label(axiom). [clausify(827)]. 1.52/2.15 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),A)),hAPP(nat,int,semiring_1_of_nat(int),B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)). [resolve(1081,a,1072,a)]. 1.52/2.15 1082 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),one_one(A))) # label(fact_442_zero__le__one) # label(axiom). [clausify(849)]. 1.52/2.15 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),one_one(nat))). [resolve(1082,a,1045,a)]. 1.52/2.15 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),one_one(int))). [resolve(1082,a,1072,a)]. 1.52/2.15 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),one_one(real))). [resolve(1082,a,1077,a)]. 1.52/2.15 1083 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(nat,A,semiring_1_of_nat(A),B))) # label(fact_457_zero__le__imp__of__nat) # label(axiom). [clausify(861)]. 1.52/2.15 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),hAPP(nat,nat,semiring_1_of_nat(nat),A))). [resolve(1083,a,1045,a)]. 1.52/2.15 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),A))). [resolve(1083,a,1072,a)]. 1.52/2.15 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,semiring_1_of_nat(real),A))). [resolve(1083,a,1077,a)]. 1.52/2.15 1084 -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_391_power__decreasing) # label(axiom). [clausify(872)]. 1.52/2.15 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),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(1084,a,1045,a)]. 1.52/2.15 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),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(1084,a,1072,a)]. 1.52/2.17 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(1084,a,1077,a)]. 1.52/2.17 1085 -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)),times_times(A,B,hAPP(nat,A,power_power(A,B),C)))) # label(fact_496_power__less__power__Suc) # label(axiom). [clausify(874)]. 1.52/2.17 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)),times_times(nat,A,hAPP(nat,nat,power_power(nat,A),B)))). [resolve(1085,a,1045,a)]. 1.52/2.17 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)),times_times(int,A,hAPP(nat,int,power_power(int,A),B)))). [resolve(1085,a,1072,a)]. 1.52/2.17 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)),times_times(real,A,hAPP(nat,real,power_power(real,A),B)))). [resolve(1085,a,1077,a)]. 1.52/2.17 1086 -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),plus_plus(A,B,D))) # label(fact_176_pos__add__strict) # label(axiom). [clausify(888)]. 1.52/2.17 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),plus_plus(nat,A,C))). [resolve(1086,a,1045,a)]. 1.52/2.17 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),plus_plus(int,A,C))). [resolve(1086,a,1072,a)]. 1.52/2.17 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),plus_plus(real,A,C))). [resolve(1086,a,1077,a)]. 1.52/2.17 1087 -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_181_power__inject__exp) # label(axiom). [clausify(984)]. 1.52/2.17 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(1087,a,1045,a)]. 1.52/2.17 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(1087,a,1072,a)]. 1.52/2.17 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(1087,a,1077,a)]. 1.52/2.17 1088 -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_181_power__inject__exp) # label(axiom). [clausify(984)]. 1.52/2.17 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(1088,a,1045,a)]. 1.52/2.17 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(1088,a,1072,a)]. 1.52/2.25 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(1088,a,1077,a)]. 1.52/2.25 1089 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(nat,A,semiring_1_of_nat(A),B))) # label(fact_458_of__nat__0__le__iff) # label(axiom). [clausify(988)]. 1.52/2.25 1090 -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_547_power2__le__imp__le) # label(axiom). [clausify(989)]. 1.52/2.25 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(1090,a,1045,a)]. 1.52/2.25 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(1090,a,1072,a)]. 1.52/2.25 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(1090,a,1077,a)]. 1.52/2.25 1091 -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)),times_times(A,B,C))) # label(fact_431_less__1__mult) # label(axiom). [clausify(990)]. 1.52/2.25 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)),times_times(nat,A,B))). [resolve(1091,a,1045,a)]. 1.52/2.25 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)),times_times(int,A,B))). [resolve(1091,a,1072,a)]. 1.52/2.25 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)),times_times(real,A,B))). [resolve(1091,a,1077,a)]. 1.52/2.25 1092 -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),hAPP(nat,A,semiring_1_of_nat(A),B)),hAPP(nat,A,semiring_1_of_nat(A),C))) # label(fact_168_less__imp__of__nat__less) # label(axiom). [clausify(994)]. 1.52/2.25 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),hAPP(nat,int,semiring_1_of_nat(int),A)),hAPP(nat,int,semiring_1_of_nat(int),B))). [resolve(1092,a,1072,a)]. 1.52/2.25 1093 ordere453448008miring(nat) # label(arity_Nat_Onat___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 1.52/2.25 1094 -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),times_times(A,C,B)),zero_zero(A))) # label(fact_280_mult__nonneg__nonpos2) # label(axiom). [clausify(7)]. 1.52/2.29 1095 -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),times_times(A,B,C)),zero_zero(A))) # label(fact_281_mult__nonneg__nonpos) # label(axiom). [clausify(79)]. 1.52/2.29 1096 -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)),times_times(A,B,C))) # label(fact_284_mult__nonneg__nonneg) # label(axiom). [clausify(527)]. 1.52/2.29 1097 -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),times_times(A,B,C)),zero_zero(A))) # label(fact_269_split__mult__neg__le) # label(axiom). [clausify(536)]. 1.52/2.29 1098 -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),times_times(A,C,B)),zero_zero(A))) # label(fact_269_split__mult__neg__le) # label(axiom). [clausify(536)]. 1.52/2.29 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),times_times(nat,B,A)),zero_zero(nat))). [resolve(1093,a,1094,a)]. 1.52/2.29 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),times_times(nat,A,B)),zero_zero(nat))). [resolve(1093,a,1095,a)]. 1.52/2.29 1099 -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),times_times(A,B,C)),zero_zero(A))) # label(fact_279_mult__nonpos__nonneg) # label(axiom). [clausify(814)]. 1.52/2.29 1100 ordere453448008miring(real) # label(arity_RealDef_Oreal___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 1.52/2.29 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),times_times(real,B,A)),zero_zero(real))). [resolve(1100,a,1094,a)]. 1.52/2.29 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),times_times(real,A,B)),zero_zero(real))). [resolve(1100,a,1095,a)]. 1.52/2.29 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)),times_times(real,A,B))). [resolve(1100,a,1096,a)]. 1.52/2.29 1101 ordere453448008miring(int) # label(arity_Int_Oint___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 1.52/2.29 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),times_times(int,B,A)),zero_zero(int))). [resolve(1101,a,1094,a)]. 1.52/2.29 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),times_times(int,A,B)),zero_zero(int))). [resolve(1101,a,1095,a)]. 2.11/2.33 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)),times_times(int,A,B))). [resolve(1101,a,1096,a)]. 2.11/2.33 1102 ordere216010020id_add(nat) # label(arity_Nat_Onat___Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 2.11/2.33 1103 -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),plus_plus(A,B,C)),zero_zero(A))) # label(fact_197_add__neg__neg) # label(axiom). [clausify(9)]. 2.11/2.33 1104 -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)),plus_plus(A,B,C))) # label(fact_488_add__pos__nonneg) # label(axiom). [clausify(41)]. 2.11/2.33 1105 -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)),plus_plus(A,B,C))) # label(fact_434_add__nonneg__nonneg) # label(axiom). [clausify(107)]. 2.11/2.33 1106 -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),plus_plus(A,B,D))) # label(fact_491_add__strict__increasing2) # label(axiom). [clausify(158)]. 2.11/2.33 1107 ordere216010020id_add(int) # label(arity_Int_Oint___Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 2.11/2.33 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),plus_plus(int,A,B)),zero_zero(int))). [resolve(1107,a,1103,a)]. 2.11/2.33 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_eq(int),zero_zero(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),plus_plus(int,A,B))). [resolve(1107,a,1104,a)]. 2.11/2.33 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),plus_plus(int,A,C))). [resolve(1107,a,1106,a)]. 2.11/2.33 1108 -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),plus_plus(A,B,C)),zero_zero(A))) # label(fact_438_add__nonpos__nonpos) # label(axiom). [clausify(267)]. 2.11/2.33 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),zero_zero(nat))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),zero_zero(nat))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),plus_plus(nat,A,B)),zero_zero(nat))). [resolve(1108,a,1102,a)]. 2.11/2.33 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),plus_plus(int,A,B)),zero_zero(int))). [resolve(1108,a,1107,a)]. 2.11/2.33 1109 -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)),plus_plus(A,B,C))) # label(fact_489_add__nonneg__pos) # label(axiom). [clausify(302)]. 2.17/2.39 1110 -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),plus_plus(A,B,C)),zero_zero(A))) # label(fact_492_add__neg__nonpos) # label(axiom). [clausify(394)]. 2.17/2.39 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),plus_plus(int,A,B)),zero_zero(int))). [resolve(1110,a,1107,a)]. 2.17/2.39 1111 -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)),plus_plus(A,B,C))) # label(fact_196_add__pos__pos) # label(axiom). [clausify(410)]. 2.17/2.39 1112 -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),plus_plus(A,B,D))) # label(fact_436_add__increasing) # label(axiom). [clausify(605)]. 2.17/2.39 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),plus_plus(nat,A,C))). [resolve(1112,a,1102,a)]. 2.17/2.39 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),plus_plus(int,A,C))). [resolve(1112,a,1107,a)]. 2.17/2.39 1113 -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),plus_plus(A,B,D))) # label(fact_490_add__strict__increasing) # label(axiom). [clausify(714)]. 2.17/2.39 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),plus_plus(nat,A,C))). [resolve(1113,a,1102,a)]. 2.17/2.39 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),plus_plus(int,A,C))). [resolve(1113,a,1107,a)]. 2.17/2.39 1114 -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),plus_plus(A,B,C)),zero_zero(A))) # label(fact_493_add__nonpos__neg) # label(axiom). [clausify(794)]. 2.17/2.39 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),plus_plus(int,A,B)),zero_zero(int))). [resolve(1114,a,1107,a)]. 2.17/2.39 1115 -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)) | plus_plus(A,B,C) != zero_zero(A) | ti(A,B) = zero_zero(A) # label(fact_435_add__nonneg__eq__0__iff) # label(axiom). [clausify(925)]. 2.17/2.39 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)) | plus_plus(nat,A,B) != zero_zero(nat) | ti(nat,A) = zero_zero(nat). [resolve(1115,a,1102,a)]. 2.17/2.39 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)) | plus_plus(int,A,B) != zero_zero(int) | ti(int,A) = zero_zero(int). [resolve(1115,a,1107,a)]. 2.18/2.40 1116 -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)) | plus_plus(A,B,C) != zero_zero(A) | ti(A,C) = zero_zero(A) # label(fact_435_add__nonneg__eq__0__iff) # label(axiom). [clausify(925)]. 2.18/2.40 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)) | plus_plus(nat,A,B) != zero_zero(nat) | ti(nat,B) = zero_zero(nat). [resolve(1116,a,1102,a)]. 2.18/2.40 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)) | plus_plus(int,A,B) != zero_zero(int) | ti(int,B) = zero_zero(int). [resolve(1116,a,1107,a)]. 2.18/2.40 1117 -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)) | plus_plus(A,B,C) = zero_zero(A) | ti(A,B) != zero_zero(A) | ti(A,C) != zero_zero(A) # label(fact_435_add__nonneg__eq__0__iff) # label(axiom). [clausify(925)]. 2.18/2.40 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)) | plus_plus(nat,A,B) = zero_zero(nat) | ti(nat,A) != zero_zero(nat) | ti(nat,B) != zero_zero(nat). [resolve(1117,a,1102,a)]. 2.18/2.40 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)) | plus_plus(int,A,B) = zero_zero(int) | ti(int,A) != zero_zero(int) | ti(int,B) != zero_zero(int). [resolve(1117,a,1107,a)]. 2.18/2.40 1118 -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),plus_plus(A,D,B))) # label(fact_437_add__increasing2) # label(axiom). [clausify(981)]. 2.18/2.40 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),plus_plus(nat,C,A))). [resolve(1118,a,1102,a)]. 2.18/2.40 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),plus_plus(int,C,A))). [resolve(1118,a,1107,a)]. 2.18/2.40 1119 ordere216010020id_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 2.18/2.40 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),plus_plus(real,A,B)),zero_zero(real))). [resolve(1119,a,1103,a)]. 2.18/2.40 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),plus_plus(real,A,B))). [resolve(1119,a,1104,a)]. 2.18/2.40 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)),plus_plus(real,A,B))). [resolve(1119,a,1105,a)]. 2.18/2.40 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),plus_plus(real,A,C))). [resolve(1119,a,1106,a)]. 2.18/2.40 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),plus_plus(real,A,B)),zero_zero(real))). [resolve(1119,a,1108,a)]. 2.20/2.46 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),plus_plus(real,A,B)),zero_zero(real))). [resolve(1119,a,1110,a)]. 2.20/2.46 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),plus_plus(real,A,C))). [resolve(1119,a,1112,a)]. 2.20/2.46 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),plus_plus(real,A,C))). [resolve(1119,a,1113,a)]. 2.20/2.46 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),plus_plus(real,A,B)),zero_zero(real))). [resolve(1119,a,1114,a)]. 2.20/2.46 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)) | plus_plus(real,A,B) != zero_zero(real) | ti(real,A) = zero_zero(real). [resolve(1119,a,1115,a)]. 2.20/2.46 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)) | plus_plus(real,A,B) != zero_zero(real) | ti(real,B) = zero_zero(real). [resolve(1119,a,1116,a)]. 2.20/2.46 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)) | plus_plus(real,A,B) = zero_zero(real) | ti(real,A) != zero_zero(real) | ti(real,B) != zero_zero(real). [resolve(1119,a,1117,a)]. 2.20/2.46 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),plus_plus(real,C,A))). [resolve(1119,a,1118,a)]. 2.20/2.46 1120 -mult_zero(A) | times_times(A,B,C) = ti(A,times_times(A,B,C)) # label(tsy_c_Groups_Otimes__class_Otimes_4_res) # label(axiom). [clausify(151)]. 2.20/2.46 1121 mult_zero(nat) # label(arity_Nat_Onat___Rings_Omult__zero) # label(axiom). [assumption]. 2.20/2.46 1122 -mult_zero(A) | times_times(A,zero_zero(A),B) = zero_zero(A) # label(fact_328_mult__zero__left) # label(axiom). [clausify(402)]. 2.20/2.46 Derived: times_times(nat,zero_zero(nat),A) = zero_zero(nat). [resolve(1122,a,1121,a)]. 2.20/2.46 1123 mult_zero(real) # label(arity_RealDef_Oreal___Rings_Omult__zero) # label(axiom). [assumption]. 2.20/2.46 Derived: times_times(real,zero_zero(real),A) = zero_zero(real). [resolve(1123,a,1122,a)]. 2.20/2.46 1124 -power(A) | -mult_zero(A) | -zero_neq_one(A) | -no_zero_divisors(A) | number_number_of(nat,B) = zero_zero(nat) | ti(A,C) != zero_zero(A) | hAPP(nat,A,power_power(A,C),number_number_of(nat,B)) = zero_zero(A) # label(fact_86_power__eq__0__iff__number__of) # label(axiom). [clausify(533)]. 2.20/2.46 Derived: -power(nat) | -zero_neq_one(nat) | -no_zero_divisors(nat) | number_number_of(nat,A) = zero_zero(nat) | ti(nat,B) != zero_zero(nat) | hAPP(nat,nat,power_power(nat,B),number_number_of(nat,A)) = zero_zero(nat). [resolve(1124,b,1121,a)]. 2.20/2.46 Derived: -power(real) | -zero_neq_one(real) | -no_zero_divisors(real) | number_number_of(nat,A) = zero_zero(nat) | ti(real,B) != zero_zero(real) | hAPP(nat,real,power_power(real,B),number_number_of(nat,A)) = zero_zero(real). [resolve(1124,b,1123,a)]. 2.29/2.51 1125 -power(A) | -mult_zero(A) | -zero_neq_one(A) | -no_zero_divisors(A) | number_number_of(nat,B) != zero_zero(nat) | hAPP(nat,A,power_power(A,C),number_number_of(nat,B)) != zero_zero(A) # label(fact_86_power__eq__0__iff__number__of) # label(axiom). [clausify(533)]. 2.29/2.51 Derived: -power(nat) | -zero_neq_one(nat) | -no_zero_divisors(nat) | number_number_of(nat,A) != zero_zero(nat) | hAPP(nat,nat,power_power(nat,B),number_number_of(nat,A)) != zero_zero(nat). [resolve(1125,b,1121,a)]. 2.29/2.51 Derived: -power(real) | -zero_neq_one(real) | -no_zero_divisors(real) | number_number_of(nat,A) != zero_zero(nat) | hAPP(nat,real,power_power(real,B),number_number_of(nat,A)) != zero_zero(real). [resolve(1125,b,1123,a)]. 2.29/2.51 1126 -power(A) | -mult_zero(A) | -zero_neq_one(A) | -no_zero_divisors(A) | ti(A,B) = zero_zero(A) | hAPP(nat,A,power_power(A,B),number_number_of(nat,C)) != zero_zero(A) # label(fact_86_power__eq__0__iff__number__of) # label(axiom). [clausify(533)]. 2.29/2.51 Derived: -power(nat) | -zero_neq_one(nat) | -no_zero_divisors(nat) | ti(nat,A) = zero_zero(nat) | hAPP(nat,nat,power_power(nat,A),number_number_of(nat,B)) != zero_zero(nat). [resolve(1126,b,1121,a)]. 2.29/2.51 Derived: -power(real) | -zero_neq_one(real) | -no_zero_divisors(real) | ti(real,A) = zero_zero(real) | hAPP(nat,real,power_power(real,A),number_number_of(nat,B)) != zero_zero(real). [resolve(1126,b,1123,a)]. 2.29/2.51 1127 mult_zero(int) # label(arity_Int_Oint___Rings_Omult__zero) # label(axiom). [assumption]. 2.29/2.51 Derived: times_times(int,zero_zero(int),A) = zero_zero(int). [resolve(1127,a,1122,a)]. 2.29/2.51 Derived: -power(int) | -zero_neq_one(int) | -no_zero_divisors(int) | number_number_of(nat,A) = zero_zero(nat) | ti(int,B) != zero_zero(int) | hAPP(nat,int,power_power(int,B),number_number_of(nat,A)) = zero_zero(int). [resolve(1127,a,1124,b)]. 2.29/2.51 Derived: -power(int) | -zero_neq_one(int) | -no_zero_divisors(int) | number_number_of(nat,A) != zero_zero(nat) | hAPP(nat,int,power_power(int,B),number_number_of(nat,A)) != zero_zero(int). [resolve(1127,a,1125,b)]. 2.29/2.51 Derived: -power(int) | -zero_neq_one(int) | -no_zero_divisors(int) | ti(int,A) = zero_zero(int) | hAPP(nat,int,power_power(int,A),number_number_of(nat,B)) != zero_zero(int). [resolve(1127,a,1126,b)]. 2.29/2.51 1128 -mult_zero(A) | times_times(A,ti(A,B),C) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_4_arg1) # label(axiom). [clausify(741)]. 2.29/2.51 1129 -mult_zero(A) | times_times(A,B,ti(A,C)) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_4_arg2) # label(axiom). [clausify(884)]. 2.29/2.51 1130 -power(A) | -no_zero_divisors(A) | -zero_neq_one(A) | -mult_zero(A) | ti(A,B) != zero_zero(A) | zero_zero(nat) = C | hAPP(nat,A,power_power(A,B),C) = zero_zero(A) # label(fact_185_power__eq__0__iff) # label(axiom). [clausify(946)]. 2.29/2.51 Derived: -power(nat) | -no_zero_divisors(nat) | -zero_neq_one(nat) | ti(nat,A) != zero_zero(nat) | zero_zero(nat) = B | hAPP(nat,nat,power_power(nat,A),B) = zero_zero(nat). [resolve(1130,d,1121,a)]. 2.29/2.51 Derived: -power(real) | -no_zero_divisors(real) | -zero_neq_one(real) | ti(real,A) != zero_zero(real) | zero_zero(nat) = B | hAPP(nat,real,power_power(real,A),B) = zero_zero(real). [resolve(1130,d,1123,a)]. 2.29/2.51 Derived: -power(int) | -no_zero_divisors(int) | -zero_neq_one(int) | ti(int,A) != zero_zero(int) | zero_zero(nat) = B | hAPP(nat,int,power_power(int,A),B) = zero_zero(int). [resolve(1130,d,1127,a)]. 2.29/2.51 1131 -power(A) | -no_zero_divisors(A) | -zero_neq_one(A) | -mult_zero(A) | ti(A,B) = zero_zero(A) | hAPP(nat,A,power_power(A,B),C) != zero_zero(A) # label(fact_185_power__eq__0__iff) # label(axiom). [clausify(946)]. 2.29/2.51 Derived: -power(nat) | -no_zero_divisors(nat) | -zero_neq_one(nat) | ti(nat,A) = zero_zero(nat) | hAPP(nat,nat,power_power(nat,A),B) != zero_zero(nat). [resolve(1131,d,1121,a)]. 2.29/2.51 Derived: -power(real) | -no_zero_divisors(real) | -zero_neq_one(real) | ti(real,A) = zero_zero(real) | hAPP(nat,real,power_power(real,A),B) != zero_zero(real). [resolve(1131,d,1123,a)]. 2.29/2.51 Derived: -power(int) | -no_zero_divisors(int) | -zero_neq_one(int) | ti(int,A) = zero_zero(int) | hAPP(nat,int,power_power(int,A),B) != zero_zero(int). [resolve(1131,d,1127,a)]. 2.37/2.67 1132 -power(A) | -no_zero_divisors(A) | -zero_neq_one(A) | -mult_zero(A) | zero_zero(nat) != B | hAPP(nat,A,power_power(A,C),B) != zero_zero(A) # label(fact_185_power__eq__0__iff) # label(axiom). [clausify(946)]. 2.37/2.67 Derived: -power(nat) | -no_zero_divisors(nat) | -zero_neq_one(nat) | zero_zero(nat) != A | hAPP(nat,nat,power_power(nat,B),A) != zero_zero(nat). [resolve(1132,d,1121,a)]. 2.37/2.67 Derived: -power(real) | -no_zero_divisors(real) | -zero_neq_one(real) | zero_zero(nat) != A | hAPP(nat,real,power_power(real,B),A) != zero_zero(real). [resolve(1132,d,1123,a)]. 2.37/2.67 Derived: -power(int) | -no_zero_divisors(int) | -zero_neq_one(int) | zero_zero(nat) != A | hAPP(nat,int,power_power(int,B),A) != zero_zero(int). [resolve(1132,d,1127,a)]. 2.37/2.67 1133 -mult_zero(A) | times_times(A,B,zero_zero(A)) = zero_zero(A) # label(fact_329_mult__zero__right) # label(axiom). [clausify(949)]. 2.37/2.67 Derived: times_times(nat,A,zero_zero(nat)) = zero_zero(nat). [resolve(1133,a,1121,a)]. 2.37/2.67 Derived: times_times(real,A,zero_zero(real)) = zero_zero(real). [resolve(1133,a,1123,a)]. 2.37/2.67 Derived: times_times(int,A,zero_zero(int)) = zero_zero(int). [resolve(1133,a,1127,a)]. 2.37/2.67 1134 -comm_semiring(A) | times_times(A,plus_plus(A,B,C),D) = plus_plus(A,times_times(A,B,D),times_times(A,C,D)) # label(fact_339_comm__semiring__class_Odistrib) # label(axiom). [clausify(938)]. 2.37/2.67 1135 comm_semiring(real) # label(arity_RealDef_Oreal___Rings_Ocomm__semiring) # label(axiom). [assumption]. 2.37/2.67 1136 comm_semiring(nat) # label(arity_Nat_Onat___Rings_Ocomm__semiring) # label(axiom). [assumption]. 2.37/2.67 1137 comm_semiring(int) # label(arity_Int_Oint___Rings_Ocomm__semiring) # label(axiom). [assumption]. 2.37/2.67 Derived: times_times(real,plus_plus(real,A,B),C) = plus_plus(real,times_times(real,A,C),times_times(real,B,C)). [resolve(1134,a,1135,a)]. 2.37/2.67 Derived: times_times(nat,plus_plus(nat,A,B),C) = plus_plus(nat,times_times(nat,A,C),times_times(nat,B,C)). [resolve(1134,a,1136,a)]. 2.37/2.67 Derived: times_times(int,plus_plus(int,A,B),C) = plus_plus(int,times_times(int,A,C),times_times(int,B,C)). [resolve(1134,a,1137,a)]. 2.37/2.67 1138 linordered_idom(int) # label(arity_Int_Oint___Rings_Olinordered__idom) # label(axiom). [assumption]. 2.37/2.67 1139 -linordered_idom(A) | hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))) = abs_abs(A,hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls))))) # label(fact_964_abs__power2) # label(axiom). [clausify(13)]. 2.37/2.67 Derived: hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))) = abs_abs(int,hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls))))). [resolve(1138,a,1139,a)]. 2.37/2.67 1140 -linordered_idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(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))))))) | ti(A,C) != zero_zero(A) | ti(A,B) != zero_zero(A) # label(fact_97_sum__power2__gt__zero__iff) # label(axiom). [clausify(127)]. 2.37/2.67 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(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))))))) | ti(int,B) != zero_zero(int) | ti(int,A) != zero_zero(int). [resolve(1140,a,1138,a)]. 2.37/2.67 1141 -linordered_idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(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))))))) | ti(A,C) = zero_zero(A) # label(fact_97_sum__power2__gt__zero__iff) # label(axiom). [clausify(127)]. 2.37/2.67 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(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))))))) | ti(int,B) = zero_zero(int). [resolve(1141,a,1138,a)]. 2.37/2.67 1142 -linordered_idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(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))))))) | ti(A,B) = zero_zero(A) # label(fact_97_sum__power2__gt__zero__iff) # label(axiom). [clausify(127)]. 2.37/2.68 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(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))))))) | ti(int,A) = zero_zero(int). [resolve(1142,a,1138,a)]. 2.37/2.68 1143 -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,abs_abs(A,B)),C))) # label(fact_926_zero__le__power__abs) # label(axiom). [clausify(150)]. 2.37/2.68 1144 -linordered_idom(A) | ti(A,B) != zero_zero(A) | ti(A,C) != zero_zero(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(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_553_sum__power2__le__zero__iff) # label(axiom). [clausify(153)]. 2.37/2.68 Derived: ti(int,A) != zero_zero(int) | ti(int,B) != zero_zero(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(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(1144,a,1138,a)]. 2.37/2.68 1145 -linordered_idom(A) | ti(A,B) = zero_zero(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(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_553_sum__power2__le__zero__iff) # label(axiom). [clausify(153)]. 2.37/2.68 Derived: ti(int,A) = zero_zero(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(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(1145,a,1138,a)]. 2.37/2.68 1146 -linordered_idom(A) | ti(A,B) = zero_zero(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(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_553_sum__power2__le__zero__iff) # label(axiom). [clausify(153)]. 2.37/2.68 Derived: ti(int,A) = zero_zero(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(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(1146,a,1138,a)]. 2.37/2.68 1147 -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_152_linorder__neqE__linordered__idom) # label(axiom). [clausify(156)]. 2.37/2.68 1148 -linordered_idom(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) | ti(A,B) = zero_zero(A) # label(fact_2_sum__power2__eq__zero__iff) # label(axiom). [clausify(210)]. 2.37/2.68 Derived: 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) | ti(int,A) = zero_zero(int). [resolve(1148,a,1138,a)]. 2.37/2.68 1149 -linordered_idom(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) | ti(A,C) = zero_zero(A) # label(fact_2_sum__power2__eq__zero__iff) # label(axiom). [clausify(210)]. 2.37/2.68 Derived: 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) | ti(int,B) = zero_zero(int). [resolve(1149,a,1138,a)]. 2.37/2.68 1150 -linordered_idom(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) | ti(A,B) != zero_zero(A) | ti(A,C) != zero_zero(A) # label(fact_2_sum__power2__eq__zero__iff) # label(axiom). [clausify(210)]. 2.47/2.69 Derived: 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) | ti(int,A) != zero_zero(int) | ti(int,B) != zero_zero(int). [resolve(1150,a,1138,a)]. 2.47/2.69 1151 -linordered_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),B),abs_abs(A,C))) # label(fact_921_dvd__abs__iff) # label(axiom). [clausify(235)]. 2.47/2.69 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),abs_abs(int,B))). [resolve(1151,a,1138,a)]. 2.47/2.69 1152 -linordered_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),B),abs_abs(A,C))) # label(fact_921_dvd__abs__iff) # label(axiom). [clausify(235)]. 2.47/2.69 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),abs_abs(int,B))). [resolve(1152,a,1138,a)]. 2.47/2.69 1153 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,B)),zero_zero(A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),pls)) # label(fact_49_less__special_I3_J) # label(axiom). [clausify(236)]. 2.47/2.69 Derived: -number_ring(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,A)),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)). [resolve(1153,a,1138,a)]. 2.47/2.69 1154 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,B)),zero_zero(A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),pls)) # label(fact_49_less__special_I3_J) # label(axiom). [clausify(236)]. 2.47/2.69 Derived: -number_ring(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,A)),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)). [resolve(1154,a,1138,a)]. 2.47/2.69 1155 -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),times_times(A,B,C)),B)) # label(fact_312_mult__right__le__one__le) # label(axiom). [clausify(328)]. 2.47/2.69 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),times_times(int,A,B)),A)). [resolve(1155,a,1138,a)]. 2.47/2.69 1156 -linordered_idom(A) | times_times(A,abs_abs(A,B),abs_abs(A,C)) = abs_abs(A,times_times(A,B,C)) # label(fact_951_abs__mult) # label(axiom). [clausify(333)]. 2.47/2.69 Derived: times_times(int,abs_abs(int,A),abs_abs(int,B)) = abs_abs(int,times_times(int,A,B)). [resolve(1156,a,1138,a)]. 2.47/2.69 1157 -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),times_times(A,C,B)),B)) # label(fact_313_mult__left__le__one__le) # label(axiom). [clausify(406)]. 2.47/2.69 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),times_times(int,B,A)),A)). [resolve(1157,a,1138,a)]. 2.48/2.70 1158 -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)))))) | ti(A,B) != zero_zero(A) # label(fact_95_zero__less__power2) # label(axiom). [clausify(438)]. 2.48/2.70 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)))))) | ti(int,A) != zero_zero(int). [resolve(1158,a,1138,a)]. 2.48/2.70 1159 -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)))))) | ti(A,B) = zero_zero(A) # label(fact_95_zero__less__power2) # label(axiom). [clausify(438)]. 2.48/2.70 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)))))) | ti(int,A) = zero_zero(int). [resolve(1159,a,1138,a)]. 2.48/2.70 1160 -linordered_idom(A) | hAPP(nat,A,power_power(A,abs_abs(A,B)),C) = abs_abs(A,hAPP(nat,A,power_power(A,B),C)) # label(fact_934_power__abs) # label(axiom). [clausify(447)]. 2.48/2.70 Derived: hAPP(nat,int,power_power(int,abs_abs(int,A)),B) = abs_abs(int,hAPP(nat,int,power_power(int,A),B)). [resolve(1160,a,1138,a)]. 2.48/2.70 1161 -number_ring(A) | -linordered_idom(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_257_le__number__of) # label(axiom). [clausify(494)]. 2.48/2.70 1162 -number_ring(A) | -linordered_idom(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_257_le__number__of) # label(axiom). [clausify(494)]. 2.48/2.70 1163 -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_62_less__special_I4_J) # label(axiom). [clausify(496)]. 2.48/2.70 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(1163,a,1138,a)]. 2.48/2.70 1164 -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_62_less__special_I4_J) # label(axiom). [clausify(496)]. 2.48/2.70 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(1164,a,1138,a)]. 2.48/2.70 1165 -linordered_idom(A) | -number_ring(A) | abs_abs(A,hAPP(nat,A,power_power(A,number_number_of(A,min)),B)) = one_one(A) # label(fact_958_abs__power__minus__one) # label(axiom). [clausify(534)]. 2.48/2.70 Derived: -number_ring(int) | abs_abs(int,hAPP(nat,int,power_power(int,number_number_of(int,min)),A)) = one_one(int). [resolve(1165,a,1138,a)]. 2.48/2.70 1166 -linordered_idom(A) | -number_ring(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_18_less__number__of) # label(axiom). [clausify(551)]. 2.48/2.70 1167 -linordered_idom(A) | -number_ring(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_18_less__number__of) # label(axiom). [clausify(551)]. 2.48/2.70 1168 -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),times_times(nat,number_number_of(nat,bit0(bit1(pls))),C)))) # label(fact_700_zero__le__even__power_H) # label(axiom). [clausify(555)]. 2.48/2.70 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,power_power(int,A),times_times(nat,number_number_of(nat,bit0(bit1(pls))),B)))). [resolve(1168,a,1138,a)]. 2.51/2.71 1169 -linordered_idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | times_times(A,abs_abs(A,C),B) = abs_abs(A,times_times(A,C,B)) # label(fact_924_abs__mult__pos) # label(axiom). [clausify(567)]. 2.51/2.71 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | times_times(int,abs_abs(int,B),A) = abs_abs(int,times_times(int,B,A)). [resolve(1169,a,1138,a)]. 2.51/2.71 1170 -linordered_idom(A) | times_times(A,abs_abs(A,B),abs_abs(A,B)) = times_times(A,B,B) # label(fact_952_abs__mult__self) # label(axiom). [clausify(589)]. 2.51/2.71 Derived: times_times(int,abs_abs(int,A),abs_abs(int,A)) = times_times(int,A,A). [resolve(1170,a,1138,a)]. 2.51/2.71 1171 -number_ring(A) | -linordered_idom(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),number_number_of(A,B))) # label(fact_389_le__special_I1_J) # label(axiom). [clausify(613)]. 2.51/2.71 Derived: -number_ring(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,A))). [resolve(1171,b,1138,a)]. 2.51/2.71 1172 -number_ring(A) | -linordered_idom(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),number_number_of(A,B))) # label(fact_389_le__special_I1_J) # label(axiom). [clausify(613)]. 2.51/2.71 Derived: -number_ring(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,A))). [resolve(1172,b,1138,a)]. 2.51/2.71 1173 -linordered_idom(A) | hAPP(nat,A,semiring_1_of_nat(A),B) = abs_abs(A,hAPP(nat,A,semiring_1_of_nat(A),B)) # label(fact_935_abs__of__nat) # label(axiom). [clausify(618)]. 2.51/2.71 Derived: hAPP(nat,int,semiring_1_of_nat(int),A) = abs_abs(int,hAPP(nat,int,semiring_1_of_nat(int),A)). [resolve(1173,a,1138,a)]. 2.51/2.71 1174 -linordered_idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(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_552_sum__power2__ge__zero) # label(axiom). [clausify(630)]. 2.51/2.71 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(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(1174,a,1138,a)]. 2.51/2.71 1175 -number_ring(A) | -linordered_idom(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),number_number_of(A,B))) # label(fact_50_less__special_I1_J) # label(axiom). [clausify(656)]. 2.51/2.71 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(1175,b,1138,a)]. 2.51/2.71 1176 -number_ring(A) | -linordered_idom(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),number_number_of(A,B))) # label(fact_50_less__special_I1_J) # label(axiom). [clausify(656)]. 2.51/2.71 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(1176,b,1138,a)]. 2.51/2.71 1177 -number_ring(A) | -linordered_idom(A) | abs_abs(A,number_number_of(A,min)) = one_one(A) # label(fact_929_abs__minus__one) # label(axiom). [clausify(673)]. 2.51/2.71 Derived: -number_ring(int) | abs_abs(int,number_number_of(int,min)) = one_one(int). [resolve(1177,b,1138,a)]. 2.51/2.71 1178 -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_546_zero__le__power2) # label(axiom). [clausify(724)]. 2.51/2.72 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(1178,a,1138,a)]. 2.51/2.72 1179 -linordered_idom(A) | abs_abs(A,one_one(A)) = one_one(A) # label(fact_933_abs__one) # label(axiom). [clausify(771)]. 2.51/2.72 Derived: abs_abs(int,one_one(int)) = one_one(int). [resolve(1179,a,1138,a)]. 2.51/2.72 1180 -linordered_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),abs_abs(A,B)),C)) # label(fact_920_abs__dvd__iff) # label(axiom). [clausify(790)]. 2.51/2.72 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),abs_abs(int,A)),B)). [resolve(1180,a,1138,a)]. 2.51/2.72 1181 -linordered_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),abs_abs(A,B)),C)) # label(fact_920_abs__dvd__iff) # label(axiom). [clausify(790)]. 2.51/2.72 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),abs_abs(int,A)),B)). [resolve(1181,a,1138,a)]. 2.51/2.72 1182 -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_403_le__special_I2_J) # label(axiom). [clausify(792)]. 2.51/2.72 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(1182,a,1138,a)]. 2.51/2.72 1183 -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_403_le__special_I2_J) # label(axiom). [clausify(792)]. 2.51/2.72 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(1183,a,1138,a)]. 2.51/2.72 1184 linordered_idom(real) # label(arity_RealDef_Oreal___Rings_Olinordered__idom) # label(axiom). [assumption]. 2.51/2.72 Derived: hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))) = abs_abs(real,hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls))))). [resolve(1184,a,1139,a)]. 2.51/2.72 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(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))))))) | ti(real,B) != zero_zero(real) | ti(real,A) != zero_zero(real). [resolve(1184,a,1140,a)]. 2.51/2.72 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(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))))))) | ti(real,B) = zero_zero(real). [resolve(1184,a,1141,a)]. 2.51/2.72 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(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))))))) | ti(real,A) = zero_zero(real). [resolve(1184,a,1142,a)]. 2.51/2.72 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,power_power(real,abs_abs(real,A)),B))). [resolve(1184,a,1143,a)]. 2.51/2.72 Derived: ti(real,A) != zero_zero(real) | ti(real,B) != zero_zero(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(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(1184,a,1144,a)]. 2.51/2.73 Derived: ti(real,A) = zero_zero(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(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(1184,a,1145,a)]. 2.51/2.73 Derived: ti(real,A) = zero_zero(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(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(1184,a,1146,a)]. 2.51/2.73 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(1184,a,1147,a)]. 2.51/2.73 Derived: 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) | ti(real,A) = zero_zero(real). [resolve(1184,a,1148,a)]. 2.51/2.73 Derived: 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) | ti(real,B) = zero_zero(real). [resolve(1184,a,1149,a)]. 2.51/2.73 Derived: 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) | ti(real,A) != zero_zero(real) | ti(real,B) != zero_zero(real). [resolve(1184,a,1150,a)]. 2.51/2.73 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),abs_abs(real,B))). [resolve(1184,a,1151,a)]. 2.51/2.73 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),abs_abs(real,B))). [resolve(1184,a,1152,a)]. 2.51/2.73 Derived: -number_ring(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,A)),zero_zero(real))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)). [resolve(1184,a,1153,a)]. 2.51/2.73 Derived: -number_ring(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,A)),zero_zero(real))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)). [resolve(1184,a,1154,a)]. 2.51/2.73 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),B),one_one(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),times_times(real,A,B)),A)). [resolve(1184,a,1155,a)]. 2.51/2.73 Derived: times_times(real,abs_abs(real,A),abs_abs(real,B)) = abs_abs(real,times_times(real,A,B)). [resolve(1184,a,1156,a)]. 2.51/2.73 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),B),one_one(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),times_times(real,B,A)),A)). [resolve(1184,a,1157,a)]. 2.51/2.73 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)))))) | ti(real,A) != zero_zero(real). [resolve(1184,a,1158,a)]. 2.51/2.73 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)))))) | ti(real,A) = zero_zero(real). [resolve(1184,a,1159,a)]. 2.51/2.73 Derived: hAPP(nat,real,power_power(real,abs_abs(real,A)),B) = abs_abs(real,hAPP(nat,real,power_power(real,A),B)). [resolve(1184,a,1160,a)]. 2.51/2.73 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(1184,a,1161,b)]. 2.51/2.73 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(1184,a,1162,b)]. 2.51/2.73 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(1184,a,1163,a)]. 2.51/2.73 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(1184,a,1164,a)]. 2.51/2.73 Derived: -number_ring(real) | abs_abs(real,hAPP(nat,real,power_power(real,number_number_of(real,min)),A)) = one_one(real). [resolve(1184,a,1165,a)]. 2.51/2.73 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(1184,a,1166,a)]. 2.51/2.73 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(1184,a,1167,a)]. 2.51/2.73 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,power_power(real,A),times_times(nat,number_number_of(nat,bit0(bit1(pls))),B)))). [resolve(1184,a,1168,a)]. 2.51/2.73 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | times_times(real,abs_abs(real,B),A) = abs_abs(real,times_times(real,B,A)). [resolve(1184,a,1169,a)]. 2.51/2.73 Derived: times_times(real,abs_abs(real,A),abs_abs(real,A)) = times_times(real,A,A). [resolve(1184,a,1170,a)]. 2.51/2.73 Derived: -number_ring(real) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),number_number_of(real,A))). [resolve(1184,a,1171,b)]. 2.51/2.73 Derived: -number_ring(real) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),number_number_of(real,A))). [resolve(1184,a,1172,b)]. 2.51/2.73 Derived: hAPP(nat,real,semiring_1_of_nat(real),A) = abs_abs(real,hAPP(nat,real,semiring_1_of_nat(real),A)). [resolve(1184,a,1173,a)]. 2.51/2.73 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(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(1184,a,1174,a)]. 2.51/2.73 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(1184,a,1175,b)]. 2.51/2.73 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(1184,a,1176,b)]. 2.51/2.73 Derived: -number_ring(real) | abs_abs(real,number_number_of(real,min)) = one_one(real). [resolve(1184,a,1177,b)]. 2.51/2.73 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(1184,a,1178,a)]. 2.51/2.73 Derived: abs_abs(real,one_one(real)) = one_one(real). [resolve(1184,a,1179,a)]. 2.51/2.73 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),abs_abs(real,A)),B)). [resolve(1184,a,1180,a)]. 2.51/2.73 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),abs_abs(real,A)),B)). [resolve(1184,a,1181,a)]. 2.51/2.73 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(1184,a,1182,a)]. 2.51/2.75 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(1184,a,1183,a)]. 2.51/2.75 1185 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(pls)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),number_number_of(A,B))) # label(fact_63_less__special_I2_J) # label(axiom). [clausify(806)]. 2.51/2.75 Derived: -number_ring(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(pls)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),number_number_of(int,A))). [resolve(1185,a,1138,a)]. 2.51/2.75 Derived: -number_ring(real) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(pls)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),number_number_of(real,A))). [resolve(1185,a,1184,a)]. 2.51/2.75 1186 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(pls)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),number_number_of(A,B))) # label(fact_63_less__special_I2_J) # label(axiom). [clausify(806)]. 2.51/2.75 Derived: -number_ring(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(pls)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),number_number_of(int,A))). [resolve(1186,a,1138,a)]. 2.51/2.75 Derived: -number_ring(real) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(pls)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),number_number_of(real,A))). [resolve(1186,a,1184,a)]. 2.51/2.75 1187 -number_ring(A) | -linordered_idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,semiring_1_of_nat(A),B)),hAPP(nat,A,power_power(A,number_number_of(A,bit0(bit1(pls)))),B))) # label(fact_103_of__nat__less__two__power) # label(axiom). [clausify(833)]. 2.51/2.75 Derived: -number_ring(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),A)),hAPP(nat,int,power_power(int,number_number_of(int,bit0(bit1(pls)))),A))). [resolve(1187,b,1138,a)]. 2.51/2.75 Derived: -number_ring(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,semiring_1_of_nat(real),A)),hAPP(nat,real,power_power(real,number_number_of(real,bit0(bit1(pls)))),A))). [resolve(1187,b,1184,a)]. 2.51/2.75 1188 -linordered_idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(nat,A,power_power(A,B),times_times(nat,number_number_of(nat,bit0(bit1(pls))),C))),zero_zero(A))) | ti(A,B) = zero_zero(A) # label(fact_699_even__power__le__0__imp__0) # label(axiom). [clausify(854)]. 2.51/2.75 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,power_power(int,A),times_times(nat,number_number_of(nat,bit0(bit1(pls))),B))),zero_zero(int))) | ti(int,A) = zero_zero(int). [resolve(1188,a,1138,a)]. 2.51/2.75 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,power_power(real,A),times_times(nat,number_number_of(nat,bit0(bit1(pls))),B))),zero_zero(real))) | ti(real,A) = zero_zero(real). [resolve(1188,a,1184,a)]. 2.51/2.75 1189 -linordered_idom(A) | abs_abs(A,B) != abs_abs(A,C) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),B)) # label(fact_919_dvd__if__abs__eq) # label(axiom). [clausify(866)]. 2.51/2.75 Derived: abs_abs(int,A) != abs_abs(int,B) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),A)). [resolve(1189,a,1138,a)]. 2.51/2.75 Derived: abs_abs(real,A) != abs_abs(real,B) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),B),A)). [resolve(1189,a,1184,a)]. 2.51/2.75 1190 -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_404_le__special_I4_J) # label(axiom). [clausify(930)]. 2.51/2.75 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(1190,a,1138,a)]. 2.51/2.75 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(1190,a,1184,a)]. 2.51/2.75 1191 -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_404_le__special_I4_J) # label(axiom). [clausify(930)]. 2.51/2.75 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(1191,a,1138,a)]. 2.51/2.75 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(1191,a,1184,a)]. 2.51/2.75 1192 -linordered_idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(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_98_not__sum__power2__lt__zero) # label(axiom). [clausify(956)]. 2.51/2.75 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(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(1192,a,1138,a)]. 2.51/2.75 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(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(1192,a,1184,a)]. 2.51/2.75 1193 -linordered_idom(A) | hAPP(nat,A,power_power(A,abs_abs(A,B)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))) # label(fact_965_power2__abs) # label(axiom). [clausify(967)]. 2.51/2.75 Derived: hAPP(nat,int,power_power(int,abs_abs(int,A)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))). [resolve(1193,a,1138,a)]. 2.51/2.75 Derived: hAPP(nat,real,power_power(real,abs_abs(real,A)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))). [resolve(1193,a,1184,a)]. 2.51/2.75 1194 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,B)),zero_zero(A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),pls)) # label(fact_390_le__special_I3_J) # label(axiom). [clausify(991)]. 2.51/2.75 Derived: -number_ring(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,A)),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),pls)). [resolve(1194,a,1138,a)]. 2.51/2.75 Derived: -number_ring(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),zero_zero(real))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),pls)). [resolve(1194,a,1184,a)]. 2.51/2.75 1195 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,B)),zero_zero(A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),pls)) # label(fact_390_le__special_I3_J) # label(axiom). [clausify(991)]. 2.51/2.75 Derived: -number_ring(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,A)),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),pls)). [resolve(1195,a,1138,a)]. 2.51/2.75 Derived: -number_ring(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),zero_zero(real))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),pls)). [resolve(1195,a,1184,a)]. 2.73/2.98 1196 -linordered_idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(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_44_even__less__0__iff) # label(axiom). [clausify(1000)]. 2.73/2.98 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(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(1196,a,1138,a)]. 2.73/2.98 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(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(1196,a,1184,a)]. 2.73/2.98 1197 -linordered_idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(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_44_even__less__0__iff) # label(axiom). [clausify(1000)]. 2.73/2.98 1198 -linordered_idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),abs_abs(A,B)),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),abs_abs(A,D)),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),times_times(A,abs_abs(A,B),abs_abs(A,D))),times_times(A,C,E))) # label(fact_943_abs__mult__less) # label(axiom). [clausify(1002)]. 2.73/2.98 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),abs_abs(int,A)),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),abs_abs(int,C)),D)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),times_times(int,abs_abs(int,A),abs_abs(int,C))),times_times(int,B,D))). [resolve(1198,a,1138,a)]. 2.73/2.98 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),abs_abs(real,A)),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),abs_abs(real,C)),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),times_times(real,abs_abs(real,A),abs_abs(real,C))),times_times(real,B,D))). [resolve(1198,a,1184,a)]. 2.73/2.98 1199 -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_96_power2__less__0) # label(axiom). [clausify(1006)]. 2.73/2.98 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(1199,a,1138,a)]. 2.73/2.98 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(1199,a,1184,a)]. 2.73/2.98 1200 -comm_monoid_add(A) | ti(A,B) = plus_plus(A,zero_zero(A),B) # label(fact_217_add__0) # label(axiom). [clausify(182)]. 2.73/2.98 1201 comm_monoid_add(nat) # label(arity_Nat_Onat___Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 2.73/2.98 1202 comm_monoid_add(real) # label(arity_RealDef_Oreal___Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 2.73/2.98 Derived: ti(nat,A) = plus_plus(nat,zero_zero(nat),A). [resolve(1200,a,1201,a)]. 2.73/2.98 Derived: ti(real,A) = plus_plus(real,zero_zero(real),A). [resolve(1200,a,1202,a)]. 2.73/2.98 1203 -comm_monoid_add(A) | ti(A,B) = plus_plus(A,B,zero_zero(A)) # label(fact_213_add_Ocomm__neutral) # label(axiom). [clausify(383)]. 2.73/2.98 Derived: ti(nat,A) = plus_plus(nat,A,zero_zero(nat)). [resolve(1203,a,1201,a)]. 2.73/2.98 Derived: ti(real,A) = plus_plus(real,A,zero_zero(real)). [resolve(1203,a,1202,a)]. 2.73/2.98 1204 comm_monoid_add(int) # label(arity_Int_Oint___Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 2.73/2.98 Derived: ti(int,A) = plus_plus(int,A,zero_zero(int)). [resolve(1204,a,1203,a)]. 2.73/2.98 1205 -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)) | plus_plus(A,E,F) != one_one(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,times_times(A,E,B),times_times(A,F,D))),C)) # label(fact_386_convex__bound__le) # label(axiom). [clausify(877)]. 2.90/3.10 1206 linord1278240602ring_1(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring__1) # label(axiom). [assumption]. 2.90/3.10 1207 linord1278240602ring_1(int) # label(arity_Int_Oint___Rings_Olinordered__semiring__1) # label(axiom). [assumption]. 2.90/3.10 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)) | plus_plus(real,D,E) != one_one(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),plus_plus(real,times_times(real,D,A),times_times(real,E,C))),B)). [resolve(1205,a,1206,a)]. 2.90/3.10 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)) | plus_plus(int,D,E) != one_one(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,times_times(int,D,A),times_times(int,E,C))),B)). [resolve(1205,a,1207,a)]. 2.90/3.10 1208 -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),plus_plus(A,B,D)),plus_plus(A,C,E))) # label(fact_355_add__mono) # label(axiom). [clausify(187)]. 2.90/3.10 1209 ordere779506340up_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 2.90/3.10 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),plus_plus(real,A,C)),plus_plus(real,B,D))). [resolve(1208,a,1209,a)]. 2.90/3.10 1210 -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),plus_plus(A,D,B)),plus_plus(A,D,C))) # label(fact_356_add__left__mono) # label(axiom). [clausify(423)]. 2.90/3.10 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),plus_plus(real,C,A)),plus_plus(real,C,B))). [resolve(1210,a,1209,a)]. 2.90/3.10 1211 ordere779506340up_add(nat) # label(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 2.90/3.10 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),plus_plus(nat,A,C)),plus_plus(nat,B,D))). [resolve(1211,a,1208,a)]. 2.90/3.10 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),plus_plus(nat,C,A)),plus_plus(nat,C,B))). [resolve(1211,a,1210,a)]. 2.90/3.10 1212 ordere779506340up_add(int) # label(arity_Int_Oint___Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 2.90/3.10 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),plus_plus(int,A,C)),plus_plus(int,B,D))). [resolve(1212,a,1208,a)]. 2.90/3.10 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),plus_plus(int,C,A)),plus_plus(int,C,B))). [resolve(1212,a,1210,a)]. 2.90/3.10 1213 -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),plus_plus(A,B,D)),plus_plus(A,C,D))) # label(fact_357_add__right__mono) # label(axiom). [clausify(1005)]. 2.98/3.18 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),plus_plus(real,A,C)),plus_plus(real,B,C))). [resolve(1213,a,1209,a)]. 2.98/3.18 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),plus_plus(nat,A,C)),plus_plus(nat,B,C))). [resolve(1213,a,1211,a)]. 2.98/3.18 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),plus_plus(int,A,C)),plus_plus(int,B,C))). [resolve(1213,a,1212,a)]. 2.98/3.18 1214 number_semiring(int) # label(arity_Int_Oint___Int_Onumber__semiring) # label(axiom). [assumption]. 2.98/3.18 1215 -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)) | times_times(A,number_number_of(A,B),number_number_of(A,C)) = number_number_of(A,times_times(int,B,C)) # label(fact_250_semiring__mult__number__of) # label(axiom). [clausify(19)]. 2.98/3.18 1216 -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,plus_plus(int,B,C)) = plus_plus(A,number_number_of(A,B),number_number_of(A,C)) # label(fact_528_semiring__add__number__of) # label(axiom). [clausify(218)]. 2.98/3.18 1217 -number_semiring(A) | number_number_of(A,bit0(bit1(pls))) = plus_plus(A,one_one(A),one_one(A)) # label(fact_9_semiring__one__add__one__is__two) # label(axiom). [clausify(276)]. 2.98/3.18 Derived: number_number_of(int,bit0(bit1(pls))) = plus_plus(int,one_one(int),one_one(int)). [resolve(1217,a,1214,a)]. 2.98/3.18 1218 -number_semiring(A) | number_number_of(A,hAPP(nat,int,semiring_1_of_nat(int),B)) = hAPP(nat,A,semiring_1_of_nat(A),B) # label(fact_94_number__of__int) # label(axiom). [clausify(307)]. 2.98/3.18 Derived: number_number_of(int,hAPP(nat,int,semiring_1_of_nat(int),A)) = hAPP(nat,int,semiring_1_of_nat(int),A). [resolve(1218,a,1214,a)]. 2.98/3.18 1219 -number_semiring(A) | plus_plus(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))))),times_times(A,times_times(A,number_number_of(A,bit0(bit1(pls))),B),C)) = hAPP(nat,A,power_power(A,plus_plus(A,B,C)),number_number_of(nat,bit0(bit1(pls)))) # label(fact_550_power2__sum) # label(axiom). [clausify(419)]. 2.98/3.18 Derived: plus_plus(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))))),times_times(int,times_times(int,number_number_of(int,bit0(bit1(pls))),A),B)) = hAPP(nat,int,power_power(int,plus_plus(int,A,B)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1219,a,1214,a)]. 2.98/3.18 1220 -number_semiring(A) | times_times(A,B,number_number_of(A,bit0(bit1(pls)))) = plus_plus(A,B,B) # label(fact_536_semiring__mult__2__right) # label(axiom). [clausify(428)]. 2.98/3.18 Derived: times_times(int,A,number_number_of(int,bit0(bit1(pls)))) = plus_plus(int,A,A). [resolve(1220,a,1214,a)]. 2.98/3.18 1221 -number_semiring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) | number_number_of(A,succ(B)) = plus_plus(A,one_one(A),number_number_of(A,B)) # label(fact_479_semiring__1__add__number__of) # label(axiom). [clausify(452)]. 2.98/3.18 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | number_number_of(int,succ(A)) = plus_plus(int,one_one(int),number_number_of(int,A)). [resolve(1221,a,1214,a)]. 2.98/3.18 1222 -number_semiring(A) | times_times(A,number_number_of(A,bit0(bit1(pls))),B) = plus_plus(A,B,B) # label(fact_538_semiring__mult__2) # label(axiom). [clausify(467)]. 2.98/3.18 Derived: times_times(int,number_number_of(int,bit0(bit1(pls))),A) = plus_plus(int,A,A). [resolve(1222,a,1214,a)]. 2.98/3.18 1223 -number_semiring(A) | number_number_of(A,pls) = zero_zero(A) # label(fact_81_semiring__numeral__0__eq__0) # label(axiom). [clausify(575)]. 2.98/3.18 Derived: number_number_of(int,pls) = zero_zero(int). [resolve(1223,a,1214,a)]. 2.98/3.19 1224 number_semiring(nat) # label(arity_Nat_Onat___Int_Onumber__semiring) # label(axiom). [assumption]. 2.98/3.19 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)) | times_times(nat,number_number_of(nat,A),number_number_of(nat,B)) = number_number_of(nat,times_times(int,A,B)). [resolve(1224,a,1215,a)]. 2.98/3.19 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,plus_plus(int,A,B)) = plus_plus(nat,number_number_of(nat,A),number_number_of(nat,B)). [resolve(1224,a,1216,a)]. 2.98/3.19 Derived: number_number_of(nat,bit0(bit1(pls))) = plus_plus(nat,one_one(nat),one_one(nat)). [resolve(1224,a,1217,a)]. 2.98/3.19 Derived: number_number_of(nat,hAPP(nat,int,semiring_1_of_nat(int),A)) = hAPP(nat,nat,semiring_1_of_nat(nat),A). [resolve(1224,a,1218,a)]. 2.98/3.19 Derived: plus_plus(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))))),times_times(nat,times_times(nat,number_number_of(nat,bit0(bit1(pls))),A),B)) = hAPP(nat,nat,power_power(nat,plus_plus(nat,A,B)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1224,a,1219,a)]. 2.98/3.19 Derived: times_times(nat,A,number_number_of(nat,bit0(bit1(pls)))) = plus_plus(nat,A,A). [resolve(1224,a,1220,a)]. 2.98/3.19 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | number_number_of(nat,succ(A)) = plus_plus(nat,one_one(nat),number_number_of(nat,A)). [resolve(1224,a,1221,a)]. 2.98/3.19 Derived: number_number_of(nat,pls) = zero_zero(nat). [resolve(1224,a,1223,a)]. 2.98/3.19 1225 -number_semiring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) | number_number_of(A,succ(B)) = plus_plus(A,number_number_of(A,B),one_one(A)) # label(fact_480_semiring__number__of__add__1) # label(axiom). [clausify(723)]. 2.98/3.19 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | number_number_of(int,succ(A)) = plus_plus(int,number_number_of(int,A),one_one(int)). [resolve(1225,a,1214,a)]. 2.98/3.19 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | number_number_of(nat,succ(A)) = plus_plus(nat,number_number_of(nat,A),one_one(nat)). [resolve(1225,a,1224,a)]. 2.98/3.19 1226 number_semiring(real) # label(arity_RealDef_Oreal___Int_Onumber__semiring) # label(axiom). [assumption]. 2.98/3.19 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)) | times_times(real,number_number_of(real,A),number_number_of(real,B)) = number_number_of(real,times_times(int,A,B)). [resolve(1226,a,1215,a)]. 2.98/3.19 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,plus_plus(int,A,B)) = plus_plus(real,number_number_of(real,A),number_number_of(real,B)). [resolve(1226,a,1216,a)]. 2.98/3.19 Derived: number_number_of(real,bit0(bit1(pls))) = plus_plus(real,one_one(real),one_one(real)). [resolve(1226,a,1217,a)]. 2.98/3.19 Derived: number_number_of(real,hAPP(nat,int,semiring_1_of_nat(int),A)) = hAPP(nat,real,semiring_1_of_nat(real),A). [resolve(1226,a,1218,a)]. 2.98/3.19 Derived: plus_plus(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))))),times_times(real,times_times(real,number_number_of(real,bit0(bit1(pls))),A),B)) = hAPP(nat,real,power_power(real,plus_plus(real,A,B)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1226,a,1219,a)]. 2.98/3.19 Derived: times_times(real,A,number_number_of(real,bit0(bit1(pls)))) = plus_plus(real,A,A). [resolve(1226,a,1220,a)]. 2.98/3.19 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | number_number_of(real,succ(A)) = plus_plus(real,one_one(real),number_number_of(real,A)). [resolve(1226,a,1221,a)]. 2.98/3.19 Derived: times_times(real,number_number_of(real,bit0(bit1(pls))),A) = plus_plus(real,A,A). [resolve(1226,a,1222,a)]. 3.12/3.34 Derived: number_number_of(real,pls) = zero_zero(real). [resolve(1226,a,1223,a)]. 3.12/3.34 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | number_number_of(real,succ(A)) = plus_plus(real,number_number_of(real,A),one_one(real)). [resolve(1226,a,1225,a)]. 3.12/3.34 1227 -number_semiring(A) | number_number_of(A,bit1(pls)) = one_one(A) # label(fact_101_semiring__numeral__1__eq__1) # label(axiom). [clausify(968)]. 3.12/3.34 Derived: number_number_of(int,bit1(pls)) = one_one(int). [resolve(1227,a,1214,a)]. 3.12/3.34 Derived: number_number_of(real,bit1(pls)) = one_one(real). [resolve(1227,a,1226,a)]. 3.12/3.34 1228 cancel_semigroup_add(real) # label(arity_RealDef_Oreal___Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 3.12/3.34 1229 -cancel_semigroup_add(A) | plus_plus(A,B,C) != plus_plus(A,B,D) | ti(A,C) = ti(A,D) # label(fact_201_add__left__imp__eq) # label(axiom). [clausify(22)]. 3.12/3.34 1230 -cancel_semigroup_add(A) | plus_plus(A,B,C) != plus_plus(A,D,C) | ti(A,B) = ti(A,D) # label(fact_199_add__right__imp__eq) # label(axiom). [clausify(59)]. 3.12/3.34 1231 -cancel_semigroup_add(A) | plus_plus(A,ti(A,B),C) = plus_plus(A,B,C) # label(tsy_c_Groups_Oplus__class_Oplus_2_arg1) # label(axiom). [clausify(126)]. 3.12/3.34 Derived: plus_plus(real,A,B) != plus_plus(real,A,C) | ti(real,B) = ti(real,C). [resolve(1228,a,1229,a)]. 3.12/3.34 Derived: plus_plus(real,A,B) != plus_plus(real,C,B) | ti(real,A) = ti(real,C). [resolve(1228,a,1230,a)]. 3.12/3.34 Derived: plus_plus(real,ti(real,A),B) = plus_plus(real,A,B). [resolve(1228,a,1231,a)]. 3.12/3.34 1232 cancel_semigroup_add(nat) # label(arity_Nat_Onat___Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 3.12/3.34 Derived: plus_plus(nat,A,B) != plus_plus(nat,A,C) | ti(nat,B) = ti(nat,C). [resolve(1232,a,1229,a)]. 3.12/3.34 Derived: plus_plus(nat,A,B) != plus_plus(nat,C,B) | ti(nat,A) = ti(nat,C). [resolve(1232,a,1230,a)]. 3.12/3.34 Derived: plus_plus(nat,ti(nat,A),B) = plus_plus(nat,A,B). [resolve(1232,a,1231,a)]. 3.12/3.34 1233 cancel_semigroup_add(int) # label(arity_Int_Oint___Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 3.12/3.34 Derived: plus_plus(int,A,B) != plus_plus(int,A,C) | ti(int,B) = ti(int,C). [resolve(1233,a,1229,a)]. 3.12/3.34 Derived: plus_plus(int,A,B) != plus_plus(int,C,B) | ti(int,A) = ti(int,C). [resolve(1233,a,1230,a)]. 3.12/3.34 Derived: plus_plus(int,ti(int,A),B) = plus_plus(int,A,B). [resolve(1233,a,1231,a)]. 3.12/3.34 1234 -cancel_semigroup_add(A) | ti(A,B) != ti(A,C) | plus_plus(A,B,D) = plus_plus(A,C,D) # label(fact_203_add__right__cancel) # label(axiom). [clausify(698)]. 3.12/3.34 Derived: ti(real,A) != ti(real,B) | plus_plus(real,A,C) = plus_plus(real,B,C). [resolve(1234,a,1228,a)]. 3.12/3.34 Derived: ti(nat,A) != ti(nat,B) | plus_plus(nat,A,C) = plus_plus(nat,B,C). [resolve(1234,a,1232,a)]. 3.12/3.34 Derived: ti(int,A) != ti(int,B) | plus_plus(int,A,C) = plus_plus(int,B,C). [resolve(1234,a,1233,a)]. 3.12/3.34 1235 -cancel_semigroup_add(A) | ti(A,B) = ti(A,C) | plus_plus(A,B,D) != plus_plus(A,C,D) # label(fact_203_add__right__cancel) # label(axiom). [clausify(698)]. 3.12/3.34 1236 -cancel_semigroup_add(A) | ti(A,B) != ti(A,C) | plus_plus(A,D,B) = plus_plus(A,D,C) # label(fact_204_add__left__cancel) # label(axiom). [clausify(918)]. 3.12/3.34 Derived: ti(real,A) != ti(real,B) | plus_plus(real,C,A) = plus_plus(real,C,B). [resolve(1236,a,1228,a)]. 3.12/3.34 Derived: ti(nat,A) != ti(nat,B) | plus_plus(nat,C,A) = plus_plus(nat,C,B). [resolve(1236,a,1232,a)]. 3.12/3.34 Derived: ti(int,A) != ti(int,B) | plus_plus(int,C,A) = plus_plus(int,C,B). [resolve(1236,a,1233,a)]. 3.12/3.34 1237 -cancel_semigroup_add(A) | ti(A,B) = ti(A,C) | plus_plus(A,D,B) != plus_plus(A,D,C) # label(fact_204_add__left__cancel) # label(axiom). [clausify(918)]. 3.12/3.34 1238 -cancel_semigroup_add(A) | ti(A,plus_plus(A,B,C)) = plus_plus(A,B,C) # label(tsy_c_Groups_Oplus__class_Oplus_2_res) # label(axiom). [clausify(939)]. 3.12/3.34 Derived: ti(real,plus_plus(real,A,B)) = plus_plus(real,A,B). [resolve(1238,a,1228,a)]. 3.12/3.34 Derived: ti(nat,plus_plus(nat,A,B)) = plus_plus(nat,A,B). [resolve(1238,a,1232,a)]. 3.12/3.34 Derived: ti(int,plus_plus(int,A,B)) = plus_plus(int,A,B). [resolve(1238,a,1233,a)]. 3.12/3.34 1239 -cancel_semigroup_add(A) | plus_plus(A,B,ti(A,C)) = plus_plus(A,B,C) # label(tsy_c_Groups_Oplus__class_Oplus_2_arg2) # label(axiom). [clausify(972)]. 3.34/3.55 Derived: plus_plus(real,A,ti(real,B)) = plus_plus(real,A,B). [resolve(1239,a,1228,a)]. 3.34/3.55 Derived: plus_plus(nat,A,ti(nat,B)) = plus_plus(nat,A,B). [resolve(1239,a,1232,a)]. 3.34/3.55 Derived: plus_plus(int,A,ti(int,B)) = plus_plus(int,A,B). [resolve(1239,a,1233,a)]. 3.34/3.55 1240 group_add(int) # label(arity_Int_Oint___Groups_Ogroup__add) # label(axiom). [assumption]. 3.34/3.55 1241 -group_add(A) | minus_minus(A,B,B) = zero_zero(A) # label(fact_322_diff__self) # label(axiom). [clausify(24)]. 3.34/3.55 1242 -group_add(A) | minus_minus(A,plus_plus(A,B,C),C) = ti(A,B) # label(fact_326_add__diff__cancel) # label(axiom). [clausify(40)]. 3.34/3.55 1243 -group_add(A) | minus_minus(A,ti(A,B),C) = minus_minus(A,B,C) # label(tsy_c_Groups_Ominus__class_Ominus_0_arg1) # label(axiom). [clausify(64)]. 3.34/3.55 1244 -group_add(A) | ti(A,B) != ti(A,C) | minus_minus(A,C,B) = zero_zero(A) # label(fact_320_right__minus__eq) # label(axiom). [clausify(67)]. 3.34/3.55 1245 -group_add(A) | ti(A,B) = ti(A,C) | minus_minus(A,C,B) != zero_zero(A) # label(fact_320_right__minus__eq) # label(axiom). [clausify(67)]. 3.34/3.55 Derived: minus_minus(int,A,A) = zero_zero(int). [resolve(1240,a,1241,a)]. 3.34/3.55 Derived: minus_minus(int,plus_plus(int,A,B),B) = ti(int,A). [resolve(1240,a,1242,a)]. 3.34/3.55 Derived: minus_minus(int,ti(int,A),B) = minus_minus(int,A,B). [resolve(1240,a,1243,a)]. 3.34/3.55 Derived: ti(int,A) != ti(int,B) | minus_minus(int,B,A) = zero_zero(int). [resolve(1240,a,1244,a)]. 3.34/3.55 Derived: ti(int,A) = ti(int,B) | minus_minus(int,B,A) != zero_zero(int). [resolve(1240,a,1245,a)]. 3.34/3.55 1246 -group_add(A) | minus_minus(A,B,zero_zero(A)) = ti(A,B) # label(fact_323_diff__0__right) # label(axiom). [clausify(531)]. 3.34/3.55 Derived: minus_minus(int,A,zero_zero(int)) = ti(int,A). [resolve(1246,a,1240,a)]. 3.34/3.55 1247 -group_add(A) | minus_minus(A,B,ti(A,C)) = minus_minus(A,B,C) # label(tsy_c_Groups_Ominus__class_Ominus_0_arg2) # label(axiom). [clausify(596)]. 3.34/3.55 Derived: minus_minus(int,A,ti(int,B)) = minus_minus(int,A,B). [resolve(1247,a,1240,a)]. 3.34/3.55 1248 -group_add(A) | minus_minus(A,B,C) = ti(A,minus_minus(A,B,C)) # label(tsy_c_Groups_Ominus__class_Ominus_0_res) # label(axiom). [clausify(707)]. 3.34/3.55 Derived: minus_minus(int,A,B) = ti(int,minus_minus(int,A,B)). [resolve(1248,a,1240,a)]. 3.34/3.55 1249 group_add(real) # label(arity_RealDef_Oreal___Groups_Ogroup__add) # label(axiom). [assumption]. 3.34/3.55 Derived: minus_minus(real,A,A) = zero_zero(real). [resolve(1249,a,1241,a)]. 3.34/3.55 Derived: minus_minus(real,plus_plus(real,A,B),B) = ti(real,A). [resolve(1249,a,1242,a)]. 3.34/3.55 Derived: minus_minus(real,ti(real,A),B) = minus_minus(real,A,B). [resolve(1249,a,1243,a)]. 3.34/3.55 Derived: ti(real,A) != ti(real,B) | minus_minus(real,B,A) = zero_zero(real). [resolve(1249,a,1244,a)]. 3.34/3.55 Derived: ti(real,A) = ti(real,B) | minus_minus(real,B,A) != zero_zero(real). [resolve(1249,a,1245,a)]. 3.34/3.55 Derived: minus_minus(real,A,zero_zero(real)) = ti(real,A). [resolve(1249,a,1246,a)]. 3.34/3.55 Derived: minus_minus(real,A,ti(real,B)) = minus_minus(real,A,B). [resolve(1249,a,1247,a)]. 3.34/3.55 Derived: minus_minus(real,A,B) = ti(real,minus_minus(real,A,B)). [resolve(1249,a,1248,a)]. 3.34/3.55 1250 -group_add(A) | ti(A,B) = plus_plus(A,minus_minus(A,B,C),C) # label(fact_327_diff__add__cancel) # label(axiom). [clausify(1009)]. 3.34/3.55 Derived: ti(int,A) = plus_plus(int,minus_minus(int,A,B),B). [resolve(1250,a,1240,a)]. 3.34/3.55 Derived: ti(real,A) = plus_plus(real,minus_minus(real,A,B),B). [resolve(1250,a,1249,a)]. 3.34/3.55 1251 comm_semiring_1(int) # label(arity_Int_Oint___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 3.34/3.55 1252 -comm_semiring_1(A) | plus_plus(A,plus_plus(A,B,C),D) = plus_plus(A,B,plus_plus(A,C,D)) # label(fact_207_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom). [clausify(27)]. 3.34/3.55 1253 -comm_semiring_1(A) | times_times(A,times_times(A,B,C),D) = times_times(A,B,times_times(A,C,D)) # label(fact_243_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom). [clausify(38)]. 3.34/3.55 1254 -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),times_times(A,C,D))) # label(fact_733_dvd__mult2) # label(axiom). [clausify(51)]. 3.34/3.55 1255 -comm_semiring_1(A) | plus_plus(A,plus_plus(A,B,C),D) = plus_plus(A,B,plus_plus(A,C,D)) # label(fact_208_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom). [clausify(53)]. 3.34/3.55 1256 -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),times_times(A,B,D)),times_times(A,C,E))) # label(fact_731_mult__dvd__mono) # label(axiom). [clausify(62)]. 3.34/3.55 1257 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),times_times(A,B,C)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),D)) # label(fact_728_dvd__mult__right) # label(axiom). [clausify(75)]. 3.34/3.55 1258 -comm_semiring_1(A) | times_times(A,one_one(A),B) = ti(A,B) # label(fact_351_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom). [clausify(93)]. 3.34/3.55 1259 -comm_semiring_1(A) | times_times(A,plus_plus(A,one_one(A),one_one(A)),B) = plus_plus(A,B,B) # label(fact_445_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom). [clausify(172)]. 3.34/3.55 1260 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),times_times(A,B,C)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) # label(fact_729_dvd__mult__left) # label(axiom). [clausify(173)]. 3.34/3.55 1261 -comm_semiring_1(A) | times_times(A,plus_plus(A,B,one_one(A)),C) = plus_plus(A,C,times_times(A,B,C)) # label(fact_446_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom). [clausify(214)]. 3.34/3.55 1262 -comm_semiring_1(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),times_times(nat,number_number_of(nat,bit0(bit1(pls))),C)) # label(fact_481_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom). [clausify(240)]. 3.34/3.55 1263 -comm_semiring_1(A) | times_times(A,zero_zero(A),B) = zero_zero(A) # label(fact_334_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom). [clausify(268)]. 3.34/3.55 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),B),times_times(A,D,C))) # label(fact_732_dvd__mult) # label(axiom). [clausify(309)]. 3.34/3.55 1265 -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_789_dvd__power) # label(axiom). [clausify(345)]. 3.34/3.55 1266 -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_789_dvd__power) # label(axiom). [clausify(345)]. 3.34/3.55 1267 -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),times_times(nat,C,D)) # label(fact_599_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom). [clausify(354)]. 3.34/3.55 1268 -comm_semiring_1(A) | times_times(A,B,one_one(A)) = ti(A,B) # label(fact_348_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom). [clausify(361)]. 3.34/3.55 1269 -comm_semiring_1(A) | ti(A,B) = hAPP(nat,A,power_power(A,B),one_one(nat)) # label(fact_227_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom). [clausify(367)]. 3.34/3.55 1270 -comm_semiring_1(A) | plus_plus(A,plus_plus(A,B,C),plus_plus(A,D,E)) = plus_plus(A,plus_plus(A,B,D),plus_plus(A,C,E)) # label(fact_202_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom). [clausify(372)]. 3.34/3.55 1271 -comm_semiring_1(A) | times_times(A,times_times(A,B,C),D) = times_times(A,times_times(A,B,D),C) # label(fact_244_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom). [clausify(458)]. 3.34/3.55 1272 -comm_semiring_1(A) | plus_plus(A,B,C) = plus_plus(A,C,B) # label(fact_210_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom). [clausify(469)]. 3.34/3.55 1273 -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_742_le__imp__power__dvd) # label(axiom). [clausify(481)]. 3.34/3.55 1274 -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_743_dvd__power__le) # label(axiom). [clausify(489)]. 3.34/3.55 1275 -comm_semiring_1(A) | ti(A,B) = plus_plus(A,zero_zero(A),B) # label(fact_218_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom). [clausify(499)]. 3.34/3.55 1276 -comm_semiring_1(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),zero_zero(A))) # label(fact_723_dvd__0__right) # label(axiom). [clausify(501)]. 3.34/3.55 1277 -comm_semiring_1(A) | times_times(A,B,B) = hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))) # label(fact_541_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom). [clausify(509)]. 3.34/3.55 1278 -comm_semiring_1(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),times_times(A,B,C))) # label(fact_735_dvd__triv__left) # label(axiom). [clausify(537)]. 3.34/3.55 1279 -comm_semiring_1(A) | times_times(A,times_times(A,B,C),times_times(A,D,E)) = times_times(A,B,times_times(A,C,times_times(A,D,E))) # label(fact_245_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom). [clausify(606)]. 3.34/3.55 1280 -comm_semiring_1(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),one_one(A)),B)) # label(fact_773_one__dvd) # label(axiom). [clausify(629)]. 3.34/3.55 1281 -comm_semiring_1(A) | plus_plus(A,B,plus_plus(A,C,D)) = plus_plus(A,C,plus_plus(A,B,D)) # label(fact_209_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom). [clausify(638)]. 3.34/3.55 Derived: times_times(int,times_times(int,A,B),C) = times_times(int,A,times_times(int,B,C)). [resolve(1251,a,1253,a)]. 3.34/3.55 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),times_times(int,B,C))). [resolve(1251,a,1254,a)]. 3.34/3.55 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),times_times(int,A,C)),times_times(int,B,D))). [resolve(1251,a,1256,a)]. 3.34/3.55 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),times_times(int,A,B)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),C)). [resolve(1251,a,1257,a)]. 3.34/3.55 Derived: times_times(int,one_one(int),A) = ti(int,A). [resolve(1251,a,1258,a)]. 3.34/3.55 Derived: times_times(int,plus_plus(int,one_one(int),one_one(int)),A) = plus_plus(int,A,A). [resolve(1251,a,1259,a)]. 3.34/3.55 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),times_times(int,A,B)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),C)). [resolve(1251,a,1260,a)]. 3.34/3.55 Derived: times_times(int,plus_plus(int,A,one_one(int)),B) = plus_plus(int,B,times_times(int,A,B)). [resolve(1251,a,1261,a)]. 3.34/3.55 Derived: 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),times_times(nat,number_number_of(nat,bit0(bit1(pls))),B)). [resolve(1251,a,1262,a)]. 3.34/3.55 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),times_times(int,C,B))). [resolve(1251,a,1264,a)]. 3.34/3.55 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(1251,a,1265,a)]. 3.34/3.55 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(1251,a,1266,a)]. 3.34/3.55 Derived: hAPP(nat,int,power_power(int,hAPP(nat,int,power_power(int,A),B)),C) = hAPP(nat,int,power_power(int,A),times_times(nat,B,C)). [resolve(1251,a,1267,a)]. 3.37/3.56 Derived: times_times(int,A,one_one(int)) = ti(int,A). [resolve(1251,a,1268,a)]. 3.37/3.56 Derived: ti(int,A) = hAPP(nat,int,power_power(int,A),one_one(nat)). [resolve(1251,a,1269,a)]. 3.37/3.56 Derived: plus_plus(int,plus_plus(int,A,B),plus_plus(int,C,D)) = plus_plus(int,plus_plus(int,A,C),plus_plus(int,B,D)). [resolve(1251,a,1270,a)]. 3.37/3.56 Derived: times_times(int,times_times(int,A,B),C) = times_times(int,times_times(int,A,C),B). [resolve(1251,a,1271,a)]. 3.37/3.56 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(1251,a,1273,a)]. 3.37/3.56 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(1251,a,1274,a)]. 3.37/3.56 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),zero_zero(int))). [resolve(1251,a,1276,a)]. 3.37/3.56 Derived: times_times(int,A,A) = hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))). [resolve(1251,a,1277,a)]. 3.37/3.56 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),times_times(int,A,B))). [resolve(1251,a,1278,a)]. 3.37/3.56 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),one_one(int)),A)). [resolve(1251,a,1280,a)]. 3.37/3.56 Derived: plus_plus(int,A,plus_plus(int,B,C)) = plus_plus(int,B,plus_plus(int,A,C)). [resolve(1251,a,1281,a)]. 3.37/3.56 1282 -comm_semiring_1(A) | times_times(A,B,plus_plus(A,C,D)) = plus_plus(A,times_times(A,B,C),times_times(A,B,D)) # label(fact_345_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom). [clausify(676)]. 3.37/3.56 Derived: times_times(int,A,plus_plus(int,B,C)) = plus_plus(int,times_times(int,A,B),times_times(int,A,C)). [resolve(1282,a,1251,a)]. 3.37/3.56 1283 -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),plus_plus(A,C,D))) # label(fact_774_dvd__add) # label(axiom). [clausify(680)]. 3.37/3.56 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),plus_plus(int,B,C))). [resolve(1283,a,1251,a)]. 3.37/3.56 1284 -comm_semiring_1(A) | times_times(A,times_times(A,B,C),times_times(A,D,E)) = times_times(A,times_times(A,B,D),times_times(A,C,E)) # label(fact_247_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom). [clausify(685)]. 3.37/3.56 Derived: times_times(int,times_times(int,A,B),times_times(int,C,D)) = times_times(int,times_times(int,A,C),times_times(int,B,D)). [resolve(1284,a,1251,a)]. 3.37/3.56 1285 -comm_semiring_1(A) | ti(A,B) = plus_plus(A,B,zero_zero(A)) # label(fact_214_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom). [clausify(720)]. 3.37/3.56 1286 comm_semiring_1(real) # label(arity_RealDef_Oreal___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 3.37/3.56 Derived: plus_plus(real,plus_plus(real,A,B),C) = plus_plus(real,A,plus_plus(real,B,C)). [resolve(1286,a,1252,a)]. 3.37/3.56 Derived: times_times(real,times_times(real,A,B),C) = times_times(real,A,times_times(real,B,C)). [resolve(1286,a,1253,a)]. 3.37/3.56 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),times_times(real,B,C))). [resolve(1286,a,1254,a)]. 3.37/3.56 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),times_times(real,A,C)),times_times(real,B,D))). [resolve(1286,a,1256,a)]. 3.37/3.56 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),times_times(real,A,B)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),B),C)). [resolve(1286,a,1257,a)]. 3.37/3.57 Derived: times_times(real,one_one(real),A) = ti(real,A). [resolve(1286,a,1258,a)]. 3.37/3.57 Derived: times_times(real,plus_plus(real,one_one(real),one_one(real)),A) = plus_plus(real,A,A). [resolve(1286,a,1259,a)]. 3.37/3.57 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),times_times(real,A,B)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),C)). [resolve(1286,a,1260,a)]. 3.37/3.57 Derived: times_times(real,plus_plus(real,A,one_one(real)),B) = plus_plus(real,B,times_times(real,A,B)). [resolve(1286,a,1261,a)]. 3.37/3.57 Derived: 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),times_times(nat,number_number_of(nat,bit0(bit1(pls))),B)). [resolve(1286,a,1262,a)]. 3.37/3.57 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),times_times(real,C,B))). [resolve(1286,a,1264,a)]. 3.37/3.57 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(1286,a,1265,a)]. 3.37/3.57 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(1286,a,1266,a)]. 3.37/3.57 Derived: hAPP(nat,real,power_power(real,hAPP(nat,real,power_power(real,A),B)),C) = hAPP(nat,real,power_power(real,A),times_times(nat,B,C)). [resolve(1286,a,1267,a)]. 3.37/3.57 Derived: times_times(real,A,one_one(real)) = ti(real,A). [resolve(1286,a,1268,a)]. 3.37/3.57 Derived: ti(real,A) = hAPP(nat,real,power_power(real,A),one_one(nat)). [resolve(1286,a,1269,a)]. 3.37/3.57 Derived: plus_plus(real,plus_plus(real,A,B),plus_plus(real,C,D)) = plus_plus(real,plus_plus(real,A,C),plus_plus(real,B,D)). [resolve(1286,a,1270,a)]. 3.37/3.57 Derived: times_times(real,times_times(real,A,B),C) = times_times(real,times_times(real,A,C),B). [resolve(1286,a,1271,a)]. 3.37/3.57 Derived: plus_plus(real,A,B) = plus_plus(real,B,A). [resolve(1286,a,1272,a)]. 3.37/3.57 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(1286,a,1273,a)]. 3.37/3.57 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(1286,a,1274,a)]. 3.37/3.57 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),zero_zero(real))). [resolve(1286,a,1276,a)]. 3.37/3.57 Derived: times_times(real,A,A) = hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))). [resolve(1286,a,1277,a)]. 3.37/3.57 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),times_times(real,A,B))). [resolve(1286,a,1278,a)]. 3.37/3.57 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),one_one(real)),A)). [resolve(1286,a,1280,a)]. 3.37/3.57 Derived: plus_plus(real,A,plus_plus(real,B,C)) = plus_plus(real,B,plus_plus(real,A,C)). [resolve(1286,a,1281,a)]. 3.37/3.57 Derived: times_times(real,A,plus_plus(real,B,C)) = plus_plus(real,times_times(real,A,B),times_times(real,A,C)). [resolve(1286,a,1282,a)]. 3.37/3.57 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),plus_plus(real,B,C))). [resolve(1286,a,1283,a)]. 3.37/3.57 Derived: times_times(real,times_times(real,A,B),times_times(real,C,D)) = times_times(real,times_times(real,A,C),times_times(real,B,D)). [resolve(1286,a,1284,a)]. 3.37/3.57 1287 -comm_semiring_1(A) | times_times(A,B,C) = times_times(A,C,B) # label(fact_239_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom). [clausify(759)]. 3.37/3.57 Derived: times_times(int,A,B) = times_times(int,B,A). [resolve(1287,a,1251,a)]. 3.37/3.57 1288 -comm_semiring_1(A) | times_times(A,plus_plus(A,B,one_one(A)),C) = plus_plus(A,times_times(A,B,C),C) # label(fact_447_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom). [clausify(763)]. 3.37/3.58 Derived: times_times(int,plus_plus(int,A,one_one(int)),B) = plus_plus(int,times_times(int,A,B),B). [resolve(1288,a,1251,a)]. 3.37/3.58 Derived: times_times(real,plus_plus(real,A,one_one(real)),B) = plus_plus(real,times_times(real,A,B),B). [resolve(1288,a,1286,a)]. 3.37/3.58 1289 -comm_semiring_1(A) | times_times(A,times_times(A,B,C),D) = times_times(A,B,times_times(A,C,D)) # label(fact_241_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom). [clausify(774)]. 3.37/3.58 1290 -comm_semiring_1(A) | times_times(A,B,zero_zero(A)) = zero_zero(A) # label(fact_333_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom). [clausify(777)]. 3.37/3.58 1291 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),zero_zero(A)),B)) | ti(A,B) = zero_zero(A) # label(fact_775_dvd__0__left) # label(axiom). [clausify(808)]. 3.37/3.58 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),zero_zero(int)),A)) | ti(int,A) = zero_zero(int). [resolve(1291,a,1251,a)]. 3.37/3.58 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),zero_zero(real)),A)) | ti(real,A) = zero_zero(real). [resolve(1291,a,1286,a)]. 3.37/3.58 1292 -comm_semiring_1(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),times_times(A,C,B))) # label(fact_734_dvd__triv__right) # label(axiom). [clausify(810)]. 3.37/3.58 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),times_times(int,B,A))). [resolve(1292,a,1251,a)]. 3.37/3.58 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),times_times(real,B,A))). [resolve(1292,a,1286,a)]. 3.37/3.58 1293 comm_semiring_1(nat) # label(arity_Nat_Onat___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 3.37/3.58 Derived: plus_plus(nat,plus_plus(nat,A,B),C) = plus_plus(nat,A,plus_plus(nat,B,C)). [resolve(1293,a,1252,a)]. 3.37/3.58 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),times_times(nat,B,C))). [resolve(1293,a,1254,a)]. 3.37/3.58 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),times_times(nat,A,C)),times_times(nat,B,D))). [resolve(1293,a,1256,a)]. 3.37/3.58 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),times_times(nat,A,B)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),C)). [resolve(1293,a,1257,a)]. 3.37/3.58 Derived: times_times(nat,one_one(nat),A) = ti(nat,A). [resolve(1293,a,1258,a)]. 3.37/3.58 Derived: times_times(nat,plus_plus(nat,one_one(nat),one_one(nat)),A) = plus_plus(nat,A,A). [resolve(1293,a,1259,a)]. 3.37/3.58 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),times_times(nat,A,B)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),C)). [resolve(1293,a,1260,a)]. 3.37/3.58 Derived: times_times(nat,plus_plus(nat,A,one_one(nat)),B) = plus_plus(nat,B,times_times(nat,A,B)). [resolve(1293,a,1261,a)]. 3.37/3.58 Derived: 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),times_times(nat,number_number_of(nat,bit0(bit1(pls))),B)). [resolve(1293,a,1262,a)]. 3.37/3.58 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),times_times(nat,C,B))). [resolve(1293,a,1264,a)]. 3.37/3.58 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(1293,a,1265,a)]. 3.37/3.58 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(1293,a,1266,a)]. 3.37/3.58 Derived: hAPP(nat,nat,power_power(nat,hAPP(nat,nat,power_power(nat,A),B)),C) = hAPP(nat,nat,power_power(nat,A),times_times(nat,B,C)). [resolve(1293,a,1267,a)]. 3.37/3.58 Derived: times_times(nat,A,one_one(nat)) = ti(nat,A). [resolve(1293,a,1268,a)]. 3.37/3.58 Derived: ti(nat,A) = hAPP(nat,nat,power_power(nat,A),one_one(nat)). [resolve(1293,a,1269,a)]. 3.37/3.58 Derived: plus_plus(nat,plus_plus(nat,A,B),plus_plus(nat,C,D)) = plus_plus(nat,plus_plus(nat,A,C),plus_plus(nat,B,D)). [resolve(1293,a,1270,a)]. 3.37/3.58 Derived: times_times(nat,times_times(nat,A,B),C) = times_times(nat,times_times(nat,A,C),B). [resolve(1293,a,1271,a)]. 3.37/3.58 Derived: plus_plus(nat,A,B) = plus_plus(nat,B,A). [resolve(1293,a,1272,a)]. 3.37/3.58 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(1293,a,1273,a)]. 3.37/3.58 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(1293,a,1274,a)]. 3.37/3.58 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),zero_zero(nat))). [resolve(1293,a,1276,a)]. 3.37/3.58 Derived: times_times(nat,A,A) = hAPP(nat,nat,power_power(nat,A),number_number_of(nat,bit0(bit1(pls)))). [resolve(1293,a,1277,a)]. 3.37/3.58 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),times_times(nat,A,B))). [resolve(1293,a,1278,a)]. 3.37/3.58 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),one_one(nat)),A)). [resolve(1293,a,1280,a)]. 3.37/3.58 Derived: plus_plus(nat,A,plus_plus(nat,B,C)) = plus_plus(nat,B,plus_plus(nat,A,C)). [resolve(1293,a,1281,a)]. 3.37/3.58 Derived: times_times(nat,A,plus_plus(nat,B,C)) = plus_plus(nat,times_times(nat,A,B),times_times(nat,A,C)). [resolve(1293,a,1282,a)]. 3.37/3.58 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),plus_plus(nat,B,C))). [resolve(1293,a,1283,a)]. 3.37/3.58 Derived: times_times(nat,times_times(nat,A,B),times_times(nat,C,D)) = times_times(nat,times_times(nat,A,C),times_times(nat,B,D)). [resolve(1293,a,1284,a)]. 3.37/3.58 Derived: times_times(nat,A,B) = times_times(nat,B,A). [resolve(1293,a,1287,a)]. 3.37/3.58 Derived: times_times(nat,plus_plus(nat,A,one_one(nat)),B) = plus_plus(nat,times_times(nat,A,B),B). [resolve(1293,a,1288,a)]. 3.37/3.58 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A)) | ti(nat,A) = zero_zero(nat). [resolve(1293,a,1291,a)]. 3.37/3.58 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),times_times(nat,B,A))). [resolve(1293,a,1292,a)]. 3.37/3.58 1294 -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_744_power__le__dvd) # label(axiom). [clausify(837)]. 3.37/3.58 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(1294,a,1251,a)]. 3.37/3.58 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(1294,a,1286,a)]. 3.37/3.58 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(1294,a,1293,a)]. 3.37/3.58 1295 -comm_semiring_1(A) | plus_plus(A,plus_plus(A,B,C),D) = plus_plus(A,plus_plus(A,B,D),C) # label(fact_205_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom). [clausify(841)]. 3.37/3.58 Derived: plus_plus(int,plus_plus(int,A,B),C) = plus_plus(int,plus_plus(int,A,C),B). [resolve(1295,a,1251,a)]. 3.37/3.60 Derived: plus_plus(real,plus_plus(real,A,B),C) = plus_plus(real,plus_plus(real,A,C),B). [resolve(1295,a,1286,a)]. 3.37/3.60 Derived: plus_plus(nat,plus_plus(nat,A,B),C) = plus_plus(nat,plus_plus(nat,A,C),B). [resolve(1295,a,1293,a)]. 3.37/3.60 1296 -comm_semiring_1(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),B)) # label(fact_763_dvd__refl) # label(axiom). [clausify(863)]. 3.37/3.60 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),A)). [resolve(1296,a,1251,a)]. 3.37/3.60 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),A)). [resolve(1296,a,1286,a)]. 3.37/3.60 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),A)). [resolve(1296,a,1293,a)]. 3.37/3.60 1297 -comm_semiring_1(A) | times_times(A,plus_plus(A,B,C),D) = plus_plus(A,times_times(A,B,D),times_times(A,C,D)) # label(fact_342_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom). [clausify(885)]. 3.37/3.60 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),hAPP(nat,A,power_power(A,B),D)),hAPP(nat,A,power_power(A,C),D))) # label(fact_772_dvd__power__same) # label(axiom). [clausify(890)]. 3.37/3.60 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(1298,a,1251,a)]. 3.37/3.60 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(1298,a,1286,a)]. 3.37/3.60 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(1298,a,1293,a)]. 3.37/3.60 1299 -comm_semiring_1(A) | times_times(A,plus_plus(A,B,C),D) = plus_plus(A,times_times(A,B,D),times_times(A,C,D)) # label(fact_343_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom). [clausify(899)]. 3.37/3.60 1300 -comm_semiring_1(A) | times_times(A,B,times_times(A,C,D)) = times_times(A,C,times_times(A,B,D)) # label(fact_240_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom). [clausify(914)]. 3.37/3.60 Derived: times_times(int,A,times_times(int,B,C)) = times_times(int,B,times_times(int,A,C)). [resolve(1300,a,1251,a)]. 3.37/3.60 Derived: times_times(real,A,times_times(real,B,C)) = times_times(real,B,times_times(real,A,C)). [resolve(1300,a,1286,a)]. 3.37/3.60 Derived: times_times(nat,A,times_times(nat,B,C)) = times_times(nat,B,times_times(nat,A,C)). [resolve(1300,a,1293,a)]. 3.37/3.60 1301 -comm_semiring_1(A) | times_times(A,times_times(A,B,C),times_times(A,D,E)) = times_times(A,D,times_times(A,times_times(A,B,C),E)) # label(fact_246_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom). [clausify(928)]. 3.37/3.60 1302 -comm_semiring_1(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),plus_plus(nat,C,D)) # label(fact_460_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom). [clausify(947)]. 3.37/3.60 Derived: 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),plus_plus(nat,B,C)). [resolve(1302,a,1251,a)]. 3.37/3.60 Derived: 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),plus_plus(nat,B,C)). [resolve(1302,a,1286,a)]. 3.37/3.60 Derived: 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),plus_plus(nat,B,C)). [resolve(1302,a,1293,a)]. 3.37/3.60 1303 -comm_semiring_1(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,times_times(A,B,D)),C) # label(fact_364_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom). [clausify(976)]. 3.37/3.60 Derived: 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,times_times(int,A,C)),B). [resolve(1303,a,1251,a)]. 3.58/3.80 Derived: 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,times_times(real,A,C)),B). [resolve(1303,a,1286,a)]. 3.58/3.80 Derived: 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,times_times(nat,A,C)),B). [resolve(1303,a,1293,a)]. 3.58/3.80 1304 -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_762_dvd__trans) # label(axiom). [clausify(978)]. 3.58/3.80 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),C)). [resolve(1304,a,1251,a)]. 3.58/3.80 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),B),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),C)). [resolve(1304,a,1286,a)]. 3.58/3.80 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),C)). [resolve(1304,a,1293,a)]. 3.58/3.80 1305 -comm_semiring_1(A) | hAPP(nat,A,power_power(A,B),zero_zero(nat)) = one_one(A) # label(fact_193_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom). [clausify(1008)]. 3.58/3.80 Derived: hAPP(nat,int,power_power(int,A),zero_zero(nat)) = one_one(int). [resolve(1305,a,1251,a)]. 3.58/3.80 Derived: hAPP(nat,real,power_power(real,A),zero_zero(nat)) = one_one(real). [resolve(1305,a,1286,a)]. 3.58/3.80 Derived: hAPP(nat,nat,power_power(nat,A),zero_zero(nat)) = one_one(nat). [resolve(1305,a,1293,a)]. 3.58/3.80 1306 -ab_group_add(A) | minus_minus(A,B,C) != zero_zero(A) | ti(A,C) = ti(A,B) # label(fact_321_eq__iff__diff__eq__0) # label(axiom). [clausify(73)]. 3.58/3.80 1307 ab_group_add(int) # label(arity_Int_Oint___Groups_Oab__group__add) # label(axiom). [assumption]. 3.58/3.80 1308 -ab_group_add(A) | minus_minus(A,B,C) = zero_zero(A) | ti(A,C) != ti(A,B) # label(fact_321_eq__iff__diff__eq__0) # label(axiom). [clausify(73)]. 3.58/3.80 1309 -ab_group_add(A) | minus_minus(A,B,C) != minus_minus(A,D,E) | ti(A,E) != ti(A,D) | ti(A,C) = ti(A,B) # label(fact_248_diff__eq__diff__eq) # label(axiom). [clausify(193)]. 3.58/3.80 Derived: minus_minus(int,A,B) != minus_minus(int,C,D) | ti(int,D) != ti(int,C) | ti(int,B) = ti(int,A). [resolve(1309,a,1307,a)]. 3.58/3.80 1310 -ab_group_add(A) | minus_minus(A,B,C) != minus_minus(A,D,E) | ti(A,E) = ti(A,D) | ti(A,C) != ti(A,B) # label(fact_248_diff__eq__diff__eq) # label(axiom). [clausify(193)]. 3.58/3.80 Derived: minus_minus(int,A,B) != minus_minus(int,C,D) | ti(int,D) = ti(int,C) | ti(int,B) != ti(int,A). [resolve(1310,a,1307,a)]. 3.58/3.80 1311 ab_group_add(real) # label(arity_RealDef_Oreal___Groups_Oab__group__add) # label(axiom). [assumption]. 3.58/3.80 Derived: minus_minus(real,A,B) != minus_minus(real,C,D) | ti(real,D) != ti(real,C) | ti(real,B) = ti(real,A). [resolve(1311,a,1309,a)]. 3.58/3.80 Derived: minus_minus(real,A,B) != minus_minus(real,C,D) | ti(real,D) = ti(real,C) | ti(real,B) != ti(real,A). [resolve(1311,a,1310,a)]. 3.58/3.80 1312 -ab_group_add(A) | minus_minus(A,plus_plus(A,B,C),plus_plus(A,D,E)) = plus_plus(A,minus_minus(A,B,D),minus_minus(A,C,E)) # label(fact_325_add__diff__add) # label(axiom). [clausify(838)]. 3.58/3.80 Derived: minus_minus(int,plus_plus(int,A,B),plus_plus(int,C,D)) = plus_plus(int,minus_minus(int,A,C),minus_minus(int,B,D)). [resolve(1312,a,1307,a)]. 3.58/3.80 Derived: minus_minus(real,plus_plus(real,A,B),plus_plus(real,C,D)) = plus_plus(real,minus_minus(real,A,C),minus_minus(real,B,D)). [resolve(1312,a,1311,a)]. 3.58/3.80 1313 linordered_ring(int) # label(arity_Int_Oint___Rings_Olinordered__ring) # label(axiom). [assumption]. 3.58/3.80 1314 -linordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),times_times(A,B,B))) # label(fact_294_zero__le__square) # label(axiom). [clausify(31)]. 3.76/4.05 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),times_times(int,A,A))). [resolve(1313,a,1314,a)]. 3.76/4.05 1315 linordered_ring(real) # label(arity_RealDef_Oreal___Rings_Olinordered__ring) # label(axiom). [assumption]. 3.76/4.05 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),times_times(real,A,A))). [resolve(1315,a,1314,a)]. 3.76/4.05 1316 -linordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),plus_plus(A,times_times(A,B,B),times_times(A,C,C)))) # label(fact_310_sum__squares__ge__zero) # label(axiom). [clausify(507)]. 3.76/4.05 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),plus_plus(int,times_times(int,A,A),times_times(int,B,B)))). [resolve(1316,a,1313,a)]. 3.76/4.05 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),plus_plus(real,times_times(real,A,A),times_times(real,B,B)))). [resolve(1316,a,1315,a)]. 3.76/4.05 1317 -linordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,times_times(A,B,B),times_times(A,C,C))),zero_zero(A))) # label(fact_485_not__sum__squares__lt__zero) # label(axiom). [clausify(896)]. 3.76/4.05 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,times_times(int,A,A),times_times(int,B,B))),zero_zero(int))). [resolve(1317,a,1313,a)]. 3.76/4.05 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),plus_plus(real,times_times(real,A,A),times_times(real,B,B))),zero_zero(real))). [resolve(1317,a,1315,a)]. 3.76/4.05 1318 -linordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),times_times(A,B,B)),zero_zero(A))) # label(fact_409_not__square__less__zero) # label(axiom). [clausify(996)]. 3.76/4.05 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),times_times(int,A,A)),zero_zero(int))). [resolve(1318,a,1313,a)]. 3.76/4.05 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),times_times(real,A,A)),zero_zero(real))). [resolve(1318,a,1315,a)]. 3.76/4.05 1319 -ring_11004092258visors(A) | ti(A,B) = zero_zero(A) | hAPP(nat,A,power_power(A,B),C) != zero_zero(A) # label(fact_165_field__power__not__zero) # label(axiom). [clausify(243)]. 3.76/4.05 1320 ring_11004092258visors(int) # label(arity_Int_Oint___Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 3.76/4.05 1321 ring_11004092258visors(real) # label(arity_RealDef_Oreal___Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 3.76/4.05 Derived: ti(int,A) = zero_zero(int) | hAPP(nat,int,power_power(int,A),B) != zero_zero(int). [resolve(1319,a,1320,a)]. 3.76/4.05 Derived: ti(real,A) = zero_zero(real) | hAPP(nat,real,power_power(real,A),B) != zero_zero(real). [resolve(1319,a,1321,a)]. 3.76/4.05 1322 -ring_11004092258visors(A) | ti(A,B) != zero_zero(A) | hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(A) # label(fact_5_zero__eq__power2) # label(axiom). [clausify(927)]. 3.76/4.05 Derived: ti(int,A) != zero_zero(int) | hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(int). [resolve(1322,a,1320,a)]. 3.76/4.05 Derived: ti(real,A) != zero_zero(real) | hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(real). [resolve(1322,a,1321,a)]. 3.76/4.05 1323 -ring_11004092258visors(A) | ti(A,B) = zero_zero(A) | hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))) != zero_zero(A) # label(fact_5_zero__eq__power2) # label(axiom). [clausify(927)]. 3.76/4.05 1324 ordere223160158up_add(nat) # label(arity_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 3.76/4.05 1325 -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),plus_plus(A,B,D)),plus_plus(A,C,E))) # label(fact_222_add__strict__mono) # label(axiom). [clausify(34)]. 3.76/4.05 1326 -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),plus_plus(A,B,D)),plus_plus(A,C,D))) # label(fact_224_add__strict__right__mono) # label(axiom). [clausify(63)]. 3.87/4.10 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),plus_plus(nat,A,C)),plus_plus(nat,B,C))). [resolve(1324,a,1326,a)]. 3.87/4.10 1327 ordere223160158up_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 3.87/4.10 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),plus_plus(real,A,C)),plus_plus(real,B,D))). [resolve(1327,a,1325,a)]. 3.87/4.10 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),plus_plus(real,A,C)),plus_plus(real,B,C))). [resolve(1327,a,1326,a)]. 3.87/4.10 1328 -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),plus_plus(A,D,B)),plus_plus(A,D,C))) # label(fact_223_add__strict__left__mono) # label(axiom). [clausify(350)]. 3.87/4.10 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),plus_plus(nat,C,A)),plus_plus(nat,C,B))). [resolve(1328,a,1324,a)]. 3.87/4.10 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),plus_plus(real,C,A)),plus_plus(real,C,B))). [resolve(1328,a,1327,a)]. 3.87/4.10 1329 -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),plus_plus(A,B,D)),plus_plus(A,C,E))) # label(fact_440_add__le__less__mono) # label(axiom). [clausify(491)]. 3.87/4.10 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),plus_plus(nat,A,C)),plus_plus(nat,B,D))). [resolve(1329,a,1324,a)]. 3.87/4.10 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),plus_plus(real,A,C)),plus_plus(real,B,D))). [resolve(1329,a,1327,a)]. 3.87/4.10 1330 -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),plus_plus(A,B,D)),plus_plus(A,C,E))) # label(fact_439_add__less__le__mono) # label(axiom). [clausify(821)]. 3.87/4.10 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),plus_plus(nat,A,C)),plus_plus(nat,B,D))). [resolve(1330,a,1324,a)]. 3.87/4.10 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),plus_plus(real,A,C)),plus_plus(real,B,D))). [resolve(1330,a,1327,a)]. 3.87/4.10 1331 ordere223160158up_add(int) # label(arity_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 3.87/4.10 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),plus_plus(int,A,C)),plus_plus(int,B,D))). [resolve(1331,a,1325,a)]. 3.87/4.10 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),plus_plus(int,A,C)),plus_plus(int,B,C))). [resolve(1331,a,1326,a)]. 3.87/4.10 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),plus_plus(int,C,A)),plus_plus(int,C,B))). [resolve(1331,a,1328,a)]. 4.12/4.32 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),plus_plus(int,A,C)),plus_plus(int,B,D))). [resolve(1331,a,1329,a)]. 4.12/4.32 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),D)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,A,C)),plus_plus(int,B,D))). [resolve(1331,a,1330,a)]. 4.12/4.32 1332 -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),minus_minus(A,C,D))) # label(fact_736_dvd__diff) # label(axiom). [clausify(375)]. 4.12/4.32 1333 comm_ring_1(real) # label(arity_RealDef_Oreal___Rings_Ocomm__ring__1) # label(axiom). [assumption]. 4.12/4.32 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),minus_minus(real,B,C))). [resolve(1332,a,1333,a)]. 4.12/4.32 1334 comm_ring_1(int) # label(arity_Int_Oint___Rings_Ocomm__ring__1) # label(axiom). [assumption]. 4.12/4.32 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),minus_minus(int,B,C))). [resolve(1334,a,1332,a)]. 4.12/4.32 1335 ordere236663937imp_le(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 4.12/4.32 1336 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,B,C)),plus_plus(A,B,D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) # label(fact_353_add__le__imp__le__left) # label(axiom). [clausify(36)]. 4.12/4.32 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),plus_plus(real,A,B)),plus_plus(real,A,C))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),C)). [resolve(1335,a,1336,a)]. 4.12/4.32 1337 ordere236663937imp_le(nat) # label(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 4.12/4.32 1338 ordere236663937imp_le(int) # label(arity_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 4.12/4.32 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,A,B)),plus_plus(int,A,C))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)). [resolve(1338,a,1336,a)]. 4.12/4.32 1339 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,B,D)),plus_plus(A,C,D))) # label(fact_226_add__less__cancel__right) # label(axiom). [clausify(282)]. 4.12/4.32 1340 -ordere236663937imp_le(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,B,D)),plus_plus(A,C,D))) # label(fact_226_add__less__cancel__right) # label(axiom). [clausify(282)]. 4.12/4.32 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),plus_plus(real,A,C)),plus_plus(real,B,C))). [resolve(1340,a,1335,a)]. 4.12/4.32 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),plus_plus(nat,A,C)),plus_plus(nat,B,C))). [resolve(1340,a,1337,a)]. 4.12/4.32 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),plus_plus(int,A,C)),plus_plus(int,B,C))). [resolve(1340,a,1338,a)]. 4.12/4.32 1341 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,B,C)),plus_plus(A,D,C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),D)) # label(fact_221_add__less__imp__less__right) # label(axiom). [clausify(368)]. 4.12/4.32 1342 -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),plus_plus(A,B,D)),plus_plus(A,C,D))) # label(fact_359_add__le__cancel__right) # label(axiom). [clausify(633)]. 4.22/4.46 1343 -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),plus_plus(A,B,D)),plus_plus(A,C,D))) # label(fact_359_add__le__cancel__right) # label(axiom). [clausify(633)]. 4.22/4.46 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),plus_plus(real,A,C)),plus_plus(real,B,C))). [resolve(1343,a,1335,a)]. 4.22/4.46 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),plus_plus(nat,A,C)),plus_plus(nat,B,C))). [resolve(1343,a,1337,a)]. 4.22/4.46 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),plus_plus(int,A,C)),plus_plus(int,B,C))). [resolve(1343,a,1338,a)]. 4.22/4.46 1344 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,B,C)),plus_plus(A,B,D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) # label(fact_225_add__less__cancel__left) # label(axiom). [clausify(652)]. 4.22/4.46 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),plus_plus(real,A,B)),plus_plus(real,A,C))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),C)). [resolve(1344,a,1335,a)]. 4.22/4.46 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,A,B)),plus_plus(nat,A,C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)). [resolve(1344,a,1337,a)]. 4.22/4.46 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,A,B)),plus_plus(int,A,C))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)). [resolve(1344,a,1338,a)]. 4.22/4.46 1345 -ordere236663937imp_le(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,B,C)),plus_plus(A,B,D))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) # label(fact_225_add__less__cancel__left) # label(axiom). [clausify(652)]. 4.22/4.46 1346 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,B,C)),plus_plus(A,B,D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) # label(fact_220_add__less__imp__less__left) # label(axiom). [clausify(693)]. 4.22/4.46 1347 -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),plus_plus(A,D,B)),plus_plus(A,D,C))) # label(fact_358_add__le__cancel__left) # label(axiom). [clausify(842)]. 4.22/4.46 1348 -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),plus_plus(A,D,B)),plus_plus(A,D,C))) # label(fact_358_add__le__cancel__left) # label(axiom). [clausify(842)]. 4.22/4.46 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),plus_plus(nat,C,A)),plus_plus(nat,C,B))). [resolve(1348,a,1337,a)]. 4.22/4.46 1349 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,B,C)),plus_plus(A,D,C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),D)) # label(fact_354_add__le__imp__le__right) # label(axiom). [clausify(908)]. 4.22/4.46 1350 -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),times_times(A,D,B)),times_times(A,D,C))) # label(fact_423_comm__mult__strict__left__mono) # label(axiom). [clausify(269)]. 4.22/4.46 1351 linord893533164strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 4.22/4.46 1352 linord893533164strict(int) # label(arity_Int_Oint___Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 4.22/4.46 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),times_times(int,C,A)),times_times(int,C,B))). [resolve(1352,a,1350,a)]. 4.39/4.58 1353 linord893533164strict(nat) # label(arity_Nat_Onat___Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 4.39/4.58 1354 semiring_1(real) # label(arity_RealDef_Oreal___Rings_Osemiring__1) # label(axiom). [assumption]. 4.39/4.58 1355 -semiring_1(A) | hAPP(nat,A,power_power(A,one_one(A)),number_number_of(nat,bit0(bit1(pls)))) = one_one(A) # label(fact_3_one__power2) # label(axiom). [clausify(48)]. 4.39/4.58 1356 -semiring_1(A) | hAPP(nat,A,semiring_1_of_nat(A),zero_zero(nat)) = zero_zero(A) # label(fact_188_of__nat__0) # label(axiom). [clausify(112)]. 4.39/4.58 Derived: hAPP(nat,real,power_power(real,one_one(real)),number_number_of(nat,bit0(bit1(pls)))) = one_one(real). [resolve(1354,a,1355,a)]. 4.39/4.58 Derived: hAPP(nat,real,semiring_1_of_nat(real),zero_zero(nat)) = zero_zero(real). [resolve(1354,a,1356,a)]. 4.39/4.58 1357 -semiring_1(A) | hAPP(nat,A,power_power(A,zero_zero(A)),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(A) # label(fact_4_zero__power2) # label(axiom). [clausify(143)]. 4.39/4.58 Derived: hAPP(nat,real,power_power(real,zero_zero(real)),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(real). [resolve(1357,a,1354,a)]. 4.39/4.58 1358 -semiring_1(A) | semiring_1_of_nat(A) = ti(fun(nat,A),semiring_1_of_nat(A)) # label(tsy_c_Nat_Osemiring__1__class_Oof__nat_res) # label(axiom). [clausify(286)]. 4.39/4.58 Derived: semiring_1_of_nat(real) = ti(fun(nat,real),semiring_1_of_nat(real)). [resolve(1358,a,1354,a)]. 4.39/4.58 1359 -semiring_1(A) | hAPP(nat,A,semiring_1_of_nat(A),one_one(nat)) = one_one(A) # label(fact_171_of__nat__1) # label(axiom). [clausify(504)]. 4.39/4.58 Derived: hAPP(nat,real,semiring_1_of_nat(real),one_one(nat)) = one_one(real). [resolve(1359,a,1354,a)]. 4.39/4.58 1360 -semiring_1(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | plus_plus(A,hAPP(nat,A,semiring_1_of_nat(A),nat_1(B)),hAPP(nat,A,semiring_1_of_nat(A),nat_1(B))) = hAPP(nat,A,semiring_1_of_nat(A),nat_1(times_times(int,number_number_of(int,bit0(bit1(pls))),B))) # label(fact_470_of__nat__double) # label(axiom). [clausify(627)]. 4.39/4.58 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | plus_plus(real,hAPP(nat,real,semiring_1_of_nat(real),nat_1(A)),hAPP(nat,real,semiring_1_of_nat(real),nat_1(A))) = hAPP(nat,real,semiring_1_of_nat(real),nat_1(times_times(int,number_number_of(int,bit0(bit1(pls))),A))). [resolve(1360,a,1354,a)]. 4.39/4.58 1361 semiring_1(nat) # label(arity_Nat_Onat___Rings_Osemiring__1) # label(axiom). [assumption]. 4.39/4.58 Derived: hAPP(nat,nat,power_power(nat,one_one(nat)),number_number_of(nat,bit0(bit1(pls)))) = one_one(nat). [resolve(1361,a,1355,a)]. 4.39/4.58 Derived: hAPP(nat,nat,semiring_1_of_nat(nat),zero_zero(nat)) = zero_zero(nat). [resolve(1361,a,1356,a)]. 4.39/4.58 Derived: hAPP(nat,nat,power_power(nat,zero_zero(nat)),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(nat). [resolve(1361,a,1357,a)]. 4.39/4.58 Derived: semiring_1_of_nat(nat) = ti(fun(nat,nat),semiring_1_of_nat(nat)). [resolve(1361,a,1358,a)]. 4.39/4.58 Derived: hAPP(nat,nat,semiring_1_of_nat(nat),one_one(nat)) = one_one(nat). [resolve(1361,a,1359,a)]. 4.39/4.58 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | plus_plus(nat,hAPP(nat,nat,semiring_1_of_nat(nat),nat_1(A)),hAPP(nat,nat,semiring_1_of_nat(nat),nat_1(A))) = hAPP(nat,nat,semiring_1_of_nat(nat),nat_1(times_times(int,number_number_of(int,bit0(bit1(pls))),A))). [resolve(1361,a,1360,a)]. 4.39/4.58 1362 -semiring_1(A) | plus_plus(A,hAPP(nat,A,semiring_1_of_nat(A),B),hAPP(nat,A,semiring_1_of_nat(A),C)) = hAPP(nat,A,semiring_1_of_nat(A),plus_plus(nat,B,C)) # label(fact_170_of__nat__add) # label(axiom). [clausify(731)]. 4.39/4.58 Derived: plus_plus(real,hAPP(nat,real,semiring_1_of_nat(real),A),hAPP(nat,real,semiring_1_of_nat(real),B)) = hAPP(nat,real,semiring_1_of_nat(real),plus_plus(nat,A,B)). [resolve(1362,a,1354,a)]. 4.39/4.58 Derived: plus_plus(nat,hAPP(nat,nat,semiring_1_of_nat(nat),A),hAPP(nat,nat,semiring_1_of_nat(nat),B)) = hAPP(nat,nat,semiring_1_of_nat(nat),plus_plus(nat,A,B)). [resolve(1362,a,1361,a)]. 4.48/4.72 1363 -semiring_1(A) | times_times(A,hAPP(nat,A,semiring_1_of_nat(A),B),hAPP(nat,A,semiring_1_of_nat(A),C)) = hAPP(nat,A,semiring_1_of_nat(A),times_times(nat,B,C)) # label(fact_265_of__nat__mult) # label(axiom). [clausify(855)]. 4.48/4.72 Derived: times_times(real,hAPP(nat,real,semiring_1_of_nat(real),A),hAPP(nat,real,semiring_1_of_nat(real),B)) = hAPP(nat,real,semiring_1_of_nat(real),times_times(nat,A,B)). [resolve(1363,a,1354,a)]. 4.48/4.72 Derived: times_times(nat,hAPP(nat,nat,semiring_1_of_nat(nat),A),hAPP(nat,nat,semiring_1_of_nat(nat),B)) = hAPP(nat,nat,semiring_1_of_nat(nat),times_times(nat,A,B)). [resolve(1363,a,1361,a)]. 4.48/4.72 1364 semiring_1(int) # label(arity_Int_Oint___Rings_Osemiring__1) # label(axiom). [assumption]. 4.48/4.72 Derived: hAPP(nat,int,power_power(int,one_one(int)),number_number_of(nat,bit0(bit1(pls)))) = one_one(int). [resolve(1364,a,1355,a)]. 4.48/4.72 Derived: hAPP(nat,int,power_power(int,zero_zero(int)),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(int). [resolve(1364,a,1357,a)]. 4.48/4.72 Derived: semiring_1_of_nat(int) = ti(fun(nat,int),semiring_1_of_nat(int)). [resolve(1364,a,1358,a)]. 4.48/4.72 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | plus_plus(int,hAPP(nat,int,semiring_1_of_nat(int),nat_1(A)),hAPP(nat,int,semiring_1_of_nat(int),nat_1(A))) = hAPP(nat,int,semiring_1_of_nat(int),nat_1(times_times(int,number_number_of(int,bit0(bit1(pls))),A))). [resolve(1364,a,1360,a)]. 4.48/4.72 Derived: plus_plus(int,hAPP(nat,int,semiring_1_of_nat(int),A),hAPP(nat,int,semiring_1_of_nat(int),B)) = hAPP(nat,int,semiring_1_of_nat(int),plus_plus(nat,A,B)). [resolve(1364,a,1362,a)]. 4.48/4.72 Derived: times_times(int,hAPP(nat,int,semiring_1_of_nat(int),A),hAPP(nat,int,semiring_1_of_nat(int),B)) = hAPP(nat,int,semiring_1_of_nat(int),times_times(nat,A,B)). [resolve(1364,a,1363,a)]. 4.48/4.72 1365 -semiring_1(A) | hAPP(nat,A,semiring_1_of_nat(A),hAPP(nat,nat,power_power(nat,B),C)) = hAPP(nat,A,power_power(A,hAPP(nat,A,semiring_1_of_nat(A),B)),C) # label(fact_172_of__nat__power) # label(axiom). [clausify(936)]. 4.48/4.72 Derived: hAPP(nat,real,semiring_1_of_nat(real),hAPP(nat,nat,power_power(nat,A),B)) = hAPP(nat,real,power_power(real,hAPP(nat,real,semiring_1_of_nat(real),A)),B). [resolve(1365,a,1354,a)]. 4.48/4.72 Derived: hAPP(nat,nat,semiring_1_of_nat(nat),hAPP(nat,nat,power_power(nat,A),B)) = hAPP(nat,nat,power_power(nat,hAPP(nat,nat,semiring_1_of_nat(nat),A)),B). [resolve(1365,a,1361,a)]. 4.48/4.72 1366 -linordered_semiring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),times_times(A,B,C)),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_301_mult__left__less__imp__less) # label(axiom). [clausify(312)]. 4.48/4.72 1367 linordered_semiring(nat) # label(arity_Nat_Onat___Rings_Olinordered__semiring) # label(axiom). [assumption]. 4.48/4.72 1368 -linordered_semiring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),times_times(A,B,C)),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_303_mult__right__less__imp__less) # label(axiom). [clausify(495)]. 4.48/4.72 1369 linordered_semiring(int) # label(arity_Int_Oint___Rings_Olinordered__semiring) # label(axiom). [assumption]. 4.48/4.72 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),times_times(int,A,B)),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(1369,a,1366,a)]. 4.48/4.72 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),times_times(int,A,B)),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(1369,a,1368,a)]. 4.48/4.72 1370 linordered_semiring(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring) # label(axiom). [assumption]. 4.58/4.82 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),times_times(real,A,B)),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(1370,a,1366,a)]. 4.58/4.82 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),times_times(real,A,B)),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(1370,a,1368,a)]. 4.58/4.82 1371 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),plus_plus(A,times_times(A,minus_minus(A,C,D),E),F))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,times_times(A,D,E),B)),plus_plus(A,times_times(A,C,E),F))) # label(fact_287_le__add__iff2) # label(axiom). [clausify(287)]. 4.58/4.82 1372 ordered_ring(real) # label(arity_RealDef_Oreal___Rings_Oordered__ring) # label(axiom). [assumption]. 4.58/4.82 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),plus_plus(real,times_times(real,minus_minus(real,B,C),D),E))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),plus_plus(real,times_times(real,C,D),A)),plus_plus(real,times_times(real,B,D),E))). [resolve(1371,a,1372,a)]. 4.58/4.82 1373 -ordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),plus_plus(A,times_times(A,minus_minus(A,C,D),E),F))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,times_times(A,D,E),B)),plus_plus(A,times_times(A,C,E),F))) # label(fact_287_le__add__iff2) # label(axiom). [clausify(287)]. 4.58/4.82 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),plus_plus(real,times_times(real,minus_minus(real,B,C),D),E))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),plus_plus(real,times_times(real,C,D),A)),plus_plus(real,times_times(real,B,D),E))). [resolve(1373,a,1372,a)]. 4.58/4.82 1374 -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),times_times(A,C,D)),times_times(A,B,D))) # label(fact_274_mult__right__mono__neg) # label(axiom). [clausify(446)]. 4.58/4.82 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),times_times(real,B,C)),times_times(real,A,C))). [resolve(1374,a,1372,a)]. 4.58/4.82 1375 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,times_times(A,B,C),D)),plus_plus(A,times_times(A,E,C),F))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),plus_plus(A,times_times(A,minus_minus(A,E,B),C),F))) # label(fact_296_less__add__iff2) # label(axiom). [clausify(549)]. 4.58/4.82 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),plus_plus(real,times_times(real,A,B),C)),plus_plus(real,times_times(real,D,B),E))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),plus_plus(real,times_times(real,minus_minus(real,D,A),B),E))). [resolve(1375,a,1372,a)]. 4.58/4.82 1376 -ordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,times_times(A,B,C),D)),plus_plus(A,times_times(A,E,C),F))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),plus_plus(A,times_times(A,minus_minus(A,E,B),C),F))) # label(fact_296_less__add__iff2) # label(axiom). [clausify(549)]. 4.58/4.82 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),plus_plus(real,times_times(real,A,B),C)),plus_plus(real,times_times(real,D,B),E))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),plus_plus(real,times_times(real,minus_minus(real,D,A),B),E))). [resolve(1376,a,1372,a)]. 4.58/4.82 1377 -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)),times_times(A,B,C))) # label(fact_270_split__mult__pos__le) # label(axiom). [clausify(550)]. 4.65/4.85 1378 -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)),times_times(A,B,C))) # label(fact_270_split__mult__pos__le) # label(axiom). [clausify(550)]. 4.65/4.85 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)),times_times(real,A,B))). [resolve(1378,a,1372,a)]. 4.65/4.85 1379 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,times_times(A,B,C),D)),plus_plus(A,times_times(A,E,C),F))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,times_times(A,minus_minus(A,B,E),C),D)),F)) # label(fact_295_less__add__iff1) # label(axiom). [clausify(683)]. 4.65/4.85 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),plus_plus(real,times_times(real,A,B),C)),plus_plus(real,times_times(real,D,B),E))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),plus_plus(real,times_times(real,minus_minus(real,A,D),B),C)),E)). [resolve(1379,a,1372,a)]. 4.65/4.85 1380 -ordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,times_times(A,B,C),D)),plus_plus(A,times_times(A,E,C),F))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,times_times(A,minus_minus(A,B,E),C),D)),F)) # label(fact_295_less__add__iff1) # label(axiom). [clausify(683)]. 4.65/4.85 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),plus_plus(real,times_times(real,A,B),C)),plus_plus(real,times_times(real,D,B),E))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),plus_plus(real,times_times(real,minus_minus(real,A,D),B),C)),E)). [resolve(1380,a,1372,a)]. 4.65/4.85 1381 -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),times_times(A,D,C)),times_times(A,D,B))) # label(fact_273_mult__left__mono__neg) # label(axiom). [clausify(706)]. 4.65/4.85 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),times_times(real,C,B)),times_times(real,C,A))). [resolve(1381,a,1372,a)]. 4.65/4.85 1382 -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)),times_times(A,B,C))) # label(fact_278_mult__nonpos__nonpos) # label(axiom). [clausify(835)]. 4.65/4.85 1383 ordered_ring(int) # label(arity_Int_Oint___Rings_Oordered__ring) # label(axiom). [assumption]. 4.65/4.85 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),plus_plus(int,times_times(int,minus_minus(int,B,C),D),E))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,times_times(int,C,D),A)),plus_plus(int,times_times(int,B,D),E))). [resolve(1383,a,1371,a)]. 4.65/4.85 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),plus_plus(int,times_times(int,minus_minus(int,B,C),D),E))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,times_times(int,C,D),A)),plus_plus(int,times_times(int,B,D),E))). [resolve(1383,a,1373,a)]. 4.65/4.85 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),times_times(int,B,C)),times_times(int,A,C))). [resolve(1383,a,1374,a)]. 4.65/4.85 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,times_times(int,A,B),C)),plus_plus(int,times_times(int,D,B),E))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),plus_plus(int,times_times(int,minus_minus(int,D,A),B),E))). [resolve(1383,a,1375,a)]. 4.72/4.95 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,times_times(int,A,B),C)),plus_plus(int,times_times(int,D,B),E))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),plus_plus(int,times_times(int,minus_minus(int,D,A),B),E))). [resolve(1383,a,1376,a)]. 4.72/4.95 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)),times_times(int,A,B))). [resolve(1383,a,1378,a)]. 4.72/4.95 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,times_times(int,A,B),C)),plus_plus(int,times_times(int,D,B),E))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,times_times(int,minus_minus(int,A,D),B),C)),E)). [resolve(1383,a,1379,a)]. 4.72/4.95 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,times_times(int,A,B),C)),plus_plus(int,times_times(int,D,B),E))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,times_times(int,minus_minus(int,A,D),B),C)),E)). [resolve(1383,a,1380,a)]. 4.72/4.95 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),times_times(int,C,B)),times_times(int,C,A))). [resolve(1383,a,1381,a)]. 4.72/4.95 1384 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,times_times(A,B,C),D)),plus_plus(A,times_times(A,E,C),F))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,times_times(A,minus_minus(A,B,E),C),D)),F)) # label(fact_285_le__add__iff1) # label(axiom). [clausify(922)]. 4.72/4.95 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),plus_plus(real,times_times(real,A,B),C)),plus_plus(real,times_times(real,D,B),E))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),plus_plus(real,times_times(real,minus_minus(real,A,D),B),C)),E)). [resolve(1384,a,1372,a)]. 4.72/4.95 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,times_times(int,A,B),C)),plus_plus(int,times_times(int,D,B),E))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,times_times(int,minus_minus(int,A,D),B),C)),E)). [resolve(1384,a,1383,a)]. 4.72/4.95 1385 -ordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,times_times(A,B,C),D)),plus_plus(A,times_times(A,E,C),F))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,times_times(A,minus_minus(A,B,E),C),D)),F)) # label(fact_285_le__add__iff1) # label(axiom). [clausify(922)]. 4.72/4.95 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),plus_plus(real,times_times(real,A,B),C)),plus_plus(real,times_times(real,D,B),E))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),plus_plus(real,times_times(real,minus_minus(real,A,D),B),C)),E)). [resolve(1385,a,1372,a)]. 4.72/4.95 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,times_times(int,A,B),C)),plus_plus(int,times_times(int,D,B),E))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,times_times(int,minus_minus(int,A,D),B),C)),E)). [resolve(1385,a,1383,a)]. 4.72/4.95 1386 order(nat) # label(arity_Nat_Onat___Orderings_Oorder) # label(axiom). [assumption]. 4.72/4.95 1387 -order(A) | hAPP(A,fun(A,bool),ti(fun(A,fun(A,bool)),B),C) = hAPP(A,fun(A,bool),B,C) # label(tsy_c_hAPP_0_arg1) # label(axiom). [clausify(61)]. 4.72/4.95 1388 -order(A) | ti(fun(A,fun(A,bool)),ord_less(A)) = ord_less(A) # label(tsy_c_Orderings_Oord__class_Oless_res) # label(axiom). [clausify(242)]. 4.72/4.95 Derived: ti(fun(nat,fun(nat,bool)),ord_less(nat)) = ord_less(nat). [resolve(1386,a,1388,a)]. 4.72/4.95 1389 -order(A) | hAPP(A,bool,B,ti(A,C)) = hAPP(A,bool,B,C) # label(tsy_c_hAPP_1_arg2) # label(axiom). [clausify(334)]. 4.88/5.08 1390 -order(A) | ord_less_eq(A) = ti(fun(A,fun(A,bool)),ord_less_eq(A)) # label(tsy_c_Orderings_Oord__class_Oless__eq_res) # label(axiom). [clausify(339)]. 4.88/5.08 Derived: ord_less_eq(nat) = ti(fun(nat,fun(nat,bool)),ord_less_eq(nat)). [resolve(1390,a,1386,a)]. 4.88/5.08 1391 order(int) # label(arity_Int_Oint___Orderings_Oorder) # label(axiom). [assumption]. 4.88/5.08 Derived: ti(fun(int,fun(int,bool)),ord_less(int)) = ord_less(int). [resolve(1391,a,1388,a)]. 4.88/5.08 Derived: ord_less_eq(int) = ti(fun(int,fun(int,bool)),ord_less_eq(int)). [resolve(1391,a,1390,a)]. 4.88/5.08 1392 -order(A) | hAPP(A,bool,ti(fun(A,bool),B),C) = hAPP(A,bool,B,C) # label(tsy_c_hAPP_1_arg1) # label(axiom). [clausify(471)]. 4.88/5.08 1393 -order(A) | hAPP(A,fun(A,bool),B,ti(A,C)) = hAPP(A,fun(A,bool),B,C) # label(tsy_c_hAPP_0_arg2) # label(axiom). [clausify(475)]. 4.88/5.08 1394 order(real) # label(arity_RealDef_Oreal___Orderings_Oorder) # label(axiom). [assumption]. 4.88/5.08 Derived: ti(fun(real,fun(real,bool)),ord_less(real)) = ord_less(real). [resolve(1394,a,1388,a)]. 4.88/5.08 Derived: ord_less_eq(real) = ti(fun(real,fun(real,bool)),ord_less_eq(real)). [resolve(1394,a,1390,a)]. 4.88/5.08 1395 -order(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | ti(A,C) = ti(A,B) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) # label(fact_884_order__le__neq__implies__less) # label(axiom). [clausify(623)]. 4.88/5.08 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | ti(nat,B) = ti(nat,A) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)). [resolve(1395,a,1386,a)]. 4.88/5.08 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | ti(real,B) = ti(real,A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)). [resolve(1395,a,1394,a)]. 4.88/5.08 1396 -order(A) | ti(fun(A,bool),hAPP(A,fun(A,bool),B,C)) = hAPP(A,fun(A,bool),B,C) # label(tsy_c_hAPP_0_res) # label(axiom). [clausify(832)]. 4.88/5.08 1397 -order(A) | ti(bool,hAPP(A,bool,B,C)) = hAPP(A,bool,B,C) # label(tsy_c_hAPP_1_res) # label(axiom). [clausify(921)]. 4.88/5.08 1398 order(bool) # label(arity_HOL_Obool___Orderings_Oorder) # label(axiom). [assumption]. 4.88/5.08 Derived: hAPP(bool,fun(bool,bool),ti(fun(bool,fun(bool,bool)),A),B) = hAPP(bool,fun(bool,bool),A,B). [resolve(1398,a,1387,a)]. 4.88/5.08 Derived: ti(fun(bool,fun(bool,bool)),ord_less(bool)) = ord_less(bool). [resolve(1398,a,1388,a)]. 4.88/5.08 Derived: hAPP(bool,bool,A,ti(bool,B)) = hAPP(bool,bool,A,B). [resolve(1398,a,1389,a)]. 4.88/5.08 Derived: ord_less_eq(bool) = ti(fun(bool,fun(bool,bool)),ord_less_eq(bool)). [resolve(1398,a,1390,a)]. 4.88/5.08 Derived: hAPP(bool,bool,ti(fun(bool,bool),A),B) = hAPP(bool,bool,A,B). [resolve(1398,a,1392,a)]. 4.88/5.08 Derived: hAPP(bool,fun(bool,bool),A,ti(bool,B)) = hAPP(bool,fun(bool,bool),A,B). [resolve(1398,a,1393,a)]. 4.88/5.08 Derived: -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less_eq(bool),A),B)) | ti(bool,B) = ti(bool,A) | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),ord_less(bool),A),B)). [resolve(1398,a,1395,a)]. 4.88/5.08 Derived: ti(fun(bool,bool),hAPP(bool,fun(bool,bool),A,B)) = hAPP(bool,fun(bool,bool),A,B). [resolve(1398,a,1396,a)]. 4.88/5.08 Derived: ti(bool,hAPP(bool,bool,A,B)) = hAPP(bool,bool,A,B). [resolve(1398,a,1397,a)]. 4.88/5.08 1399 linord20386208strict(int) # label(arity_Int_Oint___Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 4.88/5.08 1400 -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),times_times(A,D,B)),times_times(A,D,C))) # label(fact_422_mult__strict__left__mono) # label(axiom). [clausify(69)]. 4.88/5.08 1401 linord20386208strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 4.88/5.08 1402 -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),times_times(A,B,D)),times_times(A,C,E))) # label(fact_306_mult__strict__mono_H) # label(axiom). [clausify(450)]. 4.88/5.09 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_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),times_times(int,A,C)),times_times(int,B,D))). [resolve(1402,a,1399,a)]. 4.88/5.09 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_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),times_times(real,A,C)),times_times(real,B,D))). [resolve(1402,a,1401,a)]. 4.88/5.09 1403 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),times_times(A,B,C)),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_302_mult__less__imp__less__right) # label(axiom). [clausify(462)]. 4.88/5.09 1404 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),times_times(A,B,C)),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_299_mult__right__le__imp__le) # label(axiom). [clausify(472)]. 4.88/5.09 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),times_times(int,A,B)),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(1404,a,1399,a)]. 4.88/5.09 1405 -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),times_times(A,B,C)),zero_zero(A))) # label(fact_419_mult__neg__pos) # label(axiom). [clausify(583)]. 4.88/5.09 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),times_times(int,A,B)),zero_zero(int))). [resolve(1405,a,1399,a)]. 4.88/5.09 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),times_times(real,A,B)),zero_zero(real))). [resolve(1405,a,1401,a)]. 4.88/5.09 1406 -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),times_times(A,B,D)),times_times(A,C,D))) # label(fact_421_mult__strict__right__mono) # label(axiom). [clausify(614)]. 4.88/5.09 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),zero_zero(int)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),times_times(int,A,C)),times_times(int,B,C))). [resolve(1406,a,1399,a)]. 4.88/5.09 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),zero_zero(real)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),times_times(real,A,C)),times_times(real,B,C))). [resolve(1406,a,1401,a)]. 4.88/5.09 1407 -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),times_times(A,B,D)),times_times(A,C,E))) # label(fact_307_mult__strict__mono) # label(axiom). [clausify(637)]. 4.88/5.10 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),times_times(int,A,C)),times_times(int,B,D))). [resolve(1407,a,1399,a)]. 4.88/5.10 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),times_times(real,A,C)),times_times(real,B,D))). [resolve(1407,a,1401,a)]. 4.88/5.10 1408 linord20386208strict(nat) # label(arity_Nat_Onat___Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 4.88/5.10 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),times_times(nat,A,C)),times_times(nat,B,D))). [resolve(1408,a,1402,a)]. 4.88/5.10 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),times_times(nat,A,B)),times_times(nat,C,B))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),C)). [resolve(1408,a,1404,a)]. 4.88/5.10 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),times_times(nat,A,C)),times_times(nat,B,D))). [resolve(1408,a,1407,a)]. 4.88/5.10 1409 -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),times_times(A,B,D)),times_times(A,C,E))) # label(fact_305_mult__less__le__imp__less) # label(axiom). [clausify(672)]. 4.88/5.10 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),times_times(int,A,C)),times_times(int,B,D))). [resolve(1409,a,1399,a)]. 4.88/5.10 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),times_times(real,A,C)),times_times(real,B,D))). [resolve(1409,a,1401,a)]. 4.88/5.10 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),times_times(nat,A,C)),times_times(nat,B,D))). [resolve(1409,a,1408,a)]. 4.88/5.10 1410 -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),times_times(A,B,D)),times_times(A,C,E))) # label(fact_304_mult__le__less__imp__less) # label(axiom). [clausify(711)]. 4.88/5.10 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),times_times(int,A,C)),times_times(int,B,D))). [resolve(1410,a,1399,a)]. 4.88/5.10 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),times_times(real,A,C)),times_times(real,B,D))). [resolve(1410,a,1401,a)]. 4.88/5.10 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),times_times(nat,A,C)),times_times(nat,B,D))). [resolve(1410,a,1408,a)]. 4.88/5.10 1411 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),times_times(A,B,C)),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_298_mult__left__le__imp__le) # label(axiom). [clausify(750)]. 4.88/5.10 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),times_times(int,A,B)),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(1411,a,1399,a)]. 4.88/5.10 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),times_times(real,A,B)),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(1411,a,1401,a)]. 4.88/5.10 1412 -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),times_times(A,C,B)),zero_zero(A))) # label(fact_415_mult__pos__neg2) # label(axiom). [clausify(760)]. 4.88/5.10 1413 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(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_417_zero__less__mult__pos2) # label(axiom). [clausify(762)]. 4.88/5.10 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(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(1413,a,1399,a)]. 4.88/5.10 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(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(1413,a,1401,a)]. 4.99/5.21 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(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(1413,a,1408,a)]. 4.99/5.21 1414 -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),times_times(A,B,C)),zero_zero(A))) # label(fact_414_mult__pos__neg) # label(axiom). [clausify(804)]. 4.99/5.21 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),times_times(int,A,B)),zero_zero(int))). [resolve(1414,a,1399,a)]. 4.99/5.21 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),times_times(real,A,B)),zero_zero(real))). [resolve(1414,a,1401,a)]. 4.99/5.21 1415 -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)),times_times(A,B,C))) # label(fact_413_mult__pos__pos) # label(axiom). [clausify(818)]. 4.99/5.21 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),times_times(int,A,B))). [resolve(1415,a,1399,a)]. 4.99/5.21 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),times_times(real,A,B))). [resolve(1415,a,1401,a)]. 4.99/5.21 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)),times_times(nat,A,B))). [resolve(1415,a,1408,a)]. 4.99/5.21 1416 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),times_times(A,B,C)),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_300_mult__less__imp__less__left) # label(axiom). [clausify(906)]. 4.99/5.21 1417 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(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_416_zero__less__mult__pos) # label(axiom). [clausify(932)]. 4.99/5.21 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),times_times(int,A,B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)). [resolve(1417,a,1399,a)]. 4.99/5.21 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(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(1417,a,1401,a)]. 4.99/5.21 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(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(1417,a,1408,a)]. 4.99/5.21 1418 comm_monoid_mult(nat) # label(arity_Nat_Onat___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 5.41/5.62 1419 -comm_monoid_mult(A) | times_times(A,B,one_one(A)) = ti(A,B) # label(fact_347_mult_Ocomm__neutral) # label(axiom). [clausify(72)]. 5.41/5.62 1420 -comm_monoid_mult(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,times_times(A,B,D)),C) # label(fact_363_power__mult__distrib) # label(axiom). [clausify(272)]. 5.41/5.62 1421 comm_monoid_mult(int) # label(arity_Int_Oint___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 5.41/5.62 1422 comm_monoid_mult(real) # label(arity_RealDef_Oreal___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 5.41/5.62 1423 -comm_monoid_mult(A) | times_times(A,one_one(A),B) = ti(A,B) # label(fact_350_mult__1) # label(axiom). [clausify(998)]. 5.41/5.62 1424 -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)) | plus_plus(A,E,F) != one_one(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,times_times(A,E,B),times_times(A,F,D))),C)) # label(fact_401_convex__bound__lt) # label(axiom). [clausify(694)]. 5.41/5.62 1425 linord626643107strict(int) # label(arity_Int_Oint___Rings_Olinordered__semiring__1__strict) # label(axiom). [assumption]. 5.41/5.62 1426 linord626643107strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring__1__strict) # label(axiom). [assumption]. 5.41/5.62 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)) | plus_plus(int,D,E) != one_one(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,times_times(int,D,A),times_times(int,E,C))),B)). [resolve(1424,a,1425,a)]. 5.41/5.62 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)) | plus_plus(real,D,E) != one_one(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),plus_plus(real,times_times(real,D,A),times_times(real,E,C))),B)). [resolve(1424,a,1426,a)]. 5.41/5.62 1427 -zero(A) | ti(A,zero_zero(A)) = zero_zero(A) # label(tsy_c_Groups_Ozero__class_Ozero_res) # label(axiom). [clausify(136)]. 5.41/5.62 1428 zero(real) # label(arity_RealDef_Oreal___Groups_Ozero) # label(axiom). [assumption]. 5.41/5.62 Derived: ti(real,zero_zero(real)) = zero_zero(real). [resolve(1427,a,1428,a)]. 5.41/5.62 1429 zero(int) # label(arity_Int_Oint___Groups_Ozero) # label(axiom). [assumption]. 5.41/5.62 Derived: ti(int,zero_zero(int)) = zero_zero(int). [resolve(1429,a,1427,a)]. 5.41/5.62 1430 zero(nat) # label(arity_Nat_Onat___Groups_Ozero) # label(axiom). [assumption]. 5.41/5.62 Derived: ti(nat,zero_zero(nat)) = zero_zero(nat). [resolve(1430,a,1427,a)]. 5.41/5.62 1431 ab_semigroup_mult(nat) # label(arity_Nat_Onat___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 5.41/5.62 1432 -ab_semigroup_mult(A) | times_times(A,times_times(A,B,C),D) = times_times(A,B,times_times(A,C,D)) # label(fact_242_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom). [clausify(89)]. 5.41/5.62 1433 -ab_semigroup_mult(A) | times_times(A,ti(A,B),C) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_5_arg1) # label(axiom). [clausify(196)]. 5.41/5.62 1434 ab_semigroup_mult(real) # label(arity_RealDef_Oreal___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 5.41/5.62 1435 -ab_semigroup_mult(A) | times_times(A,B,ti(A,C)) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_5_arg2) # label(axiom). [clausify(912)]. 5.41/5.62 1436 ab_semigroup_mult(int) # label(arity_Int_Oint___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 5.41/5.62 1437 -ab_semigroup_mult(A) | times_times(A,B,C) = ti(A,times_times(A,B,C)) # label(tsy_c_Groups_Otimes__class_Otimes_5_res) # label(axiom). [clausify(999)]. 5.63/5.83 1438 -monoid_add(A) | plus_plus(A,ti(A,B),C) = plus_plus(A,B,C) # label(tsy_c_Groups_Oplus__class_Oplus_0_arg1) # label(axiom). [clausify(122)]. 5.63/5.83 1439 monoid_add(nat) # label(arity_Nat_Onat___Groups_Omonoid__add) # label(axiom). [assumption]. 5.63/5.83 1440 -monoid_add(A) | ti(A,plus_plus(A,B,C)) = plus_plus(A,B,C) # label(tsy_c_Groups_Oplus__class_Oplus_0_res) # label(axiom). [clausify(332)]. 5.63/5.83 1441 monoid_add(real) # label(arity_RealDef_Oreal___Groups_Omonoid__add) # label(axiom). [assumption]. 5.63/5.83 1442 monoid_add(int) # label(arity_Int_Oint___Groups_Omonoid__add) # label(axiom). [assumption]. 5.63/5.83 1443 -monoid_add(A) | ti(A,B) = plus_plus(A,zero_zero(A),B) # label(fact_219_add__0__left) # label(axiom). [clausify(798)]. 5.63/5.83 1444 -monoid_add(A) | plus_plus(A,B,ti(A,C)) = plus_plus(A,B,C) # label(tsy_c_Groups_Oplus__class_Oplus_0_arg2) # label(axiom). [clausify(840)]. 5.63/5.83 1445 -monoid_add(A) | ti(A,B) = plus_plus(A,B,zero_zero(A)) # label(fact_215_add__0__right) # label(axiom). [clausify(952)]. 5.63/5.83 1446 -ordere142940540dd_abs(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),abs_abs(A,B))) | ti(A,B) != zero_zero(A) # label(fact_945_zero__less__abs__iff) # label(axiom). [clausify(155)]. 5.63/5.83 1447 ordere142940540dd_abs(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__group__add__abs) # label(axiom). [assumption]. 5.63/5.83 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),abs_abs(real,A))) | ti(real,A) != zero_zero(real). [resolve(1446,a,1447,a)]. 5.63/5.83 1448 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),abs_abs(A,B))) | ti(A,B) = zero_zero(A) # label(fact_945_zero__less__abs__iff) # label(axiom). [clausify(155)]. 5.63/5.83 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),abs_abs(real,A))) | ti(real,A) = zero_zero(real). [resolve(1448,a,1447,a)]. 5.63/5.83 1449 -ordere142940540dd_abs(A) | ti(A,abs_abs(A,B)) = abs_abs(A,B) # label(tsy_c_Groups_Oabs__class_Oabs_res) # label(axiom). [clausify(171)]. 5.63/5.83 Derived: ti(real,abs_abs(real,A)) = abs_abs(real,A). [resolve(1449,a,1447,a)]. 5.63/5.83 1450 -ordere142940540dd_abs(A) | abs_abs(A,zero_zero(A)) = zero_zero(A) # label(fact_930_abs__zero) # label(axiom). [clausify(195)]. 5.63/5.83 Derived: abs_abs(real,zero_zero(real)) = zero_zero(real). [resolve(1450,a,1447,a)]. 5.63/5.83 1451 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),abs_abs(A,minus_minus(A,B,C))),plus_plus(A,abs_abs(A,B),abs_abs(A,C)))) # label(fact_928_abs__triangle__ineq4) # label(axiom). [clausify(247)]. 5.63/5.83 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),abs_abs(real,minus_minus(real,A,B))),plus_plus(real,abs_abs(real,A),abs_abs(real,B)))). [resolve(1451,a,1447,a)]. 5.63/5.83 1452 -ordere142940540dd_abs(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | ti(A,B) = abs_abs(A,B) # label(fact_947_abs__of__nonneg) # label(axiom). [clausify(254)]. 5.63/5.83 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | ti(real,A) = abs_abs(real,A). [resolve(1452,a,1447,a)]. 5.63/5.83 1453 -ordere142940540dd_abs(A) | ti(A,B) != zero_zero(A) | abs_abs(A,B) = zero_zero(A) # label(fact_931_abs__eq__0) # label(axiom). [clausify(284)]. 5.63/5.83 Derived: ti(real,A) != zero_zero(real) | abs_abs(real,A) = zero_zero(real). [resolve(1453,a,1447,a)]. 5.63/5.83 1454 -ordere142940540dd_abs(A) | ti(A,B) = zero_zero(A) | abs_abs(A,B) != zero_zero(A) # label(fact_931_abs__eq__0) # label(axiom). [clausify(284)]. 5.63/5.83 Derived: ti(real,A) = zero_zero(real) | abs_abs(real,A) != zero_zero(real). [resolve(1454,a,1447,a)]. 5.63/5.83 1455 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),abs_abs(A,minus_minus(A,abs_abs(A,B),abs_abs(A,C)))),abs_abs(A,minus_minus(A,B,C)))) # label(fact_941_abs__triangle__ineq3) # label(axiom). [clausify(321)]. 5.63/5.83 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),abs_abs(real,minus_minus(real,abs_abs(real,A),abs_abs(real,B)))),abs_abs(real,minus_minus(real,A,B)))). [resolve(1455,a,1447,a)]. 5.63/5.83 1456 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),abs_abs(A,minus_minus(A,plus_plus(A,B,C),plus_plus(A,D,E)))),plus_plus(A,abs_abs(A,minus_minus(A,B,D)),abs_abs(A,minus_minus(A,C,E))))) # label(fact_927_abs__diff__triangle__ineq) # label(axiom). [clausify(388)]. 5.63/5.85 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),abs_abs(real,minus_minus(real,plus_plus(real,A,B),plus_plus(real,C,D)))),plus_plus(real,abs_abs(real,minus_minus(real,A,C)),abs_abs(real,minus_minus(real,B,D))))). [resolve(1456,a,1447,a)]. 5.63/5.85 1457 -ordere142940540dd_abs(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | ti(A,B) = abs_abs(A,B) # label(fact_944_abs__of__pos) # label(axiom). [clausify(451)]. 5.63/5.85 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | ti(real,A) = abs_abs(real,A). [resolve(1457,a,1447,a)]. 5.63/5.85 1458 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),minus_minus(A,abs_abs(A,B),abs_abs(A,C))),abs_abs(A,minus_minus(A,C,B)))) # label(fact_939_abs__triangle__ineq2__sym) # label(axiom). [clausify(482)]. 5.63/5.85 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),minus_minus(real,abs_abs(real,A),abs_abs(real,B))),abs_abs(real,minus_minus(real,B,A)))). [resolve(1458,a,1447,a)]. 5.63/5.85 1459 -ordere142940540dd_abs(A) | plus_plus(A,abs_abs(A,B),abs_abs(A,C)) = abs_abs(A,plus_plus(A,abs_abs(A,B),abs_abs(A,C))) # label(fact_932_abs__add__abs) # label(axiom). [clausify(529)]. 5.63/5.85 Derived: plus_plus(real,abs_abs(real,A),abs_abs(real,B)) = abs_abs(real,plus_plus(real,abs_abs(real,A),abs_abs(real,B))). [resolve(1459,a,1447,a)]. 5.63/5.85 1460 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),minus_minus(A,abs_abs(A,B),abs_abs(A,C))),abs_abs(A,minus_minus(A,B,C)))) # label(fact_940_abs__triangle__ineq2) # label(axiom). [clausify(643)]. 5.63/5.85 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),minus_minus(real,abs_abs(real,A),abs_abs(real,B))),abs_abs(real,minus_minus(real,A,B)))). [resolve(1460,a,1447,a)]. 5.63/5.85 1461 -ordere142940540dd_abs(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),abs_abs(A,B)),zero_zero(A))) | ti(A,B) = zero_zero(A) # label(fact_948_abs__le__zero__iff) # label(axiom). [clausify(650)]. 5.63/5.85 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),abs_abs(real,A)),zero_zero(real))) | ti(real,A) = zero_zero(real). [resolve(1461,a,1447,a)]. 5.63/5.85 1462 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),abs_abs(A,B)),zero_zero(A))) | ti(A,B) != zero_zero(A) # label(fact_948_abs__le__zero__iff) # label(axiom). [clausify(650)]. 5.63/5.85 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),abs_abs(real,A)),zero_zero(real))) | ti(real,A) != zero_zero(real). [resolve(1462,a,1447,a)]. 5.63/5.85 1463 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),abs_abs(A,plus_plus(A,B,C))),plus_plus(A,abs_abs(A,B),abs_abs(A,C)))) # label(fact_942_abs__triangle__ineq) # label(axiom). [clausify(678)]. 5.63/5.85 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),abs_abs(real,plus_plus(real,A,B))),plus_plus(real,abs_abs(real,A),abs_abs(real,B)))). [resolve(1463,a,1447,a)]. 5.63/5.85 1464 -ordere142940540dd_abs(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),abs_abs(A,B)),zero_zero(A))) # label(fact_946_abs__not__less__zero) # label(axiom). [clausify(728)]. 5.63/5.85 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),abs_abs(real,A)),zero_zero(real))). [resolve(1464,a,1447,a)]. 5.63/5.85 1465 -ordere142940540dd_abs(A) | abs_abs(A,ti(A,B)) = abs_abs(A,B) # label(tsy_c_Groups_Oabs__class_Oabs_arg1) # label(axiom). [clausify(739)]. 5.63/5.85 Derived: abs_abs(real,ti(real,A)) = abs_abs(real,A). [resolve(1465,a,1447,a)]. 5.63/5.85 1466 -ordere142940540dd_abs(A) | abs_abs(A,abs_abs(A,B)) = abs_abs(A,B) # label(fact_937_abs__idempotent) # label(axiom). [clausify(744)]. 5.63/5.85 Derived: abs_abs(real,abs_abs(real,A)) = abs_abs(real,A). [resolve(1466,a,1447,a)]. 5.63/5.85 1467 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),abs_abs(A,B))) # label(fact_953_abs__ge__self) # label(axiom). [clausify(765)]. 5.63/5.87 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),abs_abs(real,A))). [resolve(1467,a,1447,a)]. 5.63/5.87 1468 -ordere142940540dd_abs(A) | abs_abs(A,minus_minus(A,B,C)) = abs_abs(A,minus_minus(A,C,B)) # label(fact_950_abs__minus__commute) # label(axiom). [clausify(776)]. 5.63/5.87 Derived: abs_abs(real,minus_minus(real,A,B)) = abs_abs(real,minus_minus(real,B,A)). [resolve(1468,a,1447,a)]. 5.63/5.87 1469 ordere142940540dd_abs(int) # label(arity_Int_Oint___Groups_Oordered__ab__group__add__abs) # label(axiom). [assumption]. 5.63/5.87 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),abs_abs(int,A))) | ti(int,A) != zero_zero(int). [resolve(1469,a,1446,a)]. 5.63/5.87 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),abs_abs(int,A))) | ti(int,A) = zero_zero(int). [resolve(1469,a,1448,a)]. 5.63/5.87 Derived: ti(int,abs_abs(int,A)) = abs_abs(int,A). [resolve(1469,a,1449,a)]. 5.63/5.87 Derived: abs_abs(int,zero_zero(int)) = zero_zero(int). [resolve(1469,a,1450,a)]. 5.63/5.87 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),abs_abs(int,minus_minus(int,A,B))),plus_plus(int,abs_abs(int,A),abs_abs(int,B)))). [resolve(1469,a,1451,a)]. 5.63/5.87 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | ti(int,A) = abs_abs(int,A). [resolve(1469,a,1452,a)]. 5.63/5.87 Derived: ti(int,A) != zero_zero(int) | abs_abs(int,A) = zero_zero(int). [resolve(1469,a,1453,a)]. 5.63/5.87 Derived: ti(int,A) = zero_zero(int) | abs_abs(int,A) != zero_zero(int). [resolve(1469,a,1454,a)]. 5.63/5.87 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),abs_abs(int,minus_minus(int,abs_abs(int,A),abs_abs(int,B)))),abs_abs(int,minus_minus(int,A,B)))). [resolve(1469,a,1455,a)]. 5.63/5.87 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),abs_abs(int,minus_minus(int,plus_plus(int,A,B),plus_plus(int,C,D)))),plus_plus(int,abs_abs(int,minus_minus(int,A,C)),abs_abs(int,minus_minus(int,B,D))))). [resolve(1469,a,1456,a)]. 5.63/5.87 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | ti(int,A) = abs_abs(int,A). [resolve(1469,a,1457,a)]. 5.63/5.87 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),minus_minus(int,abs_abs(int,A),abs_abs(int,B))),abs_abs(int,minus_minus(int,B,A)))). [resolve(1469,a,1458,a)]. 5.63/5.87 Derived: plus_plus(int,abs_abs(int,A),abs_abs(int,B)) = abs_abs(int,plus_plus(int,abs_abs(int,A),abs_abs(int,B))). [resolve(1469,a,1459,a)]. 5.63/5.87 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),minus_minus(int,abs_abs(int,A),abs_abs(int,B))),abs_abs(int,minus_minus(int,A,B)))). [resolve(1469,a,1460,a)]. 5.63/5.87 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),abs_abs(int,A)),zero_zero(int))) | ti(int,A) = zero_zero(int). [resolve(1469,a,1461,a)]. 5.63/5.87 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),abs_abs(int,A)),zero_zero(int))) | ti(int,A) != zero_zero(int). [resolve(1469,a,1462,a)]. 5.63/5.87 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),abs_abs(int,plus_plus(int,A,B))),plus_plus(int,abs_abs(int,A),abs_abs(int,B)))). [resolve(1469,a,1463,a)]. 5.63/5.87 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),abs_abs(int,A)),zero_zero(int))). [resolve(1469,a,1464,a)]. 5.63/5.87 Derived: abs_abs(int,ti(int,A)) = abs_abs(int,A). [resolve(1469,a,1465,a)]. 5.63/5.87 Derived: abs_abs(int,abs_abs(int,A)) = abs_abs(int,A). [resolve(1469,a,1466,a)]. 5.63/5.87 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),abs_abs(int,A))). [resolve(1469,a,1467,a)]. 5.63/5.87 Derived: abs_abs(int,minus_minus(int,A,B)) = abs_abs(int,minus_minus(int,B,A)). [resolve(1469,a,1468,a)]. 5.63/5.87 1470 -ordere142940540dd_abs(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),abs_abs(A,B)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_954_abs__le__D1) # label(axiom). [clausify(942)]. 5.63/5.87 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),abs_abs(real,A)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)). [resolve(1470,a,1447,a)]. 5.73/5.98 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),abs_abs(int,A)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(1470,a,1469,a)]. 5.73/5.98 1471 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),abs_abs(A,B))) # label(fact_949_abs__ge__zero) # label(axiom). [clausify(948)]. 5.73/5.98 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),abs_abs(real,A))). [resolve(1471,a,1447,a)]. 5.73/5.98 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),abs_abs(int,A))). [resolve(1471,a,1469,a)]. 5.73/5.98 1472 linord219039673up_add(real) # label(arity_RealDef_Oreal___Groups_Olinordered__ab__group__add) # label(axiom). [assumption]. 5.73/5.98 1473 -linord219039673up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,B,B)),zero_zero(A))) # label(fact_433_double__add__le__zero__iff__single__add__le__zero) # label(axiom). [clausify(97)]. 5.73/5.98 1474 -linord219039673up_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,B,B)),zero_zero(A))) # label(fact_433_double__add__le__zero__iff__single__add__le__zero) # label(axiom). [clausify(97)]. 5.73/5.98 1475 -linord219039673up_add(A) | ti(A,B) != zero_zero(A) | plus_plus(A,B,B) = zero_zero(A) # label(fact_65_double__eq__0__iff) # label(axiom). [clausify(142)]. 5.73/5.98 1476 -linord219039673up_add(A) | ti(A,B) = zero_zero(A) | plus_plus(A,B,B) != zero_zero(A) # label(fact_65_double__eq__0__iff) # label(axiom). [clausify(142)]. 5.73/5.98 1477 -linord219039673up_add(A) | plus_plus(A,B,B) != zero_zero(A) | ti(A,B) = zero_zero(A) # label(fact_216_double__zero__sym) # label(axiom). [clausify(288)]. 5.73/5.98 1478 -linord219039673up_add(A) | plus_plus(A,B,B) = zero_zero(A) | ti(A,B) != zero_zero(A) # label(fact_216_double__zero__sym) # label(axiom). [clausify(288)]. 5.73/5.98 1479 -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)),plus_plus(A,B,B))) # label(fact_194_zero__less__double__add__iff__zero__less__single__add) # label(axiom). [clausify(298)]. 5.73/5.98 1480 -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)),plus_plus(A,B,B))) # label(fact_194_zero__less__double__add__iff__zero__less__single__add) # label(axiom). [clausify(298)]. 5.73/5.98 1481 -linord219039673up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(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_432_zero__le__double__add__iff__zero__le__single__add) # label(axiom). [clausify(317)]. 5.73/5.98 1482 -linord219039673up_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(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_432_zero__le__double__add__iff__zero__le__single__add) # label(axiom). [clausify(317)]. 5.73/5.98 1483 -linord219039673up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(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_195_double__add__less__zero__iff__single__add__less__zero) # label(axiom). [clausify(353)]. 5.73/5.98 1484 -linord219039673up_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(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_195_double__add__less__zero__iff__single__add__less__zero) # label(axiom). [clausify(353)]. 5.73/5.98 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),plus_plus(real,A,A)),zero_zero(real))). [resolve(1472,a,1474,a)]. 5.73/5.98 Derived: ti(real,A) != zero_zero(real) | plus_plus(real,A,A) = zero_zero(real). [resolve(1472,a,1475,a)]. 5.90/6.10 Derived: ti(real,A) = zero_zero(real) | plus_plus(real,A,A) != zero_zero(real). [resolve(1472,a,1476,a)]. 5.90/6.10 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)),plus_plus(real,A,A))). [resolve(1472,a,1480,a)]. 5.90/6.10 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),plus_plus(real,A,A))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)). [resolve(1472,a,1481,a)]. 5.90/6.10 1485 linord219039673up_add(int) # label(arity_Int_Oint___Groups_Olinordered__ab__group__add) # label(axiom). [assumption]. 5.90/6.10 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,A,A)),zero_zero(int))). [resolve(1485,a,1474,a)]. 5.90/6.10 Derived: ti(int,A) != zero_zero(int) | plus_plus(int,A,A) = zero_zero(int). [resolve(1485,a,1475,a)]. 5.90/6.10 Derived: ti(int,A) = zero_zero(int) | plus_plus(int,A,A) != zero_zero(int). [resolve(1485,a,1476,a)]. 5.90/6.10 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)),plus_plus(int,A,A))). [resolve(1485,a,1480,a)]. 5.90/6.10 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),plus_plus(int,A,A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)). [resolve(1485,a,1481,a)]. 5.90/6.10 1486 -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),times_times(A,B,D)),times_times(A,C,E))) # label(fact_271_mult__mono) # label(axiom). [clausify(300)]. 5.90/6.10 1487 ordered_semiring(nat) # label(arity_Nat_Onat___Rings_Oordered__semiring) # label(axiom). [assumption]. 5.90/6.10 1488 ordered_semiring(real) # label(arity_RealDef_Oreal___Rings_Oordered__semiring) # label(axiom). [assumption]. 5.90/6.10 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),times_times(nat,A,C)),times_times(nat,B,D))). [resolve(1486,a,1487,a)]. 5.90/6.10 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),times_times(real,A,C)),times_times(real,B,D))). [resolve(1486,a,1488,a)]. 5.90/6.10 1489 -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),times_times(A,B,D)),times_times(A,C,D))) # label(fact_277_mult__right__mono) # label(axiom). [clausify(443)]. 5.90/6.10 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),times_times(real,A,C)),times_times(real,B,C))). [resolve(1489,a,1488,a)]. 5.90/6.10 1490 ordered_semiring(int) # label(arity_Int_Oint___Rings_Oordered__semiring) # label(axiom). [assumption]. 5.90/6.10 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),times_times(int,A,C)),times_times(int,B,D))). [resolve(1490,a,1486,a)]. 6.03/6.22 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),times_times(int,A,C)),times_times(int,B,C))). [resolve(1490,a,1489,a)]. 6.03/6.22 1491 -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),times_times(A,B,D)),times_times(A,C,E))) # label(fact_272_mult__mono_H) # label(axiom). [clausify(880)]. 6.03/6.22 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),times_times(nat,A,C)),times_times(nat,B,D))). [resolve(1491,a,1487,a)]. 6.03/6.22 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),times_times(real,A,C)),times_times(real,B,D))). [resolve(1491,a,1488,a)]. 6.03/6.22 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),times_times(int,A,C)),times_times(int,B,D))). [resolve(1491,a,1490,a)]. 6.03/6.22 1492 -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),times_times(A,D,B)),times_times(A,D,C))) # label(fact_276_mult__left__mono) # label(axiom). [clausify(933)]. 6.03/6.22 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),times_times(real,C,A)),times_times(real,C,B))). [resolve(1492,a,1488,a)]. 6.03/6.22 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),times_times(int,C,A)),times_times(int,C,B))). [resolve(1492,a,1490,a)]. 6.03/6.22 1493 linord581940658strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__ring__strict) # label(axiom). [assumption]. 6.03/6.22 1494 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(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_290_mult__le__0__iff) # label(axiom). [clausify(161)]. 6.03/6.22 1495 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(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_290_mult__le__0__iff) # label(axiom). [clausify(161)]. 6.03/6.22 1496 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(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_290_mult__le__0__iff) # label(axiom). [clausify(161)]. 6.03/6.22 1497 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(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_290_mult__le__0__iff) # label(axiom). [clausify(161)]. 6.03/6.22 1498 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(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_290_mult__le__0__iff) # label(axiom). [clausify(161)]. 6.03/6.22 1499 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(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_290_mult__le__0__iff) # label(axiom). [clausify(161)]. 6.03/6.22 1500 -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),times_times(A,D,B)),times_times(A,C,B))) # label(fact_410_mult__less__cancel__right__disj) # label(axiom). [clausify(198)]. 6.03/6.22 1501 -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),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),times_times(A,B,D)),times_times(A,C,D))) # label(fact_410_mult__less__cancel__right__disj) # label(axiom). [clausify(198)]. 6.03/6.22 1502 -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),times_times(A,C,B)),times_times(A,D,B))) # label(fact_410_mult__less__cancel__right__disj) # label(axiom). [clausify(198)]. 6.03/6.22 1503 -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),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),times_times(A,C,B)),times_times(A,D,B))) # label(fact_410_mult__less__cancel__right__disj) # label(axiom). [clausify(198)]. 6.03/6.22 1504 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),times_times(A,C,D)),times_times(A,B,D))) # label(fact_410_mult__less__cancel__right__disj) # label(axiom). [clausify(198)]. 6.03/6.22 1505 -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),zero_zero(A)),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),times_times(A,C,D)),times_times(A,B,D))) # label(fact_410_mult__less__cancel__right__disj) # label(axiom). [clausify(198)]. 6.03/6.22 1506 -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),times_times(A,B,C)),times_times(A,B,D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) # label(fact_412_mult__less__cancel__left__pos) # label(axiom). [clausify(201)]. 6.03/6.22 1507 -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),times_times(A,B,C)),times_times(A,B,D))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) # label(fact_412_mult__less__cancel__left__pos) # label(axiom). [clausify(201)]. 6.03/6.22 1508 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,times_times(A,B,B),times_times(A,C,C))),zero_zero(A))) | ti(A,B) = zero_zero(A) # label(fact_311_sum__squares__le__zero__iff) # label(axiom). [clausify(248)]. 6.03/6.22 1509 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,times_times(A,B,B),times_times(A,C,C))),zero_zero(A))) | ti(A,C) = zero_zero(A) # label(fact_311_sum__squares__le__zero__iff) # label(axiom). [clausify(248)]. 6.03/6.22 1510 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,times_times(A,B,B),times_times(A,C,C))),zero_zero(A))) | ti(A,B) != zero_zero(A) | ti(A,C) != zero_zero(A) # label(fact_311_sum__squares__le__zero__iff) # label(axiom). [clausify(248)]. 6.03/6.22 1511 -linord581940658strict(A) | ti(A,B) = zero_zero(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),plus_plus(A,times_times(A,B,B),times_times(A,C,C)))) # label(fact_486_sum__squares__gt__zero__iff) # label(axiom). [clausify(275)]. 6.03/6.22 1512 -linord581940658strict(A) | ti(A,B) = zero_zero(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),plus_plus(A,times_times(A,C,C),times_times(A,B,B)))) # label(fact_486_sum__squares__gt__zero__iff) # label(axiom). [clausify(275)]. 6.03/6.22 1513 -linord581940658strict(A) | ti(A,B) != zero_zero(A) | ti(A,C) != zero_zero(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),plus_plus(A,times_times(A,B,B),times_times(A,C,C)))) # label(fact_486_sum__squares__gt__zero__iff) # label(axiom). [clausify(275)]. 6.03/6.22 1514 -linord581940658strict(A) | ti(A,B) != zero_zero(A) | ti(A,C) != zero_zero(A) | plus_plus(A,times_times(A,C,C),times_times(A,B,B)) = zero_zero(A) # label(fact_428_sum__squares__eq__zero__iff) # label(axiom). [clausify(305)]. 6.03/6.22 1515 -linord581940658strict(A) | ti(A,B) = zero_zero(A) | plus_plus(A,times_times(A,C,C),times_times(A,B,B)) != zero_zero(A) # label(fact_428_sum__squares__eq__zero__iff) # label(axiom). [clausify(305)]. 6.03/6.22 1516 -linord581940658strict(A) | ti(A,B) = zero_zero(A) | plus_plus(A,times_times(A,B,B),times_times(A,C,C)) != zero_zero(A) # label(fact_428_sum__squares__eq__zero__iff) # label(axiom). [clausify(305)]. 6.03/6.22 1517 -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),times_times(A,B,C)),times_times(A,B,D))) # label(fact_309_mult__le__cancel__left__pos) # label(axiom). [clausify(315)]. 6.03/6.22 1518 -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),times_times(A,B,C)),times_times(A,B,D))) # label(fact_309_mult__le__cancel__left__pos) # label(axiom). [clausify(315)]. 6.03/6.22 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(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(1493,a,1495,a)]. 6.03/6.22 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(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(1493,a,1496,a)]. 6.03/6.22 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),times_times(real,C,A)),times_times(real,B,A))). [resolve(1493,a,1500,a)]. 6.03/6.22 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),times_times(real,B,A)),times_times(real,C,A))). [resolve(1493,a,1502,a)]. 6.03/6.22 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),times_times(real,B,A)),times_times(real,C,A))). [resolve(1493,a,1503,a)]. 6.03/6.23 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),times_times(real,B,C)),times_times(real,A,C))). [resolve(1493,a,1504,a)]. 6.03/6.23 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),times_times(real,B,C)),times_times(real,A,C))). [resolve(1493,a,1505,a)]. 6.03/6.23 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),times_times(real,A,B)),times_times(real,A,C))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),C)). [resolve(1493,a,1506,a)]. 6.03/6.23 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),plus_plus(real,times_times(real,A,A),times_times(real,B,B))),zero_zero(real))) | ti(real,A) = zero_zero(real). [resolve(1493,a,1508,a)]. 6.03/6.23 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),plus_plus(real,times_times(real,A,A),times_times(real,B,B))),zero_zero(real))) | ti(real,B) = zero_zero(real). [resolve(1493,a,1509,a)]. 6.03/6.23 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),plus_plus(real,times_times(real,A,A),times_times(real,B,B))),zero_zero(real))) | ti(real,A) != zero_zero(real) | ti(real,B) != zero_zero(real). [resolve(1493,a,1510,a)]. 6.03/6.23 Derived: ti(real,A) = zero_zero(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),plus_plus(real,times_times(real,A,A),times_times(real,B,B)))). [resolve(1493,a,1511,a)]. 6.03/6.23 Derived: ti(real,A) = zero_zero(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),plus_plus(real,times_times(real,B,B),times_times(real,A,A)))). [resolve(1493,a,1512,a)]. 6.03/6.23 Derived: ti(real,A) != zero_zero(real) | ti(real,B) != zero_zero(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),plus_plus(real,times_times(real,A,A),times_times(real,B,B)))). [resolve(1493,a,1513,a)]. 6.03/6.23 Derived: ti(real,A) != zero_zero(real) | ti(real,B) != zero_zero(real) | plus_plus(real,times_times(real,B,B),times_times(real,A,A)) = zero_zero(real). [resolve(1493,a,1514,a)]. 6.03/6.23 Derived: ti(real,A) = zero_zero(real) | plus_plus(real,times_times(real,B,B),times_times(real,A,A)) != zero_zero(real). [resolve(1493,a,1515,a)]. 6.03/6.23 Derived: ti(real,A) = zero_zero(real) | plus_plus(real,times_times(real,A,A),times_times(real,B,B)) != zero_zero(real). [resolve(1493,a,1516,a)]. 6.03/6.23 1519 -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),times_times(A,B,D)),times_times(A,B,C))) # label(fact_308_mult__le__cancel__left__neg) # label(axiom). [clausify(362)]. 6.03/6.23 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),times_times(real,A,C)),times_times(real,A,B))). [resolve(1519,a,1493,a)]. 6.03/6.23 1520 -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),times_times(A,B,D)),times_times(A,B,C))) # label(fact_308_mult__le__cancel__left__neg) # label(axiom). [clausify(362)]. 6.03/6.23 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),times_times(real,A,C)),times_times(real,A,B))). [resolve(1520,a,1493,a)]. 6.03/6.24 1521 -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),times_times(A,D,C)),times_times(A,D,B))) # label(fact_425_mult__strict__left__mono__neg) # label(axiom). [clausify(420)]. 6.03/6.24 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),times_times(real,C,B)),times_times(real,C,A))). [resolve(1521,a,1493,a)]. 6.03/6.24 1522 -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)),times_times(A,B,C))) # label(fact_420_mult__neg__neg) # label(axiom). [clausify(487)]. 6.03/6.24 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)),times_times(real,A,B))). [resolve(1522,a,1493,a)]. 6.03/6.24 1523 -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)),times_times(A,C,B))) # label(fact_291_zero__le__mult__iff) # label(axiom). [clausify(564)]. 6.03/6.24 1524 -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)),times_times(A,C,B))) # label(fact_291_zero__le__mult__iff) # label(axiom). [clausify(564)]. 6.03/6.24 1525 -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)),times_times(A,C,B))) # label(fact_291_zero__le__mult__iff) # label(axiom). [clausify(564)]. 6.03/6.24 1526 -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)),times_times(A,C,B))) # label(fact_291_zero__le__mult__iff) # label(axiom). [clausify(564)]. 6.03/6.24 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)),times_times(real,B,A))). [resolve(1526,a,1493,a)]. 6.03/6.24 1527 -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)),times_times(A,B,C))) # label(fact_291_zero__le__mult__iff) # label(axiom). [clausify(564)]. 6.03/6.24 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)),times_times(real,A,B))). [resolve(1527,a,1493,a)]. 6.03/6.24 1528 -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)),times_times(A,B,C))) # label(fact_291_zero__le__mult__iff) # label(axiom). [clausify(564)]. 6.03/6.25 1529 linord581940658strict(int) # label(arity_Int_Oint___Rings_Olinordered__ring__strict) # label(axiom). [assumption]. 6.03/6.25 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(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(1529,a,1494,a)]. 6.03/6.25 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(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(1529,a,1495,a)]. 6.03/6.25 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(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(1529,a,1496,a)]. 6.03/6.25 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(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(1529,a,1497,a)]. 6.03/6.25 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),times_times(int,C,A)),times_times(int,B,A))). [resolve(1529,a,1500,a)]. 6.03/6.25 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),times_times(int,B,A)),times_times(int,C,A))). [resolve(1529,a,1502,a)]. 6.03/6.25 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),times_times(int,B,A)),times_times(int,C,A))). [resolve(1529,a,1503,a)]. 6.03/6.25 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),times_times(int,B,C)),times_times(int,A,C))). [resolve(1529,a,1504,a)]. 6.03/6.25 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),times_times(int,B,C)),times_times(int,A,C))). [resolve(1529,a,1505,a)]. 6.03/6.25 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),times_times(int,A,B)),times_times(int,A,C))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)). [resolve(1529,a,1506,a)]. 6.03/6.25 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,times_times(int,A,A),times_times(int,B,B))),zero_zero(int))) | ti(int,A) = zero_zero(int). [resolve(1529,a,1508,a)]. 6.03/6.25 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,times_times(int,A,A),times_times(int,B,B))),zero_zero(int))) | ti(int,B) = zero_zero(int). [resolve(1529,a,1509,a)]. 6.03/6.25 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,times_times(int,A,A),times_times(int,B,B))),zero_zero(int))) | ti(int,A) != zero_zero(int) | ti(int,B) != zero_zero(int). [resolve(1529,a,1510,a)]. 6.03/6.25 Derived: ti(int,A) = zero_zero(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),plus_plus(int,times_times(int,A,A),times_times(int,B,B)))). [resolve(1529,a,1511,a)]. 6.03/6.25 Derived: ti(int,A) = zero_zero(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),plus_plus(int,times_times(int,B,B),times_times(int,A,A)))). [resolve(1529,a,1512,a)]. 6.03/6.25 Derived: ti(int,A) != zero_zero(int) | ti(int,B) != zero_zero(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),plus_plus(int,times_times(int,A,A),times_times(int,B,B)))). [resolve(1529,a,1513,a)]. 6.03/6.26 Derived: ti(int,A) != zero_zero(int) | ti(int,B) != zero_zero(int) | plus_plus(int,times_times(int,B,B),times_times(int,A,A)) = zero_zero(int). [resolve(1529,a,1514,a)]. 6.03/6.26 Derived: ti(int,A) = zero_zero(int) | plus_plus(int,times_times(int,B,B),times_times(int,A,A)) != zero_zero(int). [resolve(1529,a,1515,a)]. 6.03/6.26 Derived: ti(int,A) = zero_zero(int) | plus_plus(int,times_times(int,A,A),times_times(int,B,B)) != zero_zero(int). [resolve(1529,a,1516,a)]. 6.03/6.26 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),times_times(int,A,B)),times_times(int,A,C))). [resolve(1529,a,1517,a)]. 6.03/6.26 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),times_times(int,A,C)),times_times(int,A,B))). [resolve(1529,a,1519,a)]. 6.03/6.26 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),times_times(int,A,C)),times_times(int,A,B))). [resolve(1529,a,1520,a)]. 6.03/6.26 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),times_times(int,C,B)),times_times(int,C,A))). [resolve(1529,a,1521,a)]. 6.03/6.26 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)),times_times(int,A,B))). [resolve(1529,a,1522,a)]. 6.03/6.26 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)),times_times(int,B,A))). [resolve(1529,a,1525,a)]. 6.03/6.26 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)),times_times(int,B,A))). [resolve(1529,a,1526,a)]. 6.03/6.26 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)),times_times(int,A,B))). [resolve(1529,a,1527,a)]. 6.03/6.26 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)),times_times(int,A,B))). [resolve(1529,a,1528,a)]. 6.03/6.26 1530 -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),times_times(A,B,C)),times_times(A,B,D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),C)) # label(fact_418_mult__less__cancel__left__neg) # label(axiom). [clausify(867)]. 6.03/6.26 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),times_times(real,A,B)),times_times(real,A,C))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),B)). [resolve(1530,a,1493,a)]. 6.03/6.26 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),times_times(int,A,B)),times_times(int,A,C))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),B)). [resolve(1530,a,1529,a)]. 6.03/6.26 1531 -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),times_times(A,B,C)),times_times(A,B,D))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),C)) # label(fact_418_mult__less__cancel__left__neg) # label(axiom). [clausify(867)]. 6.03/6.26 1532 -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),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),times_times(A,D,B)),times_times(A,D,C))) # label(fact_411_mult__less__cancel__left__disj) # label(axiom). [clausify(876)]. 6.03/6.26 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),times_times(real,C,A)),times_times(real,C,B))). [resolve(1532,a,1493,a)]. 6.03/6.26 1533 -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),times_times(A,D,C)),times_times(A,D,B))) # label(fact_411_mult__less__cancel__left__disj) # label(axiom). [clausify(876)]. 6.03/6.26 1534 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),times_times(A,D,B)),times_times(A,D,C))) # label(fact_411_mult__less__cancel__left__disj) # label(axiom). [clausify(876)]. 6.03/6.26 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),times_times(real,C,A)),times_times(real,C,B))). [resolve(1534,a,1493,a)]. 6.03/6.26 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),times_times(int,C,A)),times_times(int,C,B))). [resolve(1534,a,1529,a)]. 6.03/6.26 1535 -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),times_times(A,D,B)),times_times(A,D,C))) # label(fact_411_mult__less__cancel__left__disj) # label(axiom). [clausify(876)]. 6.03/6.26 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),times_times(real,C,A)),times_times(real,C,B))). [resolve(1535,a,1493,a)]. 6.03/6.26 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),times_times(int,C,A)),times_times(int,C,B))). [resolve(1535,a,1529,a)]. 6.03/6.26 1536 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),times_times(A,B,D)),times_times(A,B,C))) # label(fact_411_mult__less__cancel__left__disj) # label(axiom). [clausify(876)]. 6.03/6.26 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),times_times(real,A,C)),times_times(real,A,B))). [resolve(1536,a,1493,a)]. 6.03/6.26 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),times_times(int,A,C)),times_times(int,A,B))). [resolve(1536,a,1529,a)]. 6.15/6.37 1537 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),times_times(A,B,C)),times_times(A,B,D))) # label(fact_411_mult__less__cancel__left__disj) # label(axiom). [clausify(876)]. 6.15/6.37 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),times_times(real,A,B)),times_times(real,A,C))). [resolve(1537,a,1493,a)]. 6.15/6.37 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),times_times(int,A,B)),times_times(int,A,C))). [resolve(1537,a,1529,a)]. 6.15/6.37 1538 -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),times_times(A,C,D)),times_times(A,B,D))) # label(fact_424_mult__strict__right__mono__neg) # label(axiom). [clausify(977)]. 6.15/6.37 1539 -semiri456707255roduct(A) | ti(A,B) = zero_zero(A) | ti(A,C) = ti(A,D) | ti(A,E) != ti(A,F) | plus_plus(A,E,times_times(A,B,D)) != plus_plus(A,F,times_times(A,B,C)) # label(fact_429_add__scale__eq__noteq) # label(axiom). [clausify(170)]. 6.15/6.37 1540 semiri456707255roduct(real) # label(arity_RealDef_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossp) # label(axiom). [assumption]. 6.15/6.37 Derived: ti(real,A) = zero_zero(real) | ti(real,B) = ti(real,C) | ti(real,D) != ti(real,E) | plus_plus(real,D,times_times(real,A,C)) != plus_plus(real,E,times_times(real,A,B)). [resolve(1539,a,1540,a)]. 6.15/6.37 1541 -semiri456707255roduct(A) | ti(A,B) = ti(A,C) | ti(A,D) = ti(A,E) | plus_plus(A,times_times(A,C,D),times_times(A,B,E)) != plus_plus(A,times_times(A,C,E),times_times(A,B,D)) # label(fact_344_crossproduct__noteq) # label(axiom). [clausify(239)]. 6.15/6.37 Derived: ti(real,A) = ti(real,B) | ti(real,C) = ti(real,D) | plus_plus(real,times_times(real,B,C),times_times(real,A,D)) != plus_plus(real,times_times(real,B,D),times_times(real,A,C)). [resolve(1541,a,1540,a)]. 6.15/6.37 1542 -semiri456707255roduct(A) | ti(A,B) != ti(A,C) | plus_plus(A,times_times(A,C,D),times_times(A,B,E)) = plus_plus(A,times_times(A,C,E),times_times(A,B,D)) # label(fact_344_crossproduct__noteq) # label(axiom). [clausify(239)]. 6.15/6.37 Derived: ti(real,A) != ti(real,B) | plus_plus(real,times_times(real,B,C),times_times(real,A,D)) = plus_plus(real,times_times(real,B,D),times_times(real,A,C)). [resolve(1542,a,1540,a)]. 6.15/6.37 1543 -semiri456707255roduct(A) | ti(A,B) != ti(A,C) | plus_plus(A,times_times(A,D,B),times_times(A,E,C)) = plus_plus(A,times_times(A,D,C),times_times(A,E,B)) # label(fact_344_crossproduct__noteq) # label(axiom). [clausify(239)]. 6.15/6.37 Derived: ti(real,A) != ti(real,B) | plus_plus(real,times_times(real,C,A),times_times(real,D,B)) = plus_plus(real,times_times(real,C,B),times_times(real,D,A)). [resolve(1543,a,1540,a)]. 6.15/6.37 1544 -semiri456707255roduct(A) | plus_plus(A,times_times(A,B,C),times_times(A,D,E)) != plus_plus(A,times_times(A,B,E),times_times(A,D,C)) | ti(A,B) = ti(A,D) | ti(A,C) = ti(A,E) # label(fact_341_crossproduct__eq) # label(axiom). [clausify(376)]. 6.15/6.37 Derived: plus_plus(real,times_times(real,A,B),times_times(real,C,D)) != plus_plus(real,times_times(real,A,D),times_times(real,C,B)) | ti(real,A) = ti(real,C) | ti(real,B) = ti(real,D). [resolve(1544,a,1540,a)]. 6.15/6.37 1545 -semiri456707255roduct(A) | plus_plus(A,times_times(A,B,C),times_times(A,D,E)) = plus_plus(A,times_times(A,B,E),times_times(A,D,C)) | ti(A,B) != ti(A,D) # label(fact_341_crossproduct__eq) # label(axiom). [clausify(376)]. 6.15/6.37 Derived: plus_plus(real,times_times(real,A,B),times_times(real,C,D)) = plus_plus(real,times_times(real,A,D),times_times(real,C,B)) | ti(real,A) != ti(real,C). [resolve(1545,a,1540,a)]. 6.24/6.50 1546 -semiri456707255roduct(A) | plus_plus(A,times_times(A,B,C),times_times(A,D,E)) = plus_plus(A,times_times(A,B,E),times_times(A,D,C)) | ti(A,C) != ti(A,E) # label(fact_341_crossproduct__eq) # label(axiom). [clausify(376)]. 6.24/6.50 1547 semiri456707255roduct(int) # label(arity_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc) # label(axiom). [assumption]. 6.24/6.50 Derived: ti(int,A) = zero_zero(int) | ti(int,B) = ti(int,C) | ti(int,D) != ti(int,E) | plus_plus(int,D,times_times(int,A,C)) != plus_plus(int,E,times_times(int,A,B)). [resolve(1547,a,1539,a)]. 6.24/6.50 Derived: ti(int,A) = ti(int,B) | ti(int,C) = ti(int,D) | plus_plus(int,times_times(int,B,C),times_times(int,A,D)) != plus_plus(int,times_times(int,B,D),times_times(int,A,C)). [resolve(1547,a,1541,a)]. 6.24/6.50 Derived: ti(int,A) != ti(int,B) | plus_plus(int,times_times(int,B,C),times_times(int,A,D)) = plus_plus(int,times_times(int,B,D),times_times(int,A,C)). [resolve(1547,a,1542,a)]. 6.24/6.50 Derived: ti(int,A) != ti(int,B) | plus_plus(int,times_times(int,C,A),times_times(int,D,B)) = plus_plus(int,times_times(int,C,B),times_times(int,D,A)). [resolve(1547,a,1543,a)]. 6.24/6.50 Derived: plus_plus(int,times_times(int,A,B),times_times(int,C,D)) != plus_plus(int,times_times(int,A,D),times_times(int,C,B)) | ti(int,A) = ti(int,C) | ti(int,B) = ti(int,D). [resolve(1547,a,1544,a)]. 6.24/6.50 Derived: plus_plus(int,times_times(int,A,B),times_times(int,C,D)) = plus_plus(int,times_times(int,A,D),times_times(int,C,B)) | ti(int,A) != ti(int,C). [resolve(1547,a,1545,a)]. 6.24/6.50 1548 -semiri456707255roduct(A) | ti(A,B) != zero_zero(A) | ti(A,C) = plus_plus(A,C,B) # label(fact_212_add__0__iff) # label(axiom). [clausify(658)]. 6.24/6.50 Derived: ti(real,A) != zero_zero(real) | ti(real,B) = plus_plus(real,B,A). [resolve(1548,a,1540,a)]. 6.24/6.50 Derived: ti(int,A) != zero_zero(int) | ti(int,B) = plus_plus(int,B,A). [resolve(1548,a,1547,a)]. 6.24/6.50 1549 -semiri456707255roduct(A) | ti(A,B) = zero_zero(A) | ti(A,C) != plus_plus(A,C,B) # label(fact_212_add__0__iff) # label(axiom). [clausify(658)]. 6.24/6.50 Derived: ti(real,A) = zero_zero(real) | ti(real,B) != plus_plus(real,B,A). [resolve(1549,a,1540,a)]. 6.24/6.50 Derived: ti(int,A) = zero_zero(int) | ti(int,B) != plus_plus(int,B,A). [resolve(1549,a,1547,a)]. 6.24/6.50 1550 semiri456707255roduct(nat) # label(arity_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc) # label(axiom). [assumption]. 6.24/6.50 Derived: ti(nat,A) = zero_zero(nat) | ti(nat,B) = ti(nat,C) | ti(nat,D) != ti(nat,E) | plus_plus(nat,D,times_times(nat,A,C)) != plus_plus(nat,E,times_times(nat,A,B)). [resolve(1550,a,1539,a)]. 6.24/6.50 Derived: ti(nat,A) = ti(nat,B) | ti(nat,C) = ti(nat,D) | plus_plus(nat,times_times(nat,B,C),times_times(nat,A,D)) != plus_plus(nat,times_times(nat,B,D),times_times(nat,A,C)). [resolve(1550,a,1541,a)]. 6.24/6.50 Derived: ti(nat,A) != ti(nat,B) | plus_plus(nat,times_times(nat,B,C),times_times(nat,A,D)) = plus_plus(nat,times_times(nat,B,D),times_times(nat,A,C)). [resolve(1550,a,1542,a)]. 6.24/6.50 Derived: ti(nat,A) != ti(nat,B) | plus_plus(nat,times_times(nat,C,A),times_times(nat,D,B)) = plus_plus(nat,times_times(nat,C,B),times_times(nat,D,A)). [resolve(1550,a,1543,a)]. 6.24/6.50 Derived: plus_plus(nat,times_times(nat,A,B),times_times(nat,C,D)) != plus_plus(nat,times_times(nat,A,D),times_times(nat,C,B)) | ti(nat,A) = ti(nat,C) | ti(nat,B) = ti(nat,D). [resolve(1550,a,1544,a)]. 6.24/6.50 Derived: plus_plus(nat,times_times(nat,A,B),times_times(nat,C,D)) = plus_plus(nat,times_times(nat,A,D),times_times(nat,C,B)) | ti(nat,A) != ti(nat,C). [resolve(1550,a,1545,a)]. 6.24/6.50 Derived: ti(nat,A) != zero_zero(nat) | ti(nat,B) = plus_plus(nat,B,A). [resolve(1550,a,1548,a)]. 6.24/6.50 Derived: ti(nat,A) = zero_zero(nat) | ti(nat,B) != plus_plus(nat,B,A). [resolve(1550,a,1549,a)]. 6.24/6.50 1551 -ordered_ab_group_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),minus_minus(A,B,C)),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) # label(fact_406_less__iff__diff__less__0) # label(axiom). [clausify(508)]. 6.24/6.50 1552 ordered_ab_group_add(int) # label(arity_Int_Oint___Groups_Oordered__ab__group__add) # label(axiom). [assumption]. 6.34/6.53 1553 -ordered_ab_group_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),minus_minus(A,B,C)),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) # label(fact_406_less__iff__diff__less__0) # label(axiom). [clausify(508)]. 6.34/6.53 1554 -ordered_ab_group_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(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_238_le__iff__diff__le__0) # label(axiom). [clausify(525)]. 6.34/6.53 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(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(1554,a,1552,a)]. 6.34/6.53 1555 -ordered_ab_group_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(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_238_le__iff__diff__le__0) # label(axiom). [clausify(525)]. 6.34/6.53 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(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(1555,a,1552,a)]. 6.34/6.53 1556 -ordered_ab_group_add(A) | minus_minus(A,B,C) != minus_minus(A,D,E) | -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)) # label(fact_324_diff__eq__diff__less) # label(axiom). [clausify(935)]. 6.34/6.53 Derived: minus_minus(int,A,B) != minus_minus(int,C,D) | -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)). [resolve(1556,a,1552,a)]. 6.34/6.53 1557 -ordered_ab_group_add(A) | minus_minus(A,B,C) != minus_minus(A,D,E) | 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)) # label(fact_324_diff__eq__diff__less) # label(axiom). [clausify(935)]. 6.34/6.53 Derived: minus_minus(int,A,B) != minus_minus(int,C,D) | 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)). [resolve(1557,a,1552,a)]. 6.34/6.53 1558 ordered_ab_group_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__group__add) # label(axiom). [assumption]. 6.34/6.53 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),minus_minus(real,A,B)),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)). [resolve(1558,a,1551,a)]. 6.34/6.53 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),minus_minus(real,A,B)),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)). [resolve(1558,a,1553,a)]. 6.34/6.53 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),minus_minus(real,A,B)),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)). [resolve(1558,a,1554,a)]. 6.34/6.53 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),minus_minus(real,A,B)),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)). [resolve(1558,a,1555,a)]. 6.34/6.53 Derived: minus_minus(real,A,B) != minus_minus(real,C,D) | -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)). [resolve(1558,a,1556,a)]. 6.34/6.53 Derived: minus_minus(real,A,B) != minus_minus(real,C,D) | 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)). [resolve(1558,a,1557,a)]. 6.34/6.53 1559 -ordered_ab_group_add(A) | minus_minus(A,B,C) != minus_minus(A,D,E) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_249_diff__eq__diff__less__eq) # label(axiom). [clausify(995)]. 6.34/6.53 Derived: minus_minus(int,A,B) != minus_minus(int,C,D) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),D)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(1559,a,1552,a)]. 6.34/6.53 Derived: minus_minus(real,A,B) != minus_minus(real,C,D) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)). [resolve(1559,a,1558,a)]. 6.41/6.64 1560 -ordered_ab_group_add(A) | minus_minus(A,B,C) != minus_minus(A,D,E) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_249_diff__eq__diff__less__eq) # label(axiom). [clausify(995)]. 6.41/6.64 Derived: minus_minus(int,A,B) != minus_minus(int,C,D) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),D)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(1560,a,1552,a)]. 6.41/6.64 Derived: minus_minus(real,A,B) != minus_minus(real,C,D) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),D)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)). [resolve(1560,a,1558,a)]. 6.41/6.64 1561 idom(int) # label(arity_Int_Oint___Rings_Oidom) # label(axiom). [assumption]. 6.41/6.64 1562 -idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),times_times(A,B,C)),times_times(A,D,C))) | ti(A,C) = zero_zero(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) # label(fact_741_dvd__mult__cancel__right) # label(axiom). [clausify(189)]. 6.41/6.64 1563 -idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),times_times(A,B,C)),times_times(A,D,C))) | ti(A,C) != zero_zero(A) # label(fact_741_dvd__mult__cancel__right) # label(axiom). [clausify(189)]. 6.41/6.64 1564 -idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),times_times(A,B,C)),times_times(A,D,C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) # label(fact_741_dvd__mult__cancel__right) # label(axiom). [clausify(189)]. 6.41/6.64 1565 -idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),times_times(A,B,C)),times_times(A,B,D))) | ti(A,B) = zero_zero(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),D)) # label(fact_740_dvd__mult__cancel__left) # label(axiom). [clausify(310)]. 6.41/6.64 1566 -idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),times_times(A,B,C)),times_times(A,B,D))) | ti(A,B) != zero_zero(A) # label(fact_740_dvd__mult__cancel__left) # label(axiom). [clausify(310)]. 6.41/6.64 1567 -idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),times_times(A,B,C)),times_times(A,B,D))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),D)) # label(fact_740_dvd__mult__cancel__left) # label(axiom). [clausify(310)]. 6.41/6.64 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),times_times(int,A,B)),times_times(int,C,B))) | ti(int,B) = zero_zero(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),C)). [resolve(1561,a,1562,a)]. 6.41/6.64 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),times_times(int,A,B)),times_times(int,C,B))) | ti(int,B) != zero_zero(int). [resolve(1561,a,1563,a)]. 6.41/6.64 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),times_times(int,A,B)),times_times(int,C,B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),C)). [resolve(1561,a,1564,a)]. 6.41/6.64 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),times_times(int,A,B)),times_times(int,A,C))) | ti(int,A) != zero_zero(int). [resolve(1561,a,1566,a)]. 6.41/6.64 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),times_times(int,A,B)),times_times(int,A,C))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),C)). [resolve(1561,a,1567,a)]. 6.41/6.64 1568 idom(real) # label(arity_RealDef_Oreal___Rings_Oidom) # label(axiom). [assumption]. 6.41/6.64 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),times_times(real,A,B)),times_times(real,C,B))) | ti(real,B) = zero_zero(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),C)). [resolve(1568,a,1562,a)]. 6.41/6.64 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),times_times(real,A,B)),times_times(real,C,B))) | ti(real,B) != zero_zero(real). [resolve(1568,a,1563,a)]. 6.41/6.64 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),times_times(real,A,B)),times_times(real,C,B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),C)). [resolve(1568,a,1564,a)]. 6.94/7.12 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),times_times(real,A,B)),times_times(real,A,C))) | ti(real,A) = zero_zero(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),B),C)). [resolve(1568,a,1565,a)]. 6.94/7.12 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),times_times(real,A,B)),times_times(real,A,C))) | ti(real,A) != zero_zero(real). [resolve(1568,a,1566,a)]. 6.94/7.12 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),times_times(real,A,B)),times_times(real,A,C))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),B),C)). [resolve(1568,a,1567,a)]. 6.94/7.12 1569 -semiring_char_0(A) | hAPP(nat,A,semiring_1_of_nat(A),B) != hAPP(nat,A,semiring_1_of_nat(A),C) | B = C # label(fact_158_of__nat__eq__iff) # label(axiom). [clausify(847)]. 6.94/7.12 1570 semiring_char_0(int) # label(arity_Int_Oint___Nat_Osemiring__char__0) # label(axiom). [assumption]. 6.94/7.12 1571 semiring_char_0(nat) # label(arity_Nat_Onat___Nat_Osemiring__char__0) # label(axiom). [assumption]. 6.94/7.12 1572 semiring_char_0(real) # label(arity_RealDef_Oreal___Nat_Osemiring__char__0) # label(axiom). [assumption]. 6.94/7.12 Derived: hAPP(nat,int,semiring_1_of_nat(int),A) != hAPP(nat,int,semiring_1_of_nat(int),B) | A = B. [resolve(1569,a,1570,a)]. 6.94/7.12 Derived: hAPP(nat,nat,semiring_1_of_nat(nat),A) != hAPP(nat,nat,semiring_1_of_nat(nat),B) | A = B. [resolve(1569,a,1571,a)]. 6.94/7.12 Derived: hAPP(nat,real,semiring_1_of_nat(real),A) != hAPP(nat,real,semiring_1_of_nat(real),B) | A = B. [resolve(1569,a,1572,a)]. 6.94/7.12 1573 -semiring_char_0(A) | hAPP(nat,A,semiring_1_of_nat(A),B) = hAPP(nat,A,semiring_1_of_nat(A),C) | B != C # label(fact_158_of__nat__eq__iff) # label(axiom). [clausify(847)]. 6.94/7.12 Derived: hAPP(nat,int,semiring_1_of_nat(int),A) = hAPP(nat,int,semiring_1_of_nat(int),B) | A != B. [resolve(1573,a,1570,a)]. 6.94/7.12 Derived: hAPP(nat,nat,semiring_1_of_nat(nat),A) = hAPP(nat,nat,semiring_1_of_nat(nat),B) | A != B. [resolve(1573,a,1571,a)]. 6.94/7.12 Derived: hAPP(nat,real,semiring_1_of_nat(real),A) = hAPP(nat,real,semiring_1_of_nat(real),B) | A != B. [resolve(1573,a,1572,a)]. 6.94/7.12 1574 -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),times_times(A,D,B)),times_times(A,D,C))) # label(fact_275_comm__mult__left__mono) # label(axiom). [clausify(399)]. 6.94/7.12 1575 ordere1490568538miring(real) # label(arity_RealDef_Oreal___Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 6.94/7.12 1576 ordere1490568538miring(nat) # label(arity_Nat_Onat___Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 6.94/7.12 1577 ordere1490568538miring(int) # label(arity_Int_Oint___Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 6.94/7.12 1578 ab_semigroup_add(real) # label(arity_RealDef_Oreal___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 6.94/7.12 1579 -ab_semigroup_add(A) | plus_plus(A,plus_plus(A,B,C),D) = plus_plus(A,B,plus_plus(A,C,D)) # label(fact_206_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom). [clausify(292)]. 6.94/7.12 1580 -ab_semigroup_add(A) | ti(A,plus_plus(A,B,C)) = plus_plus(A,B,C) # label(tsy_c_Groups_Oplus__class_Oplus_1_res) # label(axiom). [clausify(436)]. 6.94/7.12 1581 ab_semigroup_add(nat) # label(arity_Nat_Onat___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 6.94/7.12 1582 -ab_semigroup_add(A) | plus_plus(A,ti(A,B),C) = plus_plus(A,B,C) # label(tsy_c_Groups_Oplus__class_Oplus_1_arg1) # label(axiom). [clausify(690)]. 6.94/7.12 1583 ab_semigroup_add(int) # label(arity_Int_Oint___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 6.94/7.12 Derived: plus_plus(int,plus_plus(int,A,B),C) = plus_plus(int,A,plus_plus(int,B,C)). [resolve(1583,a,1579,a)]. 6.94/7.12 1584 -ab_semigroup_add(A) | plus_plus(A,B,ti(A,C)) = plus_plus(A,B,C) # label(tsy_c_Groups_Oplus__class_Oplus_1_arg2) # label(axiom). [clausify(953)]. 6.94/7.12 1585 -ring_1(A) | times_times(A,plus_plus(A,B,one_one(A)),minus_minus(A,B,one_one(A))) = minus_minus(A,times_times(A,B,B),one_one(A)) # label(fact_297_real__squared__diff__one__factored) # label(axiom). [clausify(773)]. 7.27/7.48 1586 ring_1(int) # label(arity_Int_Oint___Rings_Oring__1) # label(axiom). [assumption]. 7.27/7.48 1587 ring_1(real) # label(arity_RealDef_Oreal___Rings_Oring__1) # label(axiom). [assumption]. 7.27/7.48 Derived: times_times(int,plus_plus(int,A,one_one(int)),minus_minus(int,A,one_one(int))) = minus_minus(int,times_times(int,A,A),one_one(int)). [resolve(1585,a,1586,a)]. 7.27/7.48 Derived: times_times(real,plus_plus(real,A,one_one(real)),minus_minus(real,A,one_one(real))) = minus_minus(real,times_times(real,A,A),one_one(real)). [resolve(1585,a,1587,a)]. 7.27/7.48 1588 -ring_1(A) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) | minus_minus(A,hAPP(nat,A,semiring_1_of_nat(A),C),hAPP(nat,A,semiring_1_of_nat(A),B)) = hAPP(nat,A,semiring_1_of_nat(A),minus_minus(nat,C,B)) # label(fact_682_of__nat__diff) # label(axiom). [clausify(844)]. 7.27/7.48 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | minus_minus(int,hAPP(nat,int,semiring_1_of_nat(int),B),hAPP(nat,int,semiring_1_of_nat(int),A)) = hAPP(nat,int,semiring_1_of_nat(int),minus_minus(nat,B,A)). [resolve(1588,a,1586,a)]. 7.27/7.48 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | minus_minus(real,hAPP(nat,real,semiring_1_of_nat(real),B),hAPP(nat,real,semiring_1_of_nat(real),A)) = hAPP(nat,real,semiring_1_of_nat(real),minus_minus(nat,B,A)). [resolve(1588,a,1587,a)]. 7.27/7.48 1589 -ring_n68954251visors(A) | ti(A,B) != zero_zero(A) | times_times(A,C,B) = zero_zero(A) # label(fact_330_mult__eq__0__iff) # label(axiom). [clausify(511)]. 7.27/7.48 1590 ring_n68954251visors(int) # label(arity_Int_Oint___Rings_Oring__no__zero__divisors) # label(axiom). [assumption]. 7.27/7.48 Derived: ti(int,A) != zero_zero(int) | times_times(int,B,A) = zero_zero(int). [resolve(1589,a,1590,a)]. 7.27/7.48 1591 -ring_n68954251visors(A) | ti(A,B) != zero_zero(A) | times_times(A,B,C) = zero_zero(A) # label(fact_330_mult__eq__0__iff) # label(axiom). [clausify(511)]. 7.27/7.48 Derived: ti(int,A) != zero_zero(int) | times_times(int,A,B) = zero_zero(int). [resolve(1591,a,1590,a)]. 7.27/7.48 1592 -ring_n68954251visors(A) | ti(A,B) = zero_zero(A) | ti(A,C) = zero_zero(A) | times_times(A,C,B) != zero_zero(A) # label(fact_330_mult__eq__0__iff) # label(axiom). [clausify(511)]. 7.27/7.48 Derived: ti(int,A) = zero_zero(int) | ti(int,B) = zero_zero(int) | times_times(int,B,A) != zero_zero(int). [resolve(1592,a,1590,a)]. 7.27/7.48 1593 ring_n68954251visors(real) # label(arity_RealDef_Oreal___Rings_Oring__no__zero__divisors) # label(axiom). [assumption]. 7.27/7.48 Derived: ti(real,A) != zero_zero(real) | times_times(real,B,A) = zero_zero(real). [resolve(1593,a,1589,a)]. 7.27/7.48 Derived: ti(real,A) != zero_zero(real) | times_times(real,A,B) = zero_zero(real). [resolve(1593,a,1591,a)]. 7.27/7.48 Derived: ti(real,A) = zero_zero(real) | ti(real,B) = zero_zero(real) | times_times(real,B,A) != zero_zero(real). [resolve(1593,a,1592,a)]. 7.27/7.48 1594 -one(A) | ti(A,one_one(A)) = one_one(A) # label(tsy_c_Groups_Oone__class_Oone_res) # label(axiom). [clausify(519)]. 7.27/7.48 1595 one(nat) # label(arity_Nat_Onat___Groups_Oone) # label(axiom). [assumption]. 7.27/7.48 Derived: ti(nat,one_one(nat)) = one_one(nat). [resolve(1594,a,1595,a)]. 7.27/7.48 1596 one(int) # label(arity_Int_Oint___Groups_Oone) # label(axiom). [assumption]. 7.27/7.48 Derived: ti(int,one_one(int)) = one_one(int). [resolve(1596,a,1594,a)]. 7.27/7.48 1597 one(real) # label(arity_RealDef_Oreal___Groups_Oone) # label(axiom). [assumption]. 7.27/7.48 Derived: ti(real,one_one(real)) = one_one(real). [resolve(1597,a,1594,a)]. 7.27/7.48 1598 ordered_ring_abs(real) # label(arity_RealDef_Oreal___Rings_Oordered__ring__abs) # label(axiom). [assumption]. 7.27/7.48 1599 -ordered_ring_abs(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)) | times_times(A,abs_abs(A,B),abs_abs(A,C)) = abs_abs(A,times_times(A,B,C)) # label(fact_925_abs__eq__mult) # label(axiom). [clausify(424)]. 7.27/7.48 1600 -ordered_ring_abs(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))) | times_times(A,abs_abs(A,B),abs_abs(A,C)) = abs_abs(A,times_times(A,B,C)) # label(fact_925_abs__eq__mult) # label(axiom). [clausify(424)]. 8.50/8.73 1601 -ordered_ring_abs(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)) | times_times(A,abs_abs(A,B),abs_abs(A,C)) = abs_abs(A,times_times(A,B,C)) # label(fact_925_abs__eq__mult) # label(axiom). [clausify(424)]. 8.50/8.73 1602 -ordered_ring_abs(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))) | times_times(A,abs_abs(A,B),abs_abs(A,C)) = abs_abs(A,times_times(A,B,C)) # label(fact_925_abs__eq__mult) # label(axiom). [clausify(424)]. 8.50/8.73 1603 ordered_ring_abs(int) # label(arity_Int_Oint___Rings_Oordered__ring__abs) # label(axiom). [assumption]. 8.50/8.73 1604 cancel146912293up_add(nat) # label(arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 8.50/8.73 1605 -cancel146912293up_add(A) | plus_plus(A,B,C) != plus_plus(A,B,D) | ti(A,C) = ti(A,D) # label(fact_200_add__imp__eq) # label(axiom). [clausify(514)]. 8.50/8.73 1606 cancel146912293up_add(int) # label(arity_Int_Oint___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 8.50/8.73 1607 cancel146912293up_add(real) # label(arity_RealDef_Oreal___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 8.50/8.73 1608 -ring_char_0(A) | -number_ring(A) | number_number_of(A,B) != number_number_of(A,C) | ti(int,B) = ti(int,C) # label(fact_40_eq__number__of) # label(axiom). [clausify(754)]. 8.50/8.73 1609 ring_char_0(int) # label(arity_Int_Oint___Int_Oring__char__0) # label(axiom). [assumption]. 8.50/8.73 Derived: -number_ring(int) | number_number_of(int,A) != number_number_of(int,B) | ti(int,A) = ti(int,B). [resolve(1608,a,1609,a)]. 8.50/8.73 1610 -ring_char_0(A) | -number_ring(A) | number_number_of(A,B) = number_number_of(A,C) | ti(int,B) != ti(int,C) # label(fact_40_eq__number__of) # label(axiom). [clausify(754)]. 8.50/8.73 Derived: -number_ring(int) | number_number_of(int,A) = number_number_of(int,B) | ti(int,A) != ti(int,B). [resolve(1610,a,1609,a)]. 8.50/8.73 1611 ring_char_0(real) # label(arity_RealDef_Oreal___Int_Oring__char__0) # label(axiom). [assumption]. 8.50/8.73 Derived: -number_ring(real) | number_number_of(real,A) != number_number_of(real,B) | ti(int,A) = ti(int,B). [resolve(1611,a,1608,a)]. 8.50/8.73 Derived: -number_ring(real) | number_number_of(real,A) = number_number_of(real,B) | ti(int,A) != ti(int,B). [resolve(1611,a,1610,a)]. 8.50/8.73 8.50/8.73 ============================== end predicate elimination ============= 8.50/8.73 8.50/8.73 Auto_denials: (non-Horn, no changes). 8.50/8.73 8.50/8.73 Term ordering decisions: 8.50/8.73 Function symbol KB weights: int=1. nat=1. bool=1. real=1. pls=1. min=1. m=1. zprime=1. n=1. s=1. t=1. s1=1. twoSqu658283162sum2sq=1. fFalse=1. fTrue=1. tn=1. x=1. y=1. m1=1. c1=1. c2=1. c3=1. c4=1. c5=1. c6=1. c7=1. c8=1. c9=1. c10=1. c11=1. c12=1. fun=1. number_number_of=1. ti=1. power_power=1. abs_abs=1. zcong=1. legendre=1. f3=1. f4=1. f6=1. f7=1. f10=1. f13=1. f16=1. f26=1. f29=1. zero_zero=1. ord_less_eq=1. ord_less=1. one_one=1. bit1=1. dvd_dvd=1. bit0=1. semiring_1_of_nat=1. nat_1=1. succ=1. quadRes=1. undefined=1. f1=1. f8=1. f9=1. f14=1. f15=1. f20=1. f21=1. f22=1. f24=1. f25=1. times_times=1. plus_plus=1. minus_minus=1. f2=1. f5=1. f11=1. f12=1. f17=1. f18=1. f19=1. f23=1. f27=1. f28=1. f30=1. f31=1. hAPP=1. if=1. 8.50/8.73 8.50/8.73 ============================== end of process initial clauses ======== 8.50/8.73 8.50/8.73 ============================== CLAUSES FOR SEARCH ==================== 8.50/8.73 8.50/8.73 ============================== end of clauses for search ============= 8.50/8.73 8.50/8.73 ============================== SEARCH ================================ 8.50/8.73 8.50/8.73 % Starting search at 6.42 seconds. 8.50/8.73 8.50/8.73 Low Water (keep): wt=66.000, iters=3410 8.50/8.73 8.50/8.73 Low Water (keep): wt=64.000, iters=3382 8.50/8.73 8.50/8.73 Low Water (keep): wt=60.000, iters=3344 8.50/8.73 8.50/8.73 Low Water (keep): wt=57.000, iters=3475 8.50/8.73 8.50/8.73 Low Water (keep): wt=55.000, iters=3395 8.50/8.73 8.50/8.73 Low Water (keep): wt=54.000, iters=3419 8.50/8.73 8.50/8.73 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 38 (0.00 of 6.95 sec). 8.50/8.73 8.50/8.73 Low Water (keep): wt=53.000, iters=3406 8.50/8.73 8.50/8.73 Low Water (keep): wt=52.000, iters=3396 8.50/8.73 8.50/8.73 Low Water (keep): wt=49.000, iters=3334 9.76/10.02 9.76/10.02 Low Water (keep): wt=48.000, iters=3337 9.76/10.02 9.76/10.02 Low Water (keep): wt=47.000, iters=3335 9.76/10.02 9.76/10.02 Low Water (keep): wt=45.000, iters=3335 9.76/10.02 9.76/10.02 Low Water (keep): wt=44.000, iters=3399 9.76/10.02 9.76/10.02 Low Water (keep): wt=43.000, iters=3371 9.76/10.02 9.76/10.02 Low Water (keep): wt=42.000, iters=3363 9.76/10.02 9.76/10.02 Low Water (keep): wt=41.000, iters=3382 9.76/10.02 9.76/10.02 Low Water (keep): wt=40.000, iters=3382 9.76/10.02 9.76/10.02 Low Water (keep): wt=39.000, iters=3341 9.76/10.02 9.76/10.02 Low Water (keep): wt=38.000, iters=3353 9.76/10.02 9.76/10.02 Low Water (keep): wt=37.000, iters=3352 9.76/10.02 9.76/10.02 Low Water (keep): wt=36.000, iters=3359 9.76/10.02 9.76/10.02 Low Water (keep): wt=35.000, iters=3349 9.76/10.02 9.76/10.02 Low Water (keep): wt=34.000, iters=3349 9.76/10.02 9.76/10.02 Low Water (keep): wt=33.000, iters=3429 9.76/10.02 9.76/10.02 Low Water (keep): wt=32.000, iters=3367 9.76/10.02 9.76/10.02 Low Water (keep): wt=31.000, iters=3350 9.76/10.02 9.76/10.02 Low Water (keep): wt=30.000, iters=3388 9.76/10.02 9.76/10.02 Low Water (keep): wt=29.000, iters=3383 9.76/10.02 9.76/10.02 Low Water (keep): wt=28.000, iters=3349 9.76/10.02 9.76/10.02 Low Water (keep): wt=27.000, iters=3337 9.76/10.02 9.76/10.02 ============================== PROOF ================================= 9.76/10.02 % SZS status Theorem 9.76/10.02 % SZS output start Refutation 9.76/10.02 9.76/10.02 % Proof 1 at 8.25 (+ 0.05) seconds. 9.76/10.02 % Length of proof is 76. 9.76/10.02 % Level of proof is 7. 9.76/10.02 % Maximum clause weight is 30.000. 9.76/10.02 % Given clauses 1540. 9.76/10.02 9.76/10.02 4 (all B_1_1 all X_a (number(X_a) -> number_number_of(X_a,B_1_1) = number_number_of(X_a,ti(int,B_1_1)))) # label(tsy_c_Int_Onumber__class_Onumber__of_arg1) # label(axiom) # label(non_clause). [assumption]. 9.76/10.02 23 (all W nat_1(number_number_of(int,W)) = number_number_of(nat,W)) # label(fact_426_nat__number__of) # label(axiom) # label(non_clause). [assumption]. 9.76/10.02 128 (all K_1 ti(int,K_1) = plus_plus(int,pls,K_1)) # label(fact_76_add__Pls) # label(axiom) # label(non_clause). [assumption]. 9.76/10.02 212 (all B_1_1 nat_1(B_1_1) = nat_1(ti(int,B_1_1))) # label(tsy_c_Int_Onat_arg1) # label(axiom) # label(non_clause). [assumption]. 9.76/10.02 259 (all V_1 all W number_number_of(int,plus_plus(int,V_1,W)) = plus_plus(int,number_number_of(int,V_1),number_number_of(int,W))) # label(fact_17_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption]. 9.76/10.02 276 (all X_a (number_semiring(X_a) -> plus_plus(X_a,one_one(X_a),one_one(X_a)) = number_number_of(X_a,bit0(bit1(pls))))) # label(fact_9_semiring__one__add__one__is__two) # label(axiom) # label(non_clause). [assumption]. 9.76/10.02 422 (all K_1 minus_minus(int,K_1,min) = succ(K_1)) # label(fact_777_diff__bin__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.76/10.02 465 (all K_1 bit1(K_1) = plus_plus(int,plus_plus(int,one_one(int),K_1),K_1)) # label(fact_92_Bit1__def) # label(axiom) # label(non_clause). [assumption]. 9.76/10.02 647 (all K_1 plus_plus(int,K_1,K_1) = bit0(K_1)) # label(fact_78_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 9.76/10.02 687 (all Z all W plus_plus(int,W,Z) = plus_plus(int,Z,W)) # label(fact_47_zadd__commute) # label(axiom) # label(non_clause). [assumption]. 9.76/10.02 707 (all B_1_1 all B_2_1 all X_a (group_add(X_a) -> minus_minus(X_a,B_1_1,B_2_1) = ti(X_a,minus_minus(X_a,B_1_1,B_2_1)))) # label(tsy_c_Groups_Ominus__class_Ominus_0_res) # label(axiom) # label(non_clause). [assumption]. 9.76/10.02 730 (all K_1 number_number_of(int,K_1) = ti(int,K_1)) # label(fact_120_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 9.76/10.02 825 (all Z_1 all Wa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_1),Wa)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),Wa)) & ti(int,Z_1) != ti(int,Wa))) # label(fact_380_zless__le) # label(axiom) # label(non_clause). [assumption]. 9.76/10.02 891 (all L minus_minus(int,pls,bit0(L)) = bit0(minus_minus(int,pls,L))) # label(fact_645_diff__bin__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.76/10.02 898 (all K_1 plus_plus(int,K_1,one_one(int)) = succ(K_1)) # label(fact_519_succ__def) # label(axiom) # label(non_clause). [assumption]. 9.76/10.02 939 (all B_1_1 all B_2_1 all X_a (cancel_semigroup_add(X_a) -> plus_plus(X_a,B_1_1,B_2_1) = ti(X_a,plus_plus(X_a,B_1_1,B_2_1)))) # label(tsy_c_Groups_Oplus__class_Oplus_2_res) # label(axiom) # label(non_clause). [assumption]. 9.76/10.02 946 (all X_a (power(X_a) & no_zero_divisors(X_a) & zero_neq_one(X_a) & mult_zero(X_a) -> (all A_2 all Na (zero_zero(X_a) = ti(X_a,A_2) & Na != zero_zero(nat) <-> zero_zero(X_a) = hAPP(nat,X_a,power_power(X_a,A_2),Na))))) # label(fact_185_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.85/10.02 1011 -(zero_zero(int) != hAPP(nat,int,power_power(int,plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n))),number_number_of(nat,bit0(bit1(pls))))) # label(conj_0) # label(negated_conjecture) # label(non_clause). [assumption]. 9.85/10.02 1033 number(int) # label(arity_Int_Oint___Int_Onumber) # label(axiom). [assumption]. 9.85/10.02 1034 -number(A) | number_number_of(A,ti(int,B)) = number_number_of(A,B) # label(tsy_c_Int_Onumber__class_Onumber__of_arg1) # label(axiom). [clausify(4)]. 9.85/10.02 1127 mult_zero(int) # label(arity_Int_Oint___Rings_Omult__zero) # label(axiom). [assumption]. 9.85/10.02 1131 -power(A) | -no_zero_divisors(A) | -zero_neq_one(A) | -mult_zero(A) | ti(A,B) = zero_zero(A) | hAPP(nat,A,power_power(A,B),C) != zero_zero(A) # label(fact_185_power__eq__0__iff) # label(axiom). [clausify(946)]. 9.85/10.02 1214 number_semiring(int) # label(arity_Int_Oint___Int_Onumber__semiring) # label(axiom). [assumption]. 9.85/10.02 1217 -number_semiring(A) | number_number_of(A,bit0(bit1(pls))) = plus_plus(A,one_one(A),one_one(A)) # label(fact_9_semiring__one__add__one__is__two) # label(axiom). [clausify(276)]. 9.85/10.02 1233 cancel_semigroup_add(int) # label(arity_Int_Oint___Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 9.85/10.02 1238 -cancel_semigroup_add(A) | ti(A,plus_plus(A,B,C)) = plus_plus(A,B,C) # label(tsy_c_Groups_Oplus__class_Oplus_2_res) # label(axiom). [clausify(939)]. 9.85/10.02 1240 group_add(int) # label(arity_Int_Oint___Groups_Ogroup__add) # label(axiom). [assumption]. 9.85/10.02 1248 -group_add(A) | minus_minus(A,B,C) = ti(A,minus_minus(A,B,C)) # label(tsy_c_Groups_Ominus__class_Ominus_0_res) # label(axiom). [clausify(707)]. 9.85/10.02 1617 zero_neq_one(int) # label(arity_Int_Oint___Rings_Ozero__neq__one) # label(axiom). [assumption]. 9.85/10.02 1628 power(int) # label(arity_Int_Oint___Power_Opower) # label(axiom). [assumption]. 9.85/10.02 1631 nat_1(number_number_of(int,A)) = number_number_of(nat,A) # label(fact_426_nat__number__of) # label(axiom). [clausify(23)]. 9.85/10.02 1643 pls = zero_zero(int) # label(fact_73_Pls__def) # label(axiom). [assumption]. 9.85/10.02 1783 ti(int,A) = plus_plus(int,pls,A) # label(fact_76_add__Pls) # label(axiom). [clausify(128)]. 9.85/10.02 1784 plus_plus(int,zero_zero(int),A) = ti(int,A). [copy(1783),rewrite([1643(4)]),flip(a)]. 9.85/10.02 1901 nat_1(ti(int,A)) = nat_1(A) # label(tsy_c_Int_Onat_arg1) # label(axiom). [clausify(212)]. 9.85/10.02 1951 number_number_of(int,plus_plus(int,A,B)) = plus_plus(int,number_number_of(int,A),number_number_of(int,B)) # label(fact_17_plus__numeral__code_I9_J) # label(axiom). [clausify(259)]. 9.85/10.02 1952 plus_plus(int,number_number_of(int,A),number_number_of(int,B)) = number_number_of(int,plus_plus(int,A,B)). [copy(1951),flip(a)]. 9.85/10.02 2141 succ(A) = minus_minus(int,A,min) # label(fact_777_diff__bin__simps_I2_J) # label(axiom). [clausify(422)]. 9.85/10.02 2154 zero_zero(int) != one_one(int) # label(fact_74_int__0__neq__1) # label(axiom). [assumption]. 9.85/10.02 2155 one_one(int) != zero_zero(int). [copy(2154),flip(a)]. 9.85/10.02 2202 bit1(A) = plus_plus(int,plus_plus(int,one_one(int),A),A) # label(fact_92_Bit1__def) # label(axiom). [clausify(465)]. 9.85/10.02 2238 no_zero_divisors(int) # label(arity_Int_Oint___Rings_Ono__zero__divisors) # label(axiom). [assumption]. 9.85/10.02 2329 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n)))) # label(fact_0_n1pos) # label(axiom). [assumption]. 9.85/10.02 2451 bit0(A) = plus_plus(int,A,A) # label(fact_78_Bit0__def) # label(axiom). [clausify(647)]. 9.85/10.02 2508 plus_plus(int,A,B) = plus_plus(int,B,A) # label(fact_47_zadd__commute) # label(axiom). [clausify(687)]. 9.85/10.02 2552 number_number_of(int,A) = ti(int,A) # label(fact_120_number__of__is__id) # label(axiom). [clausify(730)]. 9.85/10.02 2553 ti(int,A) = number_number_of(int,A). [copy(2552),flip(a)]. 9.85/10.02 2604 zero_zero(int) = number_number_of(int,pls) # label(fact_19_zero__is__num__zero) # label(axiom). [assumption]. 9.85/10.02 2605 number_number_of(int,zero_zero(int)) = zero_zero(int). [copy(2604),rewrite([1643(4)]),flip(a)]. 9.85/10.02 2701 -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | ti(int,B) != ti(int,A) # label(fact_380_zless__le) # label(axiom). [clausify(825)]. 9.85/10.02 2702 -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | number_number_of(int,B) != number_number_of(int,A). [copy(2701),rewrite([2553(13),2553(15)])]. 9.85/10.02 2783 bit0(minus_minus(int,pls,A)) = minus_minus(int,pls,bit0(A)) # label(fact_645_diff__bin__simps_I3_J) # label(axiom). [clausify(891)]. 9.85/10.02 2784 plus_plus(int,minus_minus(int,zero_zero(int),A),minus_minus(int,zero_zero(int),A)) = minus_minus(int,zero_zero(int),plus_plus(int,A,A)). [copy(2783),rewrite([1643(2),2451(5),1643(12),2451(14)])]. 9.85/10.02 2793 succ(A) = plus_plus(int,A,one_one(int)) # label(fact_519_succ__def) # label(axiom). [clausify(898)]. 9.85/10.02 2794 plus_plus(int,A,one_one(int)) = minus_minus(int,A,min). [copy(2793),rewrite([2141(1)]),flip(a)]. 9.85/10.02 2917 hAPP(nat,int,power_power(int,plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n))),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(int) # label(conj_0) # label(negated_conjecture). [clausify(1011)]. 9.85/10.02 2918 hAPP(nat,int,power_power(int,plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n))),number_number_of(nat,number_number_of(int,minus_minus(int,zero_zero(int),plus_plus(int,min,min))))) = zero_zero(int). [copy(2917),rewrite([1643(16),2202(18),2508(22),2794(22),2508(24),1784(24),2553(22),2451(23),1952(31),2784(28)])]. 9.85/10.02 2993 number_number_of(int,ti(int,A)) = number_number_of(int,A). [resolve(1033,a,1034,a)]. 9.85/10.02 2994 number_number_of(int,number_number_of(int,A)) = number_number_of(int,A). [copy(2993),rewrite([2553(3)])]. 9.85/10.02 3228 -power(int) | -no_zero_divisors(int) | -zero_neq_one(int) | ti(int,A) = zero_zero(int) | hAPP(nat,int,power_power(int,A),B) != zero_zero(int). [resolve(1131,d,1127,a)]. 9.85/10.02 3229 number_number_of(int,A) = zero_zero(int) | hAPP(nat,int,power_power(int,A),B) != zero_zero(int). [copy(3228),rewrite([2553(8)]),unit_del(a,1628),unit_del(b,2238),unit_del(c,1617)]. 9.85/10.02 3446 number_number_of(int,bit0(bit1(pls))) = plus_plus(int,one_one(int),one_one(int)). [resolve(1217,a,1214,a)]. 9.85/10.02 3447 number_number_of(int,minus_minus(int,zero_zero(int),plus_plus(int,min,min))) = minus_minus(int,one_one(int),min). [copy(3446),rewrite([1643(2),2202(4),2508(8),2794(8),2508(10),1784(10),2553(8),2451(9),1952(17),2784(14),2994(12),2794(16)])]. 9.85/10.02 3517 ti(int,plus_plus(int,A,B)) = plus_plus(int,A,B). [resolve(1238,a,1233,a)]. 9.85/10.02 3518 number_number_of(int,plus_plus(int,A,B)) = plus_plus(int,A,B). [copy(3517),rewrite([2553(4)])]. 9.85/10.02 3534 minus_minus(int,A,B) = ti(int,minus_minus(int,A,B)). [resolve(1248,a,1240,a)]. 9.85/10.02 3535 number_number_of(int,minus_minus(int,A,B)) = minus_minus(int,A,B). [copy(3534),rewrite([2553(6)]),flip(a)]. 9.85/10.02 4216 one_one(int) = c_0. [new_symbol(2155)]. 9.85/10.02 4385 number_number_of(nat,A) = nat_1(A). [back_rewrite(1901),rewrite([2553(2),1631(3)])]. 9.85/10.02 4454 hAPP(nat,int,power_power(int,plus_plus(int,c_0,hAPP(nat,int,semiring_1_of_nat(int),n))),nat_1(minus_minus(int,zero_zero(int),plus_plus(int,min,min)))) = zero_zero(int). [back_rewrite(2918),rewrite([4216(6),3535(24),4385(23)])]. 9.85/10.02 4509 minus_minus(int,zero_zero(int),plus_plus(int,min,min)) = minus_minus(int,c_0,min). [back_rewrite(3447),rewrite([3535(10),4216(11)])]. 9.85/10.02 4644 minus_minus(int,A,min) = plus_plus(int,A,c_0). [back_rewrite(2794),rewrite([4216(3)]),flip(a)]. 9.85/10.02 4664 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),plus_plus(int,c_0,hAPP(nat,int,semiring_1_of_nat(int),n)))). [back_rewrite(2329),rewrite([4216(14)])]. 9.85/10.02 4835 hAPP(nat,int,power_power(int,plus_plus(int,c_0,hAPP(nat,int,semiring_1_of_nat(int),n))),nat_1(plus_plus(int,c_0,c_0))) = zero_zero(int). [back_rewrite(4454),rewrite([4509(21),4644(17)])]. 9.85/10.02 11831 plus_plus(int,c_0,hAPP(nat,int,semiring_1_of_nat(int),n)) != zero_zero(int). [resolve(4664,a,2702,a),rewrite([3518(11),2605(13)])]. 9.85/10.02 12357 $F. [resolve(4835,a,3229,b),rewrite([3518(11)]),unit_del(a,11831)]. 9.85/10.02 9.85/10.02 % SZS output end Refutation 9.85/10.02 ============================== end of proof ========================== 9.85/10.02 9.85/10.02 ============================== STATISTICS ============================ 9.85/10.02 9.85/10.02 Given=1540. Generated=50032. Kept=9857. proofs=1. 9.85/10.02 Usable=1526. Sos=7181. Demods=909. Limbo=0, Disabled=3741. Hints=0. 9.85/10.02 Megabytes=38.25. 9.85/10.02 User_CPU=8.25, System_CPU=0.05, Wall_clock=9. 9.85/10.02 9.85/10.02 ============================== end of statistics ===================== 9.85/10.02 9.85/10.02 ============================== end of search ========================= 9.85/10.02 9.85/10.02 THEOREM PROVED 9.85/10.02 % SZS status Theorem 9.85/10.02 9.85/10.02 Exiting with 1 proof. 9.85/10.02 9.85/10.02 Process 32689 exit (max_proofs) Tue Jul 13 16:30:31 2021 9.85/10.02 Prover9 interrupted 9.87/10.02 EOF