0.00/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.00/0.12 % Command : tptp2X_and_run_prover9 %d %s 0.12/0.33 % Computer : n017.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 : 960 0.12/0.33 % DateTime : Thu Jul 2 07:35:51 EDT 2020 0.12/0.33 % CPUTime : 1.70/2.06 ============================== Prover9 =============================== 1.70/2.06 Prover9 (32) version 2009-11A, November 2009. 1.70/2.06 Process 4051 was started by sandbox2 on n017.cluster.edu, 1.70/2.06 Thu Jul 2 07:35:53 2020 1.70/2.06 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 960 -f /tmp/Prover9_3892_n017.cluster.edu". 1.70/2.06 ============================== end of head =========================== 1.70/2.06 1.70/2.06 ============================== INPUT ================================= 1.70/2.06 1.70/2.06 % Reading from file /tmp/Prover9_3892_n017.cluster.edu 1.70/2.06 1.70/2.06 set(prolog_style_variables). 1.70/2.06 set(auto2). 1.70/2.06 % set(auto2) -> set(auto). 1.70/2.06 % set(auto) -> set(auto_inference). 1.70/2.06 % set(auto) -> set(auto_setup). 1.70/2.06 % set(auto_setup) -> set(predicate_elim). 1.70/2.06 % set(auto_setup) -> assign(eq_defs, unfold). 1.70/2.06 % set(auto) -> set(auto_limits). 1.70/2.06 % set(auto_limits) -> assign(max_weight, "100.000"). 1.70/2.06 % set(auto_limits) -> assign(sos_limit, 20000). 1.70/2.06 % set(auto) -> set(auto_denials). 1.70/2.06 % set(auto) -> set(auto_process). 1.70/2.06 % set(auto2) -> assign(new_constants, 1). 1.70/2.06 % set(auto2) -> assign(fold_denial_max, 3). 1.70/2.06 % set(auto2) -> assign(max_weight, "200.000"). 1.70/2.06 % set(auto2) -> assign(max_hours, 1). 1.70/2.06 % assign(max_hours, 1) -> assign(max_seconds, 3600). 1.70/2.06 % set(auto2) -> assign(max_seconds, 0). 1.70/2.06 % set(auto2) -> assign(max_minutes, 5). 1.70/2.06 % assign(max_minutes, 5) -> assign(max_seconds, 300). 1.70/2.06 % set(auto2) -> set(sort_initial_sos). 1.70/2.06 % set(auto2) -> assign(sos_limit, -1). 1.70/2.06 % set(auto2) -> assign(lrs_ticks, 3000). 1.70/2.06 % set(auto2) -> assign(max_megs, 400). 1.70/2.06 % set(auto2) -> assign(stats, some). 1.70/2.06 % set(auto2) -> clear(echo_input). 1.70/2.06 % set(auto2) -> set(quiet). 1.70/2.06 % set(auto2) -> clear(print_initial_clauses). 1.70/2.06 % set(auto2) -> clear(print_given). 1.70/2.06 assign(lrs_ticks,-1). 1.70/2.06 assign(sos_limit,10000). 1.70/2.06 assign(order,kbo). 1.70/2.06 set(lex_order_vars). 1.70/2.06 clear(print_given). 1.70/2.06 1.70/2.06 % formulas(sos). % not echoed (1254 formulas) 1.70/2.06 1.70/2.06 ============================== end of input ========================== 1.70/2.06 1.70/2.06 % From the command line: assign(max_seconds, 960). 1.70/2.06 1.70/2.06 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 1.70/2.06 1.70/2.06 % Formulas that are not ordinary clauses: 1.70/2.06 1 (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]. 1.70/2.06 2 (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]. 1.70/2.06 3 (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]. 1.70/2.06 4 (all B_1_1 all B_2_1 ti(nat,minus_minus(nat,B_1_1,B_2_1)) = minus_minus(nat,B_1_1,B_2_1)) # label(tsy_c_Groups_Ominus__class_Ominus_1_res) # label(axiom) # label(non_clause). [assumption]. 1.70/2.06 5 (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]. 1.70/2.06 6 (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]. 1.70/2.06 7 (all X_a (semiring_1(X_a) -> (all M all N hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,power_power(nat,M),N)) = hAPP(nat,X_a,power_power(X_a,hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),N)))) # label(fact_172_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 1.70/2.06 8 (all A_1 all P_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),P_2)) -> hAPP(nat,int,power_power(int,A_1),nat_1(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)))))) # label(fact_555_Euler_Oaux__1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.06 9 (all X_a (linordered_semidom(X_a) -> (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),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))))))) # label(fact_167_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.06 10 (all K_1 all L times_times(int,bit0(K_1),L) = bit0(times_times(int,K_1,L))) # label(fact_371_mult__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.70/2.06 11 (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]. 1.70/2.06 12 (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,B_1) = ti(int,C) -> (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]. 1.70/2.06 13 (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]. 1.70/2.06 14 (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]. 1.70/2.06 15 (all X_a (linord581940658strict(X_a) -> (all Xa all Ya (ti(X_a,Xa) = zero_zero(X_a) & zero_zero(X_a) = ti(X_a,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))))))) # label(fact_311_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.06 16 (all Xa all Ya (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)))) # label(fact_873_divides__antisym) # label(axiom) # label(non_clause). [assumption]. 1.70/2.06 17 -(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)))) & ti(int,V) = minus_minus(int,x,times_times(int,R,plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n)))))) # 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]. 1.70/2.06 18 (all Z1 all Z2 all W plus_plus(int,times_times(int,Z1,W),times_times(int,Z2,W)) = times_times(int,plus_plus(int,Z1,Z2),W)) # label(fact_382_zadd__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.70/2.06 19 (all B_1_1 all B_2_1 all X_a (cancel_semigroup_add(X_a) -> plus_plus(X_a,B_1_1,ti(X_a,B_2_1)) = plus_plus(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Oplus__class_Oplus_2_arg2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.06 20 (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]. 1.70/2.06 21 (all K (ti(int,K) = pls <-> pls = bit0(K))) # label(fact_70_rel__simps_I44_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.06 22 (all P_1 all A_2 all B (-((exists D_3 (A_2 = plus_plus(nat,B,D_3) & -hBOOL(hAPP(nat,bool,P_1,D_3)))) | -hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_2),B))) <-> 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]. 1.70/2.06 23 (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]. 1.70/2.06 24 (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]. 1.70/2.06 25 (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]. 1.70/2.06 26 (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)),bit1(K2))))) # label(fact_30_less__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.06 27 (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]. 1.70/2.06 28 (all Na (zero_zero(nat) = 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))))) # label(fact_512_int__le__0__conv) # label(axiom) # label(non_clause). [assumption]. 1.70/2.06 29 (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]. 1.70/2.06 30 (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]. 1.70/2.06 31 (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,X) != ti(X_a,Y) -> 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]. 1.70/2.06 32 (all Xa all Ya (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Xa),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))))) # label(fact_148_Nat__Transfer_Otransfer__int__nat__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.06 33 (all Wa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(Wa)),zero_zero(int))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),zero_zero(int))))) # label(fact_58_bin__less__0__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.06 34 (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]. 1.70/2.06 35 (all X_a (number_semiring(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_538_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.06 36 (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]. 1.70/2.06 37 (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]. 1.70/2.06 38 (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]. 1.70/2.06 39 (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]. 1.70/2.06 40 (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]. 1.70/2.06 41 (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]. 1.70/2.06 42 (all Va all 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))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Va),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))))) # label(fact_151_less__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.70/2.06 43 (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]. 1.70/2.06 44 (all B_1_1 all B_2_1 all X_a (order(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_1_res) # label(axiom) # label(non_clause). [assumption]. 1.70/2.06 45 (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]. 1.70/2.06 46 (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]. 1.70/2.06 47 (all X_a (ring_n68954251visors(X_a) -> (all A_2 all B (zero_zero(X_a) = ti(X_a,B) | zero_zero(X_a) = ti(X_a,A_2) <-> times_times(X_a,A_2,B) = zero_zero(X_a))))) # label(fact_330_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.06 48 -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]. 1.70/2.06 49 (all X_a (comm_ring(X_a) & dvd(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]. 1.70/2.07 50 (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]. 1.70/2.07 51 (all I_2 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),N)) -> I_2 = minus_minus(nat,N,minus_minus(nat,N,I_2)))) # label(fact_636_diff__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 52 (all P_1 all A_2 all B (hBOOL(hAPP(nat,bool,P_1,minus_minus(nat,A_2,B))) <-> (all D_3 (A_2 = plus_plus(nat,B,D_3) -> hBOOL(hAPP(nat,bool,P_1,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)))))) # label(fact_500_nat__diff__split) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 53 (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]. 1.70/2.07 54 (all X_a (linordered_idom(X_a) -> (all L all K_1 (abs_abs(X_a,L) = abs_abs(X_a,K_1) -> 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]. 1.70/2.07 55 (all Xa all Na (zero_zero(int) != ti(int,Xa) | zero_zero(nat) = Na <-> 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]. 1.70/2.07 56 (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]. 1.70/2.07 57 (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,ti(X_a,B_2_1),B_3)) # label(tsy_c_If_arg2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 58 (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]. 1.70/2.07 59 (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),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_845_dvd_Oless__trans) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 60 (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]. 1.70/2.07 61 (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]. 1.70/2.07 62 (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]. 1.70/2.07 63 (all X_a (monoid_mult(X_a) -> (all A_1 all M all 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)) = hAPP(nat,X_a,power_power(X_a,A_1),plus_plus(nat,M,N))))) # label(fact_459_power__add) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 64 (all M all N hAPP(nat,int,power_power(int,hAPP(nat,int,semiring_1_of_nat(int),M)),N) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,power_power(nat,M),N))) # label(fact_20_zpower__int) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 65 (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]. 1.70/2.07 66 (all X_b (semiring(X_b) & number(X_b) -> (all A_1 all B_1 all V_1 times_times(X_b,plus_plus(X_b,A_1,B_1),number_number_of(X_b,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)))))) # label(fact_444_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 67 (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]. 1.70/2.07 68 (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]. 1.70/2.07 69 (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]. 1.70/2.07 70 (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]. 1.70/2.07 71 (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]. 1.70/2.07 72 (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]. 1.70/2.07 73 (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]. 1.70/2.07 74 (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]. 1.70/2.07 75 (all Na all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Ma)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),times_times(nat,Na,Ma)),Ma)) <-> Na = one_one(nat)))) # label(fact_832_dvd__mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 76 (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]. 1.70/2.07 77 (all L bit1(L) != pls) # label(fact_67_rel__simps_I39_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 78 (all K_1 all L bit1(minus_minus(int,K_1,L)) = minus_minus(int,bit1(K_1),bit0(L))) # label(fact_644_diff__bin__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 79 (all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_1),zero_zero(int))) <-> 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))))) # label(fact_64_odd__less__0) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 80 (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]. 1.70/2.07 81 (all B_1_1 all B_2_1 all X_a (ab_semigroup_mult(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_5_arg1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 82 (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]. 1.70/2.07 83 (all Z times_times(nat,Z,number_number_of(nat,bit0(bit1(pls)))) = plus_plus(nat,Z,Z)) # label(fact_692_nat__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 84 (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]. 1.70/2.07 85 (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]. 1.70/2.07 86 (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))))) -> one_one(int) = ti(int,X) | zero_zero(int) = ti(int,X)))) # label(fact_554_int__pos__lt__two__imp__zero__or__one) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 87 (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]. 1.70/2.07 88 (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]. 1.70/2.07 89 (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(int),bit0(K)),bit1(L_1))))) # label(fact_505_rel__simps_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 90 (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)) | M = zero_zero(nat))) # label(fact_838_divides__cases) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 91 (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]. 1.70/2.07 92 (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]. 1.70/2.07 93 (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]. 1.70/2.07 94 (all Z ti(int,Z) = times_times(int,one_one(int),Z)) # label(fact_378_zmult__1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 95 (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]. 1.70/2.07 96 (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]. 1.70/2.07 97 (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_854_dvd_Oless__not__sym) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 98 (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]. 1.70/2.07 99 (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]. 1.70/2.07 100 (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]. 1.70/2.07 101 (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),B),A_2)) <-> 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_418_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 102 (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,V_1) = 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,plus_plus(int,V_1,V_2)) = 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]. 1.70/2.07 103 (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]. 1.70/2.07 104 (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_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))) <-> zero_zero(X_a) = ti(X_a,Xa) & zero_zero(X_a) = ti(X_a,Ya))))) # label(fact_553_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 105 (all Wa all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),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),Wa),Z_1))))) # label(fact_483_nat__mono__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 106 (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]. 1.70/2.07 107 (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]. 1.70/2.07 108 (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]. 1.70/2.07 109 (all X_a (zero(X_a) -> ti(X_a,zero_zero(X_a)) = zero_zero(X_a))) # label(tsy_c_Groups_Ozero__class_Ozero_res) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 110 (all K_1 all L bit0(minus_minus(int,K_1,L)) = minus_minus(int,bit0(K_1),bit0(L))) # label(fact_591_diff__bin__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 111 (all X abs_abs(int,hAPP(nat,int,power_power(int,X),number_number_of(nat,bit1(bit1(pls))))) = hAPP(nat,int,power_power(int,abs_abs(int,X)),number_number_of(nat,bit1(bit1(pls))))) # label(fact_959_abs__power3__distrib) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 112 (all K_1 all L plus_plus(int,bit0(K_1),bit1(L)) = bit1(plus_plus(int,K_1,L))) # label(fact_91_add__Bit0__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 113 (all B_1_1 all B_2_1 all X_a (order(X_a) -> hAPP(X_a,bool,B_1_1,ti(X_a,B_2_1)) = hAPP(X_a,bool,B_1_1,B_2_1))) # label(tsy_c_hAPP_1_arg2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 114 (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]. 1.70/2.07 115 (all X all Y all Z hAPP(nat,int,power_power(int,X),plus_plus(nat,Y,Z)) = times_times(int,hAPP(nat,int,power_power(int,X),Y),hAPP(nat,int,power_power(int,X),Z))) # label(fact_478_zpower__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 116 (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,V_1),times_times(nat,number_number_of(nat,V_2),K_1)) = times_times(nat,number_number_of(nat,times_times(int,V_1,V_2)),K_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),times_times(nat,number_number_of(nat,V_2),K_1)) = zero_zero(nat)))) # label(fact_693_nat__number__of__mult__left) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 117 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all 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))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),Ya)))))) # label(fact_389_le__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 118 (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]. 1.70/2.07 119 (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]. 1.70/2.07 120 (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]. 1.70/2.07 121 (all N all M ((M = zero_zero(nat) -> zero_zero(nat) = times_times(nat,M,N)) & (M != zero_zero(nat) -> plus_plus(nat,N,times_times(nat,minus_minus(nat,M,one_one(nat)),N)) = times_times(nat,M,N)))) # label(fact_687_mult__eq__if) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 122 (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]. 1.70/2.07 123 (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),C_1),plus_plus(X_a,times_times(X_a,minus_minus(X_a,B,A_2),E_1),D))))))) # label(fact_287_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 124 (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)) -> (Na = Ma <-> minus_minus(nat,Na,K) = minus_minus(nat,Ma,K))))) # label(fact_635_eq__diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 125 (all X_a (dvd(X_a) -> dvd_dvd(X_a) = ti(fun(X_a,fun(X_a,bool)),dvd_dvd(X_a)))) # label(tsy_c_Rings_Odvd__class_Odvd_res) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 126 (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]. 1.70/2.07 127 (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]. 1.70/2.07 128 (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]. 1.70/2.07 129 (all P (ti(bool,P) = fTrue | fFalse = ti(bool,P))) # label(help_If_3_1_T) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 130 (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]. 1.70/2.07 131 (all X_a (linord581940658strict(X_a) -> (all Xa all Ya (zero_zero(X_a) = ti(X_a,Xa) & ti(X_a,Ya) = zero_zero(X_a) <-> 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]. 1.70/2.07 132 (all M hBOOL(hAPP(int,bool,zcong(M,zero_zero(int)),M))) # label(fact_816_zcong__id) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 133 (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]. 1.70/2.07 134 (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]. 1.70/2.07 135 (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_eq(int),bit1(K)),bit0(L_1))))) # label(fact_507_rel__simps_I33_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 136 (all X_a (cancel146912293up_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,C) = ti(X_a,B_1))))) # label(fact_200_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 137 (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]. 1.70/2.07 138 (all K all Ma all Na (plus_plus(nat,K,Na) = plus_plus(nat,K,Ma) <-> Na = Ma)) # label(fact_125_nat__add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 139 -(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]. 1.70/2.07 140 (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),X),Y)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) | Y = X)) # label(fact_864_dvd_Ole__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 141 (all X_b (semiring(X_b) & number(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]. 1.70/2.07 142 (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]. 1.70/2.07 143 (all B_1_1 all B_2_1 all X_a (dvd(X_a) -> hAPP(X_a,bool,B_1_1,ti(X_a,B_2_1)) = hAPP(X_a,bool,B_1_1,B_2_1))) # label(tsy_c_hAPP_3_arg2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 144 (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]. 1.70/2.07 145 (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]. 1.70/2.07 146 (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,D_1) != ti(X_a,C) & ti(X_a,A_1) = ti(X_a,B_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]. 1.70/2.07 147 (all X_a (power(X_a) & semiring_0(X_a) -> (all W ((number_number_of(nat,W) != zero_zero(nat) -> hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),number_number_of(nat,W)) = zero_zero(X_a)) & (number_number_of(nat,W) = zero_zero(nat) -> hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),number_number_of(nat,W)) = one_one(X_a)))))) # label(fact_11_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 148 (all X_a (ab_group_add(X_a) -> (all A_2 all B (ti(X_a,B) = ti(X_a,A_2) <-> zero_zero(X_a) = minus_minus(X_a,A_2,B))))) # label(fact_321_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 149 (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]. 1.70/2.07 150 (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]. 1.70/2.07 151 (all B_1_1 all B_2_1 legendre(ti(int,B_1_1),B_2_1) = legendre(B_1_1,B_2_1)) # label(tsy_c_Residues_OLegendre_arg1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 152 (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]. 1.70/2.07 153 (all B_1_1 all B_2_1 zcong(B_1_1,B_2_1) = zcong(B_1_1,ti(int,B_2_1))) # label(tsy_c_IntPrimes_Ozcong_arg2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 154 (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)) -> number_number_of(X_a,plus_plus(int,V_1,V_2)) = plus_plus(X_a,number_number_of(X_a,V_1),number_number_of(X_a,V_2))))))) # label(fact_528_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 155 (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]. 1.70/2.07 156 (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]. 1.70/2.07 157 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> one_one(X_a) = abs_abs(X_a,number_number_of(X_a,min)))) # label(fact_929_abs__minus__one) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 158 (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]. 1.70/2.07 159 (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]. 1.70/2.07 160 (all M all N all K_1 minus_minus(nat,times_times(nat,M,K_1),times_times(nat,N,K_1)) = times_times(nat,minus_minus(nat,M,N),K_1)) # label(fact_630_diff__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 161 (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]. 1.70/2.07 162 (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]. 1.70/2.07 163 (all X_a (monoid_mult(X_a) -> (all A_1 all M all N hAPP(nat,X_a,power_power(X_a,hAPP(nat,X_a,power_power(X_a,A_1),M)),N) = hAPP(nat,X_a,power_power(X_a,A_1),times_times(nat,M,N))))) # label(fact_600_power__mult) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 164 (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]. 1.70/2.07 165 (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]. 1.70/2.07 166 (all Xa all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya)) | Xa = Ya <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya)))) # label(fact_611_less__eq__real__def) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 167 (all X_b (ring(X_b) & number(X_b) -> (all A_1 all B_1 all 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))) = times_times(X_b,minus_minus(X_b,A_1,B_1),number_number_of(X_b,V_1))))) # label(fact_263_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 168 (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]. 1.70/2.07 169 (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]. 1.70/2.07 170 (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]. 1.70/2.07 171 (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]. 1.70/2.07 172 (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]. 1.70/2.07 173 (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(B_1,zero_zero(int)),P_2)) & -hBOOL(hAPP(int,bool,zcong(A_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]. 1.70/2.07 174 (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]. 1.70/2.07 175 (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),A_2),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))))))) # label(fact_238_le__iff__diff__le__0) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 176 (all X_a (number_ring(X_a) -> (all V_1 all W number_number_of(X_a,times_times(int,V_1,W)) = times_times(X_a,number_number_of(X_a,V_1),number_number_of(X_a,W))))) # label(fact_258_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 177 (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]. 1.70/2.07 178 (all K_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K_1)) -> (exists N_1 (hAPP(nat,int,semiring_1_of_nat(int),N_1) = ti(int,K_1) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)))))) # label(fact_190_zero__less__imp__eq__int) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 179 (all Ma all Na (zero_zero(nat) = Na & zero_zero(nat) = Ma <-> plus_plus(nat,Ma,Na) = zero_zero(nat))) # label(fact_154_add__is__0) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 180 (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]. 1.70/2.07 181 (all W pls = times_times(int,pls,W)) # label(fact_370_mult__Pls) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 182 (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]. 1.70/2.07 183 (all K_1 all M all N minus_minus(nat,M,N) = minus_minus(nat,plus_plus(nat,K_1,M),plus_plus(nat,K_1,N))) # label(fact_368_Nat_Odiff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 184 (all A_1 all 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),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]. 1.70/2.07 185 (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]. 1.70/2.07 186 (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]. 1.70/2.07 187 (all Xa all Ya all P_1 ((-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))) & (hBOOL(P_1) -> hAPP(nat,int,semiring_1_of_nat(int),Xa) = 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]. 1.70/2.07 188 (all Ma all K 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,Ma,K)),times_times(nat,Na,K))))) # label(fact_585_mult__le__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 189 (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]. 1.70/2.07 190 (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]. 1.70/2.07 191 (all X_a (semiri456707255roduct(X_a) -> (all C_1 all D all A_2 all B (ti(X_a,D) != ti(X_a,C_1) & ti(X_a,B) != ti(X_a,A_2) <-> plus_plus(X_a,times_times(X_a,A_2,C_1),times_times(X_a,B,D)) != plus_plus(X_a,times_times(X_a,A_2,D),times_times(X_a,B,C_1)))))) # label(fact_344_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 192 (all Xa all Ya (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)))) # label(fact_871_dvd_Oeq__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 193 (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]. 1.70/2.07 194 (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]. 1.70/2.07 195 (all B_1_1 all B_2_1 all X_a (mult_zero(X_a) -> times_times(X_a,B_1_1,B_2_1) = times_times(X_a,ti(X_a,B_1_1),B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_4_arg1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 196 (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]. 1.70/2.07 197 (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]. 1.70/2.07 198 (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(int,bool,P_1,X_1)))) <-> (exists X_1 hBOOL(hAPP(int,bool,P_1,hAPP(nat,int,semiring_1_of_nat(int),X_1)))))) # label(fact_473_transfer__int__nat__quantifiers_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 199 (all X_a (order(X_a) -> ti(fun(X_a,fun(X_a,bool)),ord_less_eq(X_a)) = ord_less_eq(X_a))) # label(tsy_c_Orderings_Oord__class_Oless__eq_res) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 200 (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]. 1.70/2.07 201 (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]. 1.70/2.07 202 (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]. 1.70/2.07 203 (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]. 1.70/2.07 204 (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]. 1.70/2.07 205 (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]. 1.70/2.07 206 (all M all N M = minus_minus(nat,plus_plus(nat,M,N),N)) # label(fact_365_diff__add__inverse2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 207 (all X_a (comm_semiring_1(X_a) -> (all X all N hAPP(nat,X_a,power_power(X_a,X),times_times(nat,number_number_of(nat,bit0(bit1(pls))),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))))) # label(fact_481_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 208 (all X_b (monoid_mult(X_b) & number(X_b) -> (all W times_times(X_b,number_number_of(X_b,W),number_number_of(X_b,W)) = hAPP(nat,X_b,power_power(X_b,number_number_of(X_b,W)),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_545_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 209 (all M all N (abs_abs(int,times_times(int,M,N)) = one_one(int) -> one_one(int) = abs_abs(int,M))) # label(fact_938_abs__zmult__eq__1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 210 (all Z_1 all 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) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),nat_1(Z_1)),Ma)))) # label(fact_835_nat__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 211 (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]. 1.70/2.07 212 (all K all L_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),minus_minus(int,K,L_1)),zero_zero(int))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L_1)))) # label(fact_647_less__bin__lemma) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 213 (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]. 1.70/2.07 214 (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]. 1.70/2.07 215 (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]. 1.70/2.07 216 (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]. 1.70/2.07 217 (all K all L_1 (bit1(K) = bit1(L_1) <-> ti(int,L_1) = ti(int,K))) # label(fact_42_rel__simps_I51_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 218 (all A_1 all B_1 (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)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1))))) # label(fact_867_dvd_Oneq__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 219 (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),A_2),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))))))) # label(fact_225_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 220 (all X_a (number_ring(X_a) -> (all V_1 number_number_of(X_a,plus_plus(int,V_1,bit1(pls))) = plus_plus(X_a,number_number_of(X_a,V_1),one_one(X_a))))) # label(fact_7_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 221 (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]. 1.70/2.07 222 (all K_1 all L bit1(K_1) != bit0(L)) # label(fact_68_rel__simps_I50_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 223 (all Ma all Na (hAPP(nat,int,semiring_1_of_nat(int),Na) = hAPP(nat,int,semiring_1_of_nat(int),Ma) <-> Na = Ma)) # label(fact_48_int__int__eq) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 224 (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]. 1.70/2.07 225 (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]. 1.70/2.07 226 (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]. 1.70/2.07 227 (all A_2 all B all C_1 (zero_zero(real) != C_1 -> (times_times(real,B,C_1) = times_times(real,A_2,C_1) <-> A_2 = B))) # label(fact_604_real__mult__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 228 (exists X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) & (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,zcong(s1,Y_1),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),Y_1),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))) -> ti(int,X_1) = ti(int,Y_1))) & 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)))) & 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)))))) # 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]. 1.70/2.07 229 (all X_a (number_semiring(X_a) -> one_one(X_a) = number_number_of(X_a,bit1(pls)))) # label(fact_101_semiring__numeral__1__eq__1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 230 (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]. 1.70/2.07 231 (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),X),Z)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z),X))))) # label(fact_856_dvd_Ole__less__trans) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 232 (all X_a (ring_1(X_a) -> (all X times_times(X_a,plus_plus(X_a,X,one_one(X_a)),minus_minus(X_a,X,one_one(X_a))) = minus_minus(X_a,times_times(X_a,X,X),one_one(X_a))))) # label(fact_297_real__squared__diff__one__factored) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 233 (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)) -> hAPP(nat,X_a,semiring_1_of_nat(X_a),nat_1(times_times(int,number_number_of(int,bit0(bit1(pls))),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))))))) # label(fact_470_of__nat__double) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 234 (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),bit1(K)),bit1(L_1))))) # label(fact_31_rel__simps_I17_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 235 (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]. 1.70/2.07 236 (all Ma all K all Na (zero_zero(nat) = K | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),Na)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),times_times(nat,Ma,K)),times_times(nat,Na,K))))) # label(fact_805_nat__mult__dvd__cancel__disj_H) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 237 (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]. 1.70/2.07 238 (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]. 1.70/2.07 239 (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]. 1.70/2.07 240 (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,I_2,minus_minus(nat,J_1,K_1)) = minus_minus(nat,plus_plus(nat,I_2,J_1),K_1))) # label(fact_670_add__diff__assoc) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 241 (all K_1 all L plus_plus(int,bit0(times_times(int,K_1,L)),L) = times_times(int,bit1(K_1),L)) # label(fact_509_mult__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 242 (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]. 1.70/2.07 243 (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 (hAPP(nat,real,power_power(real,X_1),N) = A_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)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))))))) # label(fact_229_realpow__pos__nth__unique) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 244 (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),minus_minus(nat,Ma,K)),minus_minus(nat,Na,K))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)))))) # label(fact_662_less__diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 245 (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]. 1.70/2.07 246 (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]. 1.70/2.07 247 (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]. 1.70/2.07 248 (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)) -> number_number_of(X_a,times_times(int,V_1,V_2)) = times_times(X_a,number_number_of(X_a,V_1),number_number_of(X_a,V_2))))))) # label(fact_250_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 1.70/2.07 249 (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]. 1.70/2.07 250 (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))) -> number_number_of(X_a,V_1) = hAPP(nat,X_a,semiring_1_of_nat(X_a),number_number_of(nat,V_1))))))) # label(fact_549_of__nat__number__of__lemma) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 251 (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]. 1.70/2.08 252 (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]. 1.70/2.08 253 (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]. 1.70/2.08 254 (all Z all W times_times(int,W,Z) = times_times(int,Z,W)) # label(fact_252_zmult__commute) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 255 (all X all Y all Z hAPP(nat,int,power_power(int,hAPP(nat,int,power_power(int,X),Y)),Z) = hAPP(nat,int,power_power(int,X),times_times(nat,Y,Z))) # label(fact_641_zpower__zpower) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 256 (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]. 1.70/2.08 257 (all K_1 bit0(K_1) = plus_plus(int,K_1,K_1)) # label(fact_78_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 258 (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]. 1.70/2.08 259 (all X_a (linordered_idom(X_a) & number_ring(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]. 1.70/2.08 260 (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]. 1.70/2.08 261 (all X all Y (X = Y -> 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]. 1.70/2.08 262 (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)))) <-> ti(int,Wa) = ti(int,Z_1) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z_1)))) # label(fact_60_zless__add1__eq) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 263 (all Xa all P_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_3),Xa)) <-> hBOOL(hAPP(int,bool,zcong(Xa,zero_zero(int)),P_3)))) # label(fact_801_zcong__eq__zdvd__prop) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 264 (all X_a (linordered_idom(X_a) -> (all Xa all Ya (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) = zero_zero(X_a) & ti(X_a,Xa) = zero_zero(X_a))))) # label(fact_2_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 265 (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]. 1.70/2.08 266 (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]. 1.70/2.08 267 (all Ma all Na (times_times(int,Ma,Na) = one_one(int) <-> one_one(int) = ti(int,Ma) & ti(int,Na) = one_one(int) | number_number_of(int,min) = ti(int,Ma) & number_number_of(int,min) = ti(int,Na))) # label(fact_785_zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 268 (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]. 1.70/2.08 269 (all X_a (comm_semiring_1(X_a) -> (all A_1 times_times(X_a,A_1,zero_zero(X_a)) = zero_zero(X_a)))) # label(fact_333_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 270 (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]. 1.70/2.08 271 (all X_a (semiring_1(X_a) -> hAPP(nat,X_a,semiring_1_of_nat(X_a),one_one(nat)) = one_one(X_a))) # label(fact_171_of__nat__1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 272 (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]. 1.70/2.08 273 (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]. 1.70/2.08 274 (all X_a (linord219039673up_add(X_a) -> (all A_2 (ti(X_a,A_2) = zero_zero(X_a) <-> plus_plus(X_a,A_2,A_2) = zero_zero(X_a))))) # label(fact_216_double__zero__sym) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 275 (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]. 1.70/2.08 276 (all A_2 all B all Ma (hBOOL(hAPP(int,bool,zcong(B,A_2),Ma)) <-> hBOOL(hAPP(int,bool,zcong(A_2,B),Ma)))) # label(fact_883_zcong__sym) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 277 (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]. 1.70/2.08 278 (all Na (zero_zero(int) = hAPP(nat,int,semiring_1_of_nat(int),Na) <-> zero_zero(nat) = Na)) # label(fact_27_int__eq__0__conv) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 279 (all X_a (number_ring(X_a) -> (all V_1 all W number_number_of(X_a,minus_minus(int,V_1,W)) = minus_minus(X_a,number_number_of(X_a,V_1),number_number_of(X_a,W))))) # label(fact_264_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 280 (all Z_1 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)),nat_1(Z_1))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z_1))))) # label(fact_388_nat__less__eq__zless) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 281 (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,M,N) = minus_minus(nat,minus_minus(nat,M,K_1),minus_minus(nat,N,K_1))))) # label(fact_634_Nat_Odiff__diff__eq) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 282 (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]. 1.70/2.08 283 (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]. 1.70/2.08 284 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all Xa all Ya (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),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))))))) # label(fact_257_le__number__of) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 285 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),min)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K)),min)))) # label(fact_760_rel__simps_I28_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 286 (all X_a (comm_semiring_1(X_a) -> (all Lx all Rx all Ry times_times(X_a,Lx,times_times(X_a,Rx,Ry)) = times_times(X_a,times_times(X_a,Lx,Rx),Ry)))) # label(fact_241_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 287 (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(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]. 1.70/2.08 288 (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]. 1.70/2.08 289 (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]. 1.70/2.08 290 (all Ma all Z_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),nat_1(abs_abs(int,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)))) # label(fact_962_int__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 291 (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]. 1.70/2.08 292 (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),bit1(K)),bit1(L_1))))) # label(fact_372_rel__simps_I34_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 293 (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]. 1.70/2.08 294 -(all S -(hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),S)) & 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)))) & 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)))))) # 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]. 1.70/2.08 295 (all X_a (number_ring(X_a) -> (all K_1 number_number_of(X_a,succ(K_1)) = plus_plus(X_a,one_one(X_a),number_number_of(X_a,K_1))))) # label(fact_532_number__of__succ) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 296 (all L bit1(minus_minus(int,min,L)) = minus_minus(int,min,bit0(L))) # label(fact_784_diff__bin__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 297 (all N all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) -> hAPP(nat,nat,power_power(nat,nat_1(Z)),N) = nat_1(hAPP(nat,int,power_power(int,Z),N)))) # label(fact_400_nat__power__eq) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 298 (all X_a (ring_11004092258visors(X_a) -> (all N all A_1 (ti(X_a,A_1) != zero_zero(X_a) -> zero_zero(X_a) != hAPP(nat,X_a,power_power(X_a,A_1),N))))) # label(fact_165_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 299 (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]. 1.70/2.08 300 (all Ma all K all Na (times_times(nat,Ma,K) = times_times(nat,Na,K) <-> K = zero_zero(nat) | Na = Ma)) # label(fact_598_mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 301 (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]. 1.70/2.08 302 (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]. 1.70/2.08 303 (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]. 1.70/2.08 304 (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]. 1.70/2.08 305 (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]. 1.70/2.08 306 (all W number_number_of(nat,W) = nat_1(number_number_of(int,W))) # label(fact_426_nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 307 (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),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]. 1.70/2.08 308 (all B_1_1 bit0(ti(int,B_1_1)) = bit0(B_1_1)) # label(tsy_c_Int_OBit0_arg1) # label(hypothesis) # label(non_clause). [assumption]. 1.70/2.08 309 (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]. 1.70/2.08 310 (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_1) <-> hBOOL(P_4))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) & hBOOL(P_1) <-> 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]. 1.70/2.08 311 (all X_b (number(X_b) & ring(X_b) -> (all V_1 all B_1 all C times_times(X_b,number_number_of(X_b,V_1),minus_minus(X_b,B_1,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))))) # label(fact_262_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 312 (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]. 1.70/2.08 313 (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,C_1) = ti(X_a,D) <-> ti(X_a,B) = ti(X_a,A_2)))))) # label(fact_248_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 314 (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]. 1.70/2.08 315 (all V_1 ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls)) -> one_one(nat) = plus_plus(nat,one_one(nat),number_number_of(nat,V_1))) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls)) -> number_number_of(nat,succ(V_1)) = plus_plus(nat,one_one(nat),number_number_of(nat,V_1))))) # label(fact_233_nat__1__add__number__of) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 316 (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]. 1.70/2.08 317 (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]. 1.70/2.08 318 (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]. 1.70/2.08 319 (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]. 1.70/2.08 320 (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]. 1.70/2.08 321 (all X_a (semiring_1(X_a) -> hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(X_a))) # label(fact_4_zero__power2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 322 (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]. 1.70/2.08 323 (all Xa all Ya (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) <-> Ya = zero_zero(real) & zero_zero(real) = Xa)) # label(fact_106_realpow__two__sum__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 324 (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]. 1.70/2.08 325 (all B_1_1 all X_a (ordere142940540dd_abs(X_a) -> abs_abs(X_a,ti(X_a,B_1_1)) = abs_abs(X_a,B_1_1))) # label(tsy_c_Groups_Oabs__class_Oabs_arg1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 326 (all Ma all Na (one_one(nat) = times_times(nat,Ma,Na) <-> one_one(nat) = Ma & Na = one_one(nat))) # label(fact_626_nat__mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 327 (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),M)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),N))))) # label(fact_727_zprime__zdvd__zmult__better) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 328 (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]. 1.70/2.08 329 (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]. 1.70/2.08 330 (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]. 1.70/2.08 331 (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]. 1.70/2.08 332 (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,Lx,times_times(X_a,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]. 1.70/2.08 333 (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]. 1.70/2.08 334 (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]. 1.70/2.08 335 (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_855_dvd_Oless__imp__neq) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 336 (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 (hAPP(nat,int,F,I) = ti(int,K) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),Na)))))))) # label(fact_994_int__val__lemma) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 337 (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]. 1.70/2.08 338 (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]. 1.70/2.08 339 (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)),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))))))) # label(fact_432_zero__le__double__add__iff__zero__le__single__add) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 340 (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]. 1.70/2.08 341 (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]. 1.70/2.08 342 (all X_a (comm_monoid_mult(X_a) -> (all A_1 ti(X_a,A_1) = times_times(X_a,A_1,one_one(X_a))))) # label(fact_347_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 343 (all X_a (ring_11004092258visors(X_a) -> (all A_2 (zero_zero(X_a) = ti(X_a,A_2) <-> hAPP(nat,X_a,power_power(X_a,A_2),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(X_a))))) # label(fact_5_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 344 (all K_1 bit1(K_1) != pls) # label(fact_66_rel__simps_I46_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 345 (all Na (zero_zero(nat) = Na <-> 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]. 1.70/2.08 346 (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]. 1.70/2.08 347 (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]. 1.70/2.08 348 (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]. 1.70/2.08 349 (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]. 1.70/2.08 350 (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]. 1.70/2.08 351 (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),zero_zero(X_a)),times_times(X_a,A_2,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)),A_2)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)))))) # label(fact_291_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 352 (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),pls)) <-> 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))))))) # label(fact_390_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 353 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),plus_plus(nat,Ma,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)))) # label(fact_144_add__gr__0) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 354 (all B_1_1 all B_2_1 all X_a (no_zero_divisors(X_a) -> times_times(X_a,ti(X_a,B_1_1),B_2_1) = times_times(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_3_arg1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 355 (all 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))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)))) # label(fact_162_zero__less__int__conv) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 356 (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]. 1.70/2.08 357 (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]. 1.70/2.08 358 (all B_1_1 nat_1(B_1_1) = ti(nat,nat_1(B_1_1))) # label(tsy_c_Int_Onat_res) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 359 (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]. 1.70/2.08 360 (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 (hAPP(nat,int,semiring_1_of_nat(int),N_1) = ti(int,I_1) -> hBOOL(hAPP(nat,bool,P_1,N_1)))))) # label(fact_523_split__nat) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 361 (all X_a (comm_semiring_1(X_a) -> (all A_1 all C all D_1 plus_plus(X_a,A_1,plus_plus(X_a,C,D_1)) = plus_plus(X_a,plus_plus(X_a,A_1,C),D_1)))) # label(fact_208_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 362 (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]. 1.70/2.08 363 (all Xa all Ya (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Xa),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))))) # label(fact_677_Nat__Transfer_Otransfer__int__nat__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 364 (all Ma all Z_1 (hAPP(nat,int,semiring_1_of_nat(int),Ma) = ti(int,Z_1) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1)) & Ma = nat_1(Z_1))) # label(fact_318_int__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 365 (all M zero_zero(nat) = times_times(nat,M,zero_zero(nat))) # label(fact_595_mult__0__right) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 366 (all K all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),times_times(nat,K,Ma)),times_times(nat,K,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)))) # label(fact_651_mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 367 (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]. 1.70/2.08 368 (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]. 1.70/2.08 369 (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]. 1.70/2.08 370 (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]. 1.70/2.08 371 (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]. 1.70/2.08 372 (all X_a (comm_semiring_1(X_a) -> (all M all A_1 plus_plus(X_a,M,times_times(X_a,A_1,M)) = times_times(X_a,plus_plus(X_a,A_1,one_one(X_a)),M)))) # label(fact_446_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 373 (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]. 1.70/2.08 374 (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]. 1.70/2.08 375 (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]. 1.70/2.08 376 (all Z_1 all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),nat_1(abs_abs(int,Z_1))),Ma)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Z_1),hAPP(nat,int,semiring_1_of_nat(int),Ma))))) # label(fact_963_dvd__int__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 377 (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]. 1.70/2.08 378 (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]. 1.70/2.08 379 (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]. 1.70/2.08 380 (all K_1 ti(int,K_1) = number_number_of(int,K_1)) # label(fact_120_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 381 (all Z plus_plus(int,plus_plus(int,one_one(int),Z),Z) != zero_zero(int)) # label(fact_93_odd__nonzero) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 382 (all X_a ti(X_a,undefined(X_a)) = undefined(X_a)) # label(tsy_c_HOL_Oundefined_res) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 383 (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]. 1.70/2.08 384 (all A_1 all B_1 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))))) = hAPP(nat,int,power_power(int,plus_plus(int,A_1,B_1)),number_number_of(nat,bit1(bit1(pls))))) # label(fact_234_zadd__power3) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 385 (all N (number_number_of(int,min) = hAPP(nat,int,power_power(int,number_number_of(int,min)),N) | one_one(int) = hAPP(nat,int,power_power(int,number_number_of(int,min)),N))) # label(fact_876_neg__one__power) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 386 (all P_2 all M ((zero_zero(nat) = M -> hAPP(nat,nat,power_power(nat,P_2),M) = one_one(nat)) & (zero_zero(nat) != M -> hAPP(nat,nat,power_power(nat,P_2),M) = times_times(nat,P_2,hAPP(nat,nat,power_power(nat,P_2),minus_minus(nat,M,one_one(nat))))))) # label(fact_689_power__eq__if) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 387 (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]. 1.70/2.08 388 (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]. 1.70/2.08 389 (all X_a (number_ring(X_a) -> (all A_1 ti(X_a,A_1) = plus_plus(X_a,A_1,number_number_of(X_a,pls))))) # label(fact_85_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 390 (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]. 1.70/2.08 391 (all B_1_1 ti(int,succ(B_1_1)) = succ(B_1_1)) # label(tsy_c_Int_Osucc_res) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 392 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B_1 all C plus_plus(X_a,A_1,plus_plus(X_a,B_1,C)) = plus_plus(X_a,plus_plus(X_a,A_1,B_1),C)))) # label(fact_207_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 393 (all B_1_1 all B_2_1 all X_a (semiring(X_a) -> ti(X_a,times_times(X_a,B_1_1,B_2_1)) = times_times(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_2_res) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 394 (all X_a (number_ring(X_a) -> number_number_of(X_a,bit1(pls)) = one_one(X_a))) # label(fact_12_semiring__norm_I110_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 395 (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),Ma) = hAPP(nat,X_a,power_power(X_a,A_2),Na) <-> Na = Ma))))) # label(fact_181_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 396 (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,K_1,N) = plus_plus(nat,M,L) -> 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]. 1.70/2.08 397 (all X_a (linordered_idom(X_a) -> (all X all 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)) -> 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]. 1.70/2.08 398 (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]. 1.70/2.08 399 (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),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_eq(X_a),A_2),zero_zero(X_a))))))) # label(fact_433_double__add__le__zero__iff__single__add__le__zero) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 400 (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]. 1.70/2.08 401 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),bit0(K))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),K)))) # label(fact_780_rel__simps_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 402 (all X_a (semiring_1(X_a) -> hAPP(nat,X_a,power_power(X_a,one_one(X_a)),number_number_of(nat,bit0(bit1(pls)))) = one_one(X_a))) # label(fact_3_one__power2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 403 (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]. 1.70/2.08 404 (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]. 1.70/2.08 405 (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]. 1.70/2.08 406 (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]. 1.70/2.08 407 (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]. 1.70/2.08 408 (all Ma all Ta all K (ti(int,K) != zero_zero(int) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Ma),Ta)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),times_times(int,K,Ma)),times_times(int,K,Ta)))))) # label(fact_907_zdvd__mono) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 409 (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]. 1.70/2.08 410 (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]. 1.70/2.08 411 (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]. 1.70/2.08 412 (all Ma all Na (Ma != 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)))) # label(fact_121_nat__neq__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 413 (all N times_times(nat,zero_zero(nat),N) = zero_zero(nat)) # label(fact_594_mult__0) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 414 (all K_1 plus_plus(int,plus_plus(int,one_one(int),K_1),K_1) = bit1(K_1)) # label(fact_92_Bit1__def) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 415 (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)) -> minus_minus(int,P_2,one_one(int)) = ti(int,A_1) | one_one(int) = ti(int,A_1)))))) # label(fact_880_zcong__square__zless) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 416 (all X_a (dvd(X_a) -> (all A_1 all B_1 all K_1 (A_1 = times_times(X_a,B_1,K_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]. 1.70/2.08 417 (all Xa all Wa (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Xa)) | zero_zero(nat) = number_number_of(nat,Wa) <-> 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]. 1.70/2.08 418 (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]. 1.70/2.08 419 (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]. 1.70/2.08 420 (all A_1 all B_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1)) -> (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_863_dvd_Ole__neq__trans) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 421 (all Z plus_plus(nat,Z,Z) = times_times(nat,number_number_of(nat,bit0(bit1(pls))),Z)) # label(fact_691_nat__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 422 (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]. 1.70/2.08 423 (all N all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M)) -> hAPP(nat,int,semiring_1_of_nat(int),minus_minus(nat,M,N)) = minus_minus(int,hAPP(nat,int,semiring_1_of_nat(int),M),hAPP(nat,int,semiring_1_of_nat(int),N)))) # label(fact_589_zdiff__int) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 424 (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]. 1.70/2.08 425 (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_1) <-> hBOOL(P_4))) -> ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> hBOOL(P_1)) <-> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> hBOOL(P_4))))) # label(fact_906_imp__le__cong) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 426 (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]. 1.70/2.08 427 (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]. 1.70/2.08 428 (all K all Ma all Na (times_times(nat,K,Ma) = times_times(nat,K,Na) <-> Na = Ma | K = zero_zero(nat))) # label(fact_597_mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 429 (all X_a (ordere142940540dd_abs(X_a) -> (all A_2 (zero_zero(X_a) != ti(X_a,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))))))) # label(fact_945_zero__less__abs__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 430 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 all B_1 abs_abs(X_a,minus_minus(X_a,B_1,A_1)) = abs_abs(X_a,minus_minus(X_a,A_1,B_1))))) # label(fact_950_abs__minus__commute) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 431 (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]. 1.70/2.08 432 (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]. 1.70/2.08 433 (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]. 1.70/2.08 434 (all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),abs_abs(int,Z_1)),one_one(int))) <-> zero_zero(int) = ti(int,Z_1))) # label(fact_956_zabs__less__one__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 435 (all B_1_1 all B_2_1 legendre(B_1_1,B_2_1) = legendre(B_1_1,ti(int,B_2_1))) # label(tsy_c_Residues_OLegendre_arg2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 436 (all X_a (number_semiring(X_a) -> (all X all Y 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)) = hAPP(nat,X_a,power_power(X_a,plus_plus(X_a,X,Y)),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_550_power2__sum) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 437 (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]. 1.70/2.08 438 (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]. 1.70/2.08 439 (all K_1 all L bit0(minus_minus(int,K_1,L)) = minus_minus(int,bit1(K_1),bit1(L))) # label(fact_643_diff__bin__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 440 (all M M = minus_minus(nat,M,zero_zero(nat))) # label(fact_337_minus__nat_Odiff__0) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 441 (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_850_dvd_Oless__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 442 (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]. 1.70/2.08 443 (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]. 1.70/2.08 444 (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]. 1.70/2.08 445 (all X_a (number_ring(X_a) -> zero_zero(X_a) = number_number_of(X_a,pls))) # label(fact_83_semiring__norm_I112_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 446 (all X_a (number_ring(X_a) -> (all 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)) = number_number_of(X_a,bit1(W))))) # label(fact_100_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 447 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(K)),pls)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),pls)))) # label(fact_504_rel__simps_I29_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 448 (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]. 1.70/2.08 449 (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]. 1.70/2.08 450 (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]. 1.70/2.08 451 (all X_a (power(X_a) & semiring_0(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)) & (N = zero_zero(nat) -> one_one(X_a) = hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),N)))))) # label(fact_109_power__0__left) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 452 (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]. 1.70/2.08 453 (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]. 1.70/2.08 454 (all A_1 times_times(int,A_1,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls))))) = hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit1(bit1(pls))))) # label(fact_543_cube__square) # label(axiom) # label(non_clause). [assumption]. 1.70/2.08 455 (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]. 1.70/2.08 456 (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]. 1.70/2.09 457 (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]. 1.70/2.09 458 (all K all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),plus_plus(nat,Na,K))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),Na)))) # label(fact_814_dvd__reduce) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 459 (all X_a (comm_semiring_1(X_a) -> (all A_1 times_times(X_a,zero_zero(X_a),A_1) = zero_zero(X_a)))) # label(fact_334_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 460 (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]. 1.70/2.09 461 (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]. 1.70/2.09 462 (all B_1_1 all B_2_1 all X_a (order(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_0_arg2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 463 (all D_1 all I_2 (ti(int,I_2) != zero_zero(int) -> (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]. 1.70/2.09 464 (all X_a (number_ring(X_a) -> (all V_1 all W number_number_of(X_a,times_times(int,V_1,W)) = times_times(X_a,number_number_of(X_a,V_1),number_number_of(X_a,W))))) # label(fact_259_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 465 (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]. 1.70/2.09 466 (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]. 1.70/2.09 467 (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]. 1.70/2.09 468 (all N all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) -> nat_1(hAPP(nat,int,power_power(int,X),N)) = hAPP(nat,nat,power_power(nat,nat_1(X)),N))) # label(fact_399_Nat__Transfer_Otransfer__nat__int__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 469 (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),pls)) <-> 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))))))) # label(fact_49_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 470 (all B_1_1 all B_2_1 all X_a (group_add(X_a) -> minus_minus(X_a,ti(X_a,B_1_1),B_2_1) = minus_minus(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Ominus__class_Ominus_0_arg1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 471 (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]. 1.70/2.09 472 (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]. 1.70/2.09 473 (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]. 1.70/2.09 474 (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]. 1.70/2.09 475 (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(int,bool,P_1,X_1)))) <-> (all X_1 hBOOL(hAPP(int,bool,P_1,hAPP(nat,int,semiring_1_of_nat(int),X_1)))))) # label(fact_474_transfer__int__nat__quantifiers_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 476 (all K_1 bit0(succ(K_1)) = succ(bit1(K_1))) # label(fact_515_succ__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 477 (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]. 1.70/2.09 478 (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]. 1.70/2.09 479 (all X_a all X all Y ti(X_a,X) = if(X_a,fTrue,X,Y)) # label(help_If_1_1_T) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 480 (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]. 1.70/2.09 481 (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]. 1.70/2.09 482 (all Xa all Na (Xa = one_one(nat) | zero_zero(nat) = Na <-> one_one(nat) = hAPP(nat,nat,power_power(nat,Xa),Na))) # label(fact_192_exp__eq__1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 483 (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]. 1.70/2.09 484 (all X_a (mult_zero(X_a) & no_zero_divisors(X_a) & zero_neq_one(X_a) & power(X_a) -> (all A_2 all Na (zero_zero(X_a) = hAPP(nat,X_a,power_power(X_a,A_2),Na) <-> zero_zero(X_a) = ti(X_a,A_2) & Na != zero_zero(nat))))) # label(fact_185_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 485 (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_811_zdvd__int) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 486 (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]. 1.70/2.09 487 (all V_1 all W plus_plus(int,number_number_of(int,V_1),number_number_of(int,W)) = number_number_of(int,plus_plus(int,V_1,W))) # label(fact_17_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 488 (all Ma all Na (Ma = zero_zero(nat) & zero_zero(nat) != Na <-> zero_zero(nat) = hAPP(nat,nat,power_power(nat,Ma),Na))) # label(fact_228_nat__power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 489 (all Na (Na != zero_zero(nat) <-> 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]. 1.70/2.09 490 (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,B_1_1,ti(X_a,B_2_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_5_arg2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 491 (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,one_one(int)),P_2)) | hBOOL(hAPP(int,bool,zcong(A_1,minus_minus(int,P_2,one_one(int))),P_2)))))) # label(fact_902_zcong__square) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 492 (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]. 1.70/2.09 493 (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]. 1.70/2.09 494 (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]. 1.70/2.09 495 (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]. 1.70/2.09 496 (all M all N all K_1 times_times(nat,M,times_times(nat,N,K_1)) = times_times(nat,times_times(nat,M,N),K_1)) # label(fact_577_nat__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 497 (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]. 1.70/2.09 498 (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]. 1.70/2.09 499 (all X_a (group_add(X_a) -> (all A_1 minus_minus(X_a,A_1,zero_zero(X_a)) = ti(X_a,A_1)))) # label(fact_323_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 500 (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]. 1.70/2.09 501 (all Wa all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Wa),Z_1)) <-> (exists N_1 ti(int,Z_1) = plus_plus(int,Wa,hAPP(nat,int,semiring_1_of_nat(int),N_1))))) # label(fact_477_zle__iff__zadd) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 502 (all M M = plus_plus(nat,M,zero_zero(nat))) # label(fact_155_Nat_Oadd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 503 (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]. 1.70/2.09 504 (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]. 1.70/2.09 505 (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]. 1.70/2.09 506 (all C all A_1 all B_1 (B_1 = A_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]. 1.70/2.09 507 (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]. 1.70/2.09 508 (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]. 1.70/2.09 509 (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]. 1.70/2.09 510 (all X_a (number_ring(X_a) -> (all A_1 ti(X_a,A_1) = plus_plus(X_a,number_number_of(X_a,pls),A_1)))) # label(fact_84_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 511 (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]. 1.70/2.09 512 (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]. 1.70/2.09 513 (all L bit0(L) != min) # label(fact_765_rel__simps_I42_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 514 (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]. 1.70/2.09 515 (all X_a (no_zero_divisors(X_a) -> (all B_1 all A_1 (zero_zero(X_a) != ti(X_a,A_1) -> (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]. 1.70/2.09 516 (all X_a (no_zero_divisors(X_a) -> (all A_1 all B_1 (zero_zero(X_a) = times_times(X_a,A_1,B_1) -> ti(X_a,A_1) = zero_zero(X_a) | ti(X_a,B_1) = zero_zero(X_a))))) # label(fact_332_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 517 (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(if(int,P_1,Xa,Ya)) = nat_1(Xa)))) # label(fact_268_nat__if__cong) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 518 (all A_2 all B (ti(int,B) = ti(int,A_2) <-> hBOOL(hAPP(int,bool,zcong(A_2,B),zero_zero(int))))) # label(fact_885_IntPrimes_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 519 (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]. 1.70/2.09 520 (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]. 1.70/2.09 521 (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]. 1.70/2.09 522 (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]. 1.70/2.09 523 (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]. 1.70/2.09 524 (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]. 1.70/2.09 525 (all P_1 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),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]. 1.70/2.09 526 (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]. 1.70/2.09 527 (all Na all P_1 (-hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) -> (hBOOL(hAPP(nat,bool,P_1,Na)) -> (exists K_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_2),Na)) & hBOOL(hAPP(nat,bool,P_1,plus_plus(nat,K_2,one_one(nat)))) & (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))))))))) # label(fact_976_ex__least__nat__less) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 528 (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]. 1.70/2.09 529 (all B_1_1 all B_2_1 minus_minus(nat,B_1_1,B_2_1) = minus_minus(nat,B_1_1,ti(nat,B_2_1))) # label(tsy_c_Groups_Ominus__class_Ominus_1_arg2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 530 (all K all Ma all Na (times_times(nat,K,Na) = times_times(nat,K,Ma) <-> zero_zero(nat) = K | Ma = Na)) # label(fact_980_nat__mult__eq__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 531 (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]. 1.70/2.09 532 (all K_1 succ(K_1) = minus_minus(int,K_1,min)) # label(fact_777_diff__bin__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 533 (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]. 1.70/2.09 534 (all X_a (comm_semiring_1(X_a) -> (all X hAPP(nat,X_a,power_power(X_a,X),one_one(nat)) = ti(X_a,X)))) # label(fact_227_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 535 (all X_a (linordered_idom(X_a) -> (all A_1 times_times(X_a,A_1,A_1) = times_times(X_a,abs_abs(X_a,A_1),abs_abs(X_a,A_1))))) # label(fact_952_abs__mult__self) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 536 (all X_a (group_add(X_a) -> (all A_1 all B_1 ti(X_a,A_1) = plus_plus(X_a,minus_minus(X_a,A_1,B_1),B_1)))) # label(fact_327_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 537 (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]. 1.70/2.09 538 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K1)),bit0(K2))))) # label(fact_33_less__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 539 (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]. 1.70/2.09 540 (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]. 1.70/2.09 541 (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]. 1.70/2.09 542 (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]. 1.70/2.09 543 (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]. 1.70/2.09 544 (all X_a (mult_zero(X_a) -> (all A_1 zero_zero(X_a) = times_times(X_a,zero_zero(X_a),A_1)))) # label(fact_328_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 545 (all X_a (number_ring(X_a) & linordered_idom(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]. 1.70/2.09 546 (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]. 1.70/2.09 547 (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)) -> (minus_minus(int,A_1,one_one(int)) != ti(int,X) -> 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]. 1.70/2.09 548 (all X_a (semiri456707255roduct(X_a) -> (all B all A_2 (plus_plus(X_a,B,A_2) = ti(X_a,B) <-> ti(X_a,A_2) = zero_zero(X_a))))) # label(fact_212_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 549 (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]. 1.70/2.09 550 (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 (plus_plus(nat,I_2,K_2) = J_1 & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_2)))))) # label(fact_191_less__imp__add__positive) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 551 (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]. 1.70/2.09 552 (all Xa (Xa = zero_zero(real) <-> -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),times_times(real,Xa,Xa))))) # label(fact_653_not__real__square__gt__zero) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 553 (all X_a (monoid_mult(X_a) -> (all N one_one(X_a) = hAPP(nat,X_a,power_power(X_a,one_one(X_a)),N)))) # label(fact_166_power__one) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 554 (all X_a (power(X_a) -> (all M all N ((N = zero_zero(nat) -> one_one(X_a) = hAPP(nat,X_a,power_power(X_a,M),N)) & (zero_zero(nat) != N -> hAPP(nat,X_a,power_power(X_a,M),N) = times_times(X_a,M,hAPP(nat,X_a,power_power(X_a,M),minus_minus(nat,N,one_one(nat))))))))) # label(fact_558_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 555 (all M all N (minus_minus(nat,M,N) = zero_zero(nat) -> (minus_minus(nat,N,M) = zero_zero(nat) -> N = M))) # label(fact_335_diffs0__imp__equal) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 556 (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]. 1.70/2.09 557 (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]. 1.70/2.09 558 (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]. 1.70/2.09 559 (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]. 1.70/2.09 560 (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]. 1.70/2.09 561 (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)) -> (plus_plus(nat,times_times(nat,I_1,U),Ma) = plus_plus(nat,times_times(nat,J,U),Na) <-> Na = plus_plus(nat,times_times(nat,minus_minus(nat,I_1,J),U),Ma)))) # label(fact_989_nat__eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 562 (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]. 1.70/2.09 563 (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]. 1.70/2.09 564 (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]. 1.70/2.09 565 (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_665_int__mult) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 566 (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),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),zero_zero(X_a)),C_1)) & 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),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]. 1.70/2.09 567 (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]. 1.70/2.09 568 (all Na all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma)) -> (times_times(int,Ma,Na) = one_one(int) <-> ti(int,Na) = one_one(int) & ti(int,Ma) = one_one(int)))) # label(fact_510_pos__zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 569 (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]. 1.70/2.09 570 (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,Wa) != ti(int,Z_1))) # label(fact_380_zless__le) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 571 (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]. 1.70/2.09 572 (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)) -> plus_plus(X_a,one_one(X_a),number_number_of(X_a,V_1)) = number_number_of(X_a,succ(V_1)))))) # label(fact_479_semiring__1__add__number__of) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 573 (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),plus_plus(X_a,A_2,C_1)),plus_plus(X_a,B,C_1))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B)))))) # label(fact_359_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 574 (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_675_add__diff__assoc2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 575 (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]. 1.70/2.09 576 (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]. 1.70/2.09 577 (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_eq(X_a),number_number_of(X_a,Va)),number_number_of(X_a,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))))))) # label(fact_448_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 578 (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),plus_plus(X_a,A_2,C_1)),plus_plus(X_a,B,C_1))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B)))))) # label(fact_226_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 579 (all T all A ti(T,A) = ti(T,ti(T,A))) # label(help_ti_idem) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 580 (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)) -> (plus_plus(nat,times_times(nat,J,U),Na) = plus_plus(nat,times_times(nat,I_1,U),Ma) <-> plus_plus(nat,times_times(nat,minus_minus(nat,J,I_1),U),Na) = Ma))) # label(fact_992_nat__eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 581 (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]. 1.70/2.09 582 (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]. 1.70/2.09 583 (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)) -> nat_1(plus_plus(int,Z,Z_2)) = plus_plus(nat,nat_1(Z),nat_1(Z_2))))) # label(fact_397_nat__add__distrib) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 584 (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]. 1.70/2.09 585 (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]. 1.70/2.09 586 (all B_1_1 all B_2_1 all X_a (group_add(X_a) -> ti(X_a,minus_minus(X_a,B_1_1,B_2_1)) = minus_minus(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Ominus__class_Ominus_0_res) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 587 (all Ma all Va (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,Va))) & nat_1(number_number_of(int,Va)) = Ma <-> hAPP(nat,int,semiring_1_of_nat(int),Ma) = number_number_of(int,Va))) # label(fact_398_int__eq__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 588 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]. 1.70/2.09 589 (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),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)),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),zero_zero(X_a)),A_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_925_abs__eq__mult) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 590 (all Xa all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Xa)) | Na = zero_zero(nat) <-> 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]. 1.70/2.09 591 (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]. 1.70/2.09 592 (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]. 1.70/2.09 593 (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]. 1.70/2.09 594 (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]. 1.70/2.09 595 (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]. 1.70/2.09 596 (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]. 1.70/2.09 597 (all P_1 ((all X1 hBOOL(hAPP(nat,bool,P_1,X1))) <-> (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))))))) # label(fact_292_all__nat) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 598 (all Z_1 (abs_abs(int,Z_1) = one_one(int) <-> one_one(int) = ti(int,Z_1) | number_number_of(int,min) = ti(int,Z_1))) # label(fact_957_abs__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 599 (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]. 1.70/2.09 600 (all S_1 all T_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),S_1),T_2)) -> T_2 != S_1)) # label(fact_131_less__not__refl3) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 601 (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]. 1.70/2.09 602 (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]. 1.70/2.09 603 (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),A_2),B)) <-> 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_412_mult__less__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 604 (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]. 1.70/2.09 605 (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]. 1.70/2.09 606 (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]. 1.70/2.09 607 (all X_a (monoid_mult(X_a) -> (all A_1 all N times_times(X_a,A_1,hAPP(nat,X_a,power_power(X_a,A_1),N)) = times_times(X_a,hAPP(nat,X_a,power_power(X_a,A_1),N),A_1)))) # label(fact_362_power__commutes) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 608 (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]. 1.70/2.09 609 (all X_a (linordered_semidom(X_a) -> (all 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))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)))))) # label(fact_107_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 610 (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),A_2),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_2,A_2)),zero_zero(X_a))))))) # label(fact_195_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 611 (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]. 1.70/2.09 612 (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]. 1.70/2.09 613 (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]. 1.70/2.09 614 (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),dvd_dvd(nat),nat_1(Xa)),nat_1(Ya))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Xa),Ya)))))) # label(fact_833_transfer__nat__int__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 615 (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]. 1.70/2.09 616 (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]. 1.70/2.09 617 (all A_2 all B all Ma (hBOOL(hAPP(int,bool,zcong(A_2,B),Ma)) <-> (exists K_2 ti(int,B) = plus_plus(int,A_2,times_times(int,Ma,K_2))))) # label(fact_895_zcong__iff__lin) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 618 (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]. 1.70/2.09 619 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) -> (N != M -> 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]. 1.70/2.09 620 (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]. 1.70/2.09 621 (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]. 1.70/2.09 622 (all M all N (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) -> plus_plus(nat,N,minus_minus(nat,M,N)) = M)) # label(fact_456_add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 623 (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]. 1.70/2.09 624 (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]. 1.70/2.09 625 (all X_a (one(X_a) -> (all Xa (ti(X_a,Xa) = one_one(X_a) <-> ti(X_a,Xa) = one_one(X_a))))) # label(fact_211_one__reorient) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 626 (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]. 1.70/2.09 627 (all 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)) = ti(int,Z))) # label(fact_319_nat__0__le) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 628 (all X 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))))) = hAPP(nat,real,power_power(real,times_times(real,number_number_of(real,bit0(bit1(pls))),X)),number_number_of(nat,bit0(bit1(pls))))) # label(fact_695_four__x__squared) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 629 (all K_1 all L plus_plus(int,bit0(K_1),bit0(L)) = bit0(plus_plus(int,K_1,L))) # label(fact_77_add__Bit0__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 630 (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]. 1.70/2.09 631 (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),bit1(pls)),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))))))) # label(fact_403_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 632 (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) <-> B = A_2))) # label(fact_603_real__mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 633 (all C all A_1 all B_1 (B_1 = A_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]. 1.70/2.09 634 (all X_a (comm_semiring_1(X_a) -> (all A_1 all C all D_1 plus_plus(X_a,A_1,plus_plus(X_a,C,D_1)) = plus_plus(X_a,C,plus_plus(X_a,A_1,D_1))))) # label(fact_209_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 635 (all M all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),M)) -> (hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),M)) -> zero_zero(int) = ti(int,A_1))))) # label(fact_898_zcong__zless__0) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 636 (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]. 1.70/2.09 637 (all L bit0(minus_minus(int,min,L)) = minus_minus(int,min,bit1(L))) # label(fact_783_diff__bin__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 638 (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]. 1.70/2.09 639 (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]. 1.70/2.09 640 (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]. 1.70/2.09 641 (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]. 1.70/2.09 642 (all X all N hAPP(nat,int,power_power(int,hAPP(nat,int,semiring_1_of_nat(int),X)),N) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,power_power(nat,X),N))) # label(fact_175_Nat__Transfer_Otransfer__int__nat__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 643 (all M all N (M = times_times(nat,M,N) -> one_one(nat) = N | M = zero_zero(nat))) # label(fact_657_mult__eq__self__implies__10) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 644 (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]. 1.70/2.09 645 (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]. 1.70/2.09 646 (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]. 1.70/2.09 647 (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,Ya) = ti(int,Xa) <-> nat_1(Xa) = nat_1(Ya))))) # label(fact_282_transfer__nat__int__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 648 (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]. 1.70/2.09 649 (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]. 1.70/2.09 650 (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]. 1.70/2.09 651 (all Xa all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Xa)) | Na = zero_zero(nat) <-> 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_143_nat__zero__less__power__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.09 652 (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]. 1.70/2.09 653 (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]. 1.70/2.09 654 (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]. 1.70/2.09 655 (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]. 1.70/2.09 656 (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)) -> (plus_plus(X_a,U_1,V_1) = one_one(X_a) -> 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]. 1.70/2.10 657 (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_664_zmult__int) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 658 (all A_2 all 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)) <-> hBOOL(hAPP(int,bool,zcong(A_2,minus_minus(int,P_3,one_one(int))),P_3)))) # label(fact_912_inv__not__p__minus__1__aux) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 659 (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]. 1.70/2.10 660 (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]. 1.70/2.10 661 (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]. 1.70/2.10 662 (all Ma all K 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,Ma,K)),times_times(nat,Na,K))))) # label(fact_650_mult__less__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 663 (all B_1_1 nat_1(ti(int,B_1_1)) = nat_1(B_1_1)) # label(tsy_c_Int_Onat_arg1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 664 (all K_1 bit0(K_1) != min) # label(fact_766_rel__simps_I45_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 665 (all X_a (comm_semiring_1(X_a) -> (all X all N (one_one(X_a) = ti(X_a,X) | 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),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]. 1.70/2.10 666 (all Xa all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),minus_minus(real,Xa,Ya)),zero_zero(real))))) # label(fact_584_real__le__eq__diff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 667 (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]. 1.70/2.10 668 (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]. 1.70/2.10 669 (all A_2 all 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)))) <-> abs_abs(int,B) = abs_abs(int,A_2))) # label(fact_967_power2__eq__iff__abs__eq) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 670 (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)) -> Y != X)) # label(fact_851_dvd_Oless__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 671 (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]. 1.70/2.10 672 (all X_a (cancel_semigroup_add(X_a) -> (all B all A_2 all C_1 (plus_plus(X_a,B,A_2) = plus_plus(X_a,C_1,A_2) <-> ti(X_a,B) = ti(X_a,C_1))))) # label(fact_203_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 673 (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,C),B_1) = plus_plus(X_a,plus_plus(X_a,A_1,B_1),C)))) # label(fact_205_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 674 (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]. 1.70/2.10 675 (all Xa (one_one(int) = abs_abs(int,Xa) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Xa),one_one(int))))) # label(fact_923_zdvd1__eq) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 676 (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,A_2,E_1),C_1) = plus_plus(X_a,times_times(X_a,B,E_1),D) <-> plus_plus(X_a,times_times(X_a,minus_minus(X_a,A_2,B),E_1),C_1) = ti(X_a,D))))) # label(fact_286_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 677 (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]. 1.70/2.10 678 (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]. 1.70/2.10 679 (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_810_Nat__Transfer_Otransfer__int__nat__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 680 (all X_a (number_ring(X_a) -> (all X all Y 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)) = hAPP(nat,X_a,power_power(X_a,minus_minus(X_a,X,Y)),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_520_power2__diff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 681 (all K all I_1 all J (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J)) -> (minus_minus(nat,J,I_1) = K <-> J = plus_plus(nat,K,I_1)))) # label(fact_673_le__imp__diff__is__add) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 682 (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]. 1.70/2.10 683 (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),N),M)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),M),zero_zero(int))))) # label(fact_738_zdvd__bounds) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 684 (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),ord_less_eq(nat),Ma),Na)) & Ma != Na)) # label(fact_609_nat__less__le) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 685 (all N all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M)) -> N != M)) # label(fact_130_less__not__refl2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 686 (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(B_1,zero_zero(int)),P_2)) | hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_2)))))) # label(fact_828_zcong__zprime__prod__zero) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 687 (all X_a (group_add(X_a) -> (all A_2 all B (ti(X_a,B) = ti(X_a,A_2) <-> zero_zero(X_a) = minus_minus(X_a,A_2,B))))) # label(fact_320_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 688 (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]. 1.70/2.10 689 (all X_a (linordered_idom(X_a) -> (all A_1 abs_abs(X_a,hAPP(nat,X_a,power_power(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_964_abs__power2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 690 (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]. 1.70/2.10 691 (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]. 1.70/2.10 692 (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]. 1.70/2.10 693 (all N times_times(nat,one_one(nat),N) = N) # label(fact_629_nat__mult__1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 694 (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]. 1.70/2.10 695 (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),C_1),D)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B))))))) # label(fact_249_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 696 (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]. 1.70/2.10 697 (all J all K all I_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),plus_plus(nat,I_1,K))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),minus_minus(nat,J,K)),I_1)))) # label(fact_667_le__diff__conv) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 698 -(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]. 1.70/2.10 699 (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]. 1.70/2.10 700 (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]. 1.70/2.10 701 (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(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))))))))))) # label(fact_97_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 702 (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]. 1.70/2.10 703 (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]. 1.70/2.10 704 (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]. 1.70/2.10 705 (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]. 1.70/2.10 706 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),bit1(K))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K)))) # label(fact_503_rel__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 707 (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]. 1.70/2.10 708 (all Z ti(int,Z) = times_times(int,Z,one_one(int))) # label(fact_379_zmult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 709 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),bit0(K))))) # label(fact_464_rel__simps_I21_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 710 (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]. 1.70/2.10 711 (all X_a (comm_monoid_add(X_a) -> (all A_1 ti(X_a,A_1) = plus_plus(X_a,A_1,zero_zero(X_a))))) # label(fact_213_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 712 (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]. 1.70/2.10 713 (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]. 1.70/2.10 714 (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]. 1.70/2.10 715 (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]. 1.70/2.10 716 (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]. 1.70/2.10 717 (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]. 1.70/2.10 718 (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]. 1.70/2.10 719 (all X_a (ring(X_a) -> (all X all Y all A_1 all 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)) = minus_minus(X_a,times_times(X_a,X,Y),times_times(X_a,A_1,B_1))))) # label(fact_289_mult__diff__mult) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 720 (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]. 1.70/2.10 721 (all C all A_1 all 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)) -> (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]. 1.70/2.10 722 (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]. 1.70/2.10 723 (all X_a (number_ring(X_a) -> (all A_1 ti(X_a,A_1) = times_times(X_a,number_number_of(X_a,bit1(pls)),A_1)))) # label(fact_498_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 724 (all X_a (comm_semiring_1(X_a) -> (all A_1 ti(X_a,A_1) = plus_plus(X_a,A_1,zero_zero(X_a))))) # label(fact_214_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 725 (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),abs_abs(X_a,K))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Ma),K)))))) # label(fact_921_dvd__abs__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 726 (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]. 1.70/2.10 727 (all Ma all 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)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),nat_1(Z_1))))) # label(fact_487_zless__nat__eq__int__zless) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 728 (all Va (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Va),pls)) <-> number_number_of(nat,Va) = zero_zero(nat))) # label(fact_502_eq__number__of__0) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 729 (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]. 1.70/2.10 730 (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]. 1.70/2.10 731 (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]. 1.70/2.10 732 (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]. 1.70/2.10 733 (all X_a (cancel_semigroup_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,B_1) = ti(X_a,C))))) # label(fact_201_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 734 (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]. 1.70/2.10 735 (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]. 1.70/2.10 736 (all B_1_1 all B_2_1 all X_a (cancel_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_2_res) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 737 (all Xa all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya)) & Xa != Ya)) # label(fact_610_real__less__def) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 738 (all P_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),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,P_3) = ti(int,M_1) | one_one(int) = ti(int,M_1))) <-> hBOOL(hAPP(int,bool,zprime,P_3)))) # label(fact_901_zprime__def) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 739 (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]. 1.70/2.10 740 (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]. 1.70/2.10 741 (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]. 1.70/2.10 742 (all B_1_1 all B_2_1 all X_a (power(X_a) -> times_times(X_a,B_1_1,ti(X_a,B_2_1)) = times_times(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_1_arg2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 743 (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]. 1.70/2.10 744 (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 (A_1 = hAPP(nat,real,power_power(real,R),N) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R))))))) # label(fact_230_realpow__pos__nth) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 745 (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]. 1.70/2.10 746 (all I_2 all J_1 all K_1 minus_minus(nat,minus_minus(nat,I_2,K_1),J_1) = minus_minus(nat,minus_minus(nat,I_2,J_1),K_1)) # label(fact_267_diff__commute) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 747 (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]. 1.70/2.10 748 (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]. 1.70/2.10 749 (all X_a (linordered_semidom(X_a) -> (all A_1 all N all B_1 (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)),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]. 1.70/2.10 750 (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]. 1.70/2.10 751 (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]. 1.70/2.10 752 (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]. 1.70/2.10 753 (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]. 1.70/2.10 754 (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]. 1.70/2.10 755 (all K_1 succ(bit0(K_1)) = bit1(K_1)) # label(fact_514_succ__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 756 (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]. 1.70/2.10 757 (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]. 1.70/2.10 758 (all M all N (plus_plus(nat,M,N) = M -> N = zero_zero(nat))) # label(fact_153_add__eq__self__zero) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 759 (all L_1 (min = ti(int,L_1) <-> bit1(L_1) = min)) # label(fact_770_rel__simps_I43_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 760 (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]. 1.70/2.10 761 (all Ma all Wa ((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)) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa)) -> zero_zero(nat) = Ma) <-> nat_1(Wa) = Ma)) # label(fact_394_nat__eq__iff2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 762 (all K all L_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K)),bit1(L_1))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L_1)))) # label(fact_462_rel__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 763 (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)) -> (nat_1(Z_3) = nat_1(Z_1) <-> ti(int,Z_3) = ti(int,Z_1))))) # label(fact_261_eq__nat__nat__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 764 (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]. 1.70/2.10 765 (all X_a (linord581940658strict(X_a) -> (all Xa all Ya (ti(X_a,Ya) != zero_zero(X_a) | ti(X_a,Xa) != 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]. 1.70/2.10 766 (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]. 1.70/2.10 767 (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]. 1.70/2.10 768 -(all S all W_1 -(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) & 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)))))) # 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]. 1.70/2.10 769 (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_844_dvd_Oless__asym) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 770 (all 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)) & (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)) = ti(int,Z)))) # label(fact_317_int__nat__eq) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 771 (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]. 1.70/2.10 772 (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]. 1.70/2.10 773 (all X_a (ab_semigroup_mult(X_a) -> (all A_1 all B_1 all C times_times(X_a,times_times(X_a,A_1,B_1),C) = times_times(X_a,A_1,times_times(X_a,B_1,C))))) # label(fact_242_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 774 (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]. 1.70/2.10 775 (all K_1 all L bit0(K_1) != bit1(L)) # label(fact_69_rel__simps_I49_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 776 (all Z1 all Z2 all Z3 times_times(int,Z1,times_times(int,Z2,Z3)) = times_times(int,times_times(int,Z1,Z2),Z3)) # label(fact_254_zmult__assoc) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 777 (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]. 1.70/2.10 778 (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]. 1.70/2.10 779 (all K all L_1 (bit0(L_1) = bit0(K) <-> ti(int,L_1) = ti(int,K))) # label(fact_43_rel__simps_I48_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 780 (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]. 1.70/2.10 781 (all N (zero_zero(nat) != N -> 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]. 1.70/2.10 782 (all B_1_1 all B_2_1 all X_a (mult_zero(X_a) -> times_times(X_a,B_1_1,B_2_1) = times_times(X_a,B_1_1,ti(X_a,B_2_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_4_arg2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 783 (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]. 1.70/2.10 784 (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),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_eq(X_a),A_2),B)))))) # label(fact_358_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 785 -(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]. 1.70/2.10 786 (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]. 1.70/2.10 787 (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]. 1.70/2.10 788 (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]. 1.70/2.10 789 (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)) | Xa = Ya <-> 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]. 1.70/2.10 790 (all K_1 plus_plus(int,pls,K_1) = ti(int,K_1)) # label(fact_76_add__Pls) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 791 (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]. 1.70/2.10 792 (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]. 1.70/2.10 793 (all M minus_minus(nat,M,M) = zero_zero(nat)) # label(fact_336_diff__self__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 794 (all X_a (monoid_mult(X_a) -> (all A_1 hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,bit1(bit1(pls)))) = times_times(X_a,times_times(X_a,A_1,A_1),A_1)))) # label(fact_527_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 795 (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]. 1.70/2.10 796 (all X_a (ordered_ab_group_add(X_a) -> (all A_2 all B all C_1 all D (minus_minus(X_a,C_1,D) = minus_minus(X_a,A_2,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),C_1),D))))))) # label(fact_324_diff__eq__diff__less) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 797 (all X_a (number_ring(X_a) -> zero_zero(X_a) = number_number_of(X_a,pls))) # label(fact_82_number__of__Pls) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 798 (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]. 1.70/2.10 799 (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]. 1.70/2.10 800 (all A_1 all B_1 times_times(int,plus_plus(int,A_1,B_1),minus_minus(int,A_1,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)))))) # label(fact_698_zspecial__product) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 801 (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]. 1.70/2.10 802 (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_859_dvd_Oantisym) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 803 (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]. 1.70/2.10 804 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B_1 all C plus_plus(X_a,times_times(X_a,A_1,C),times_times(X_a,B_1,C)) = times_times(X_a,plus_plus(X_a,A_1,B_1),C)))) # label(fact_343_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 805 (all Na all Ma (zero_zero(int) != ti(int,Ma) -> (one_one(int) = abs_abs(int,Na) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),times_times(int,Ma,Na)),Ma))))) # label(fact_961_zdvd__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 806 (all X all Y hAPP(nat,real,power_power(real,plus_plus(real,X,Y)),number_number_of(nat,bit0(bit1(pls)))) = 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))) # label(fact_721_real__sum__squared__expand) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 807 (all X_a (number_ring(X_a) -> (all V_1 all W all Z times_times(X_a,number_number_of(X_a,V_1),times_times(X_a,number_number_of(X_a,W),Z)) = times_times(X_a,number_number_of(X_a,times_times(int,V_1,W)),Z)))) # label(fact_260_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 808 (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]. 1.70/2.10 809 (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]. 1.70/2.10 810 (all Ma all Na (zero_zero(nat) = Ma | zero_zero(nat) = Na <-> zero_zero(nat) = times_times(nat,Ma,Na))) # label(fact_596_mult__is__0) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 811 (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]. 1.70/2.10 812 (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) = ti(fun(X_a,bool),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]. 1.70/2.10 813 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) <-> minus_minus(nat,Ma,Na) = zero_zero(nat))) # label(fact_658_diff__is__0__eq) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 814 (all X_a (zero_neq_one(X_a) -> one_one(X_a) != zero_zero(X_a))) # label(fact_163_one__neq__zero) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 815 (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]. 1.70/2.10 816 (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]. 1.70/2.10 817 (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]. 1.70/2.10 818 (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]. 1.70/2.10 819 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) -> zero_zero(nat) = minus_minus(nat,M,N))) # label(fact_659_diff__is__0__eq_H) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 820 (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]. 1.70/2.10 821 (all X_a (monoid_add(X_a) -> (all A_1 ti(X_a,A_1) = plus_plus(X_a,zero_zero(X_a),A_1)))) # label(fact_219_add__0__left) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 822 (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]. 1.70/2.10 823 (all K all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),times_times(nat,K,Ma)),times_times(nat,K,Na))) <-> K = zero_zero(nat) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),Na)))) # label(fact_984_nat__mult__dvd__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 824 (all X_a (number_ring(X_a) & linordered_idom(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)),one_one(X_a))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),bit1(pls))))))) # label(fact_404_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 825 (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]. 1.70/2.10 826 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),pls)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K)),pls)))) # label(fact_54_rel__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 827 (all Z plus_plus(int,zero_zero(int),Z) = ti(int,Z)) # label(fact_80_zadd__0) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 828 (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]. 1.70/2.10 829 (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]. 1.70/2.10 830 (all Xa all Ya (Ya = Xa <-> hAPP(nat,int,semiring_1_of_nat(int),Xa) = hAPP(nat,int,semiring_1_of_nat(int),Ya))) # label(fact_159_Nat__Transfer_Otransfer__int__nat__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 831 (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]. 1.70/2.10 832 (all Ma all 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))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)))) # label(fact_150_zless__int) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 833 (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]. 1.70/2.10 834 (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]. 1.70/2.10 835 (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]. 1.70/2.10 836 (all X all Y hAPP(nat,int,semiring_1_of_nat(int),plus_plus(nat,X,Y)) = plus_plus(int,hAPP(nat,int,semiring_1_of_nat(int),X),hAPP(nat,int,semiring_1_of_nat(int),Y))) # label(fact_174_Nat__Transfer_Otransfer__int__nat__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 837 (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]. 1.70/2.10 838 (all X_a (semiring_1(X_a) -> hAPP(nat,X_a,semiring_1_of_nat(X_a),zero_zero(nat)) = zero_zero(X_a))) # label(fact_188_of__nat__0) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 839 (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]. 1.70/2.10 840 (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]. 1.70/2.10 841 (all N minus_minus(nat,zero_zero(nat),N) = zero_zero(nat)) # label(fact_338_diff__0__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 842 (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),M)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),N)))))) # label(fact_900_zprime__zdvd__zmult) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 843 (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]. 1.70/2.10 844 (all Z plus_plus(int,Z,zero_zero(int)) = ti(int,Z)) # label(fact_79_zadd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 845 (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]. 1.70/2.10 846 (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(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Wa),Z_1)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),nat_1(Wa)),nat_1(Z_1)))))) # label(fact_688_nat__le__eq__zle) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 847 (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),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_269_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 848 (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]. 1.70/2.10 849 (all B_1_1 all B_2_1 all X_a (dvd(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_2_arg1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 850 -(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]. 1.70/2.10 851 (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]. 1.70/2.10 852 (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]. 1.70/2.10 853 (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]. 1.70/2.10 854 (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]. 1.70/2.10 855 (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]. 1.70/2.10 856 (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]. 1.70/2.10 857 (all I_2 all U_1 all J_1 all K_1 plus_plus(nat,times_times(nat,plus_plus(nat,I_2,J_1),U_1),K_1) = plus_plus(nat,times_times(nat,I_2,U_1),plus_plus(nat,times_times(nat,J_1,U_1),K_1))) # label(fact_981_left__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 858 (all B_1_1 ti(int,bit0(B_1_1)) = bit0(B_1_1)) # label(tsy_c_Int_OBit0_res) # label(hypothesis) # label(non_clause). [assumption]. 1.70/2.10 859 (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]. 1.70/2.10 860 (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_993_nat0__intermed__int__val) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 861 (all K all Na all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),Na)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),plus_plus(int,Na,times_times(int,K,Ma)))))) # label(fact_748_zdvd__reduce) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 862 (all K (bit1(K) = min <-> min = ti(int,K))) # label(fact_771_rel__simps_I47_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 863 (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,Na) = times_times(nat,K,Ma) <-> Ma = Na))) # label(fact_983_nat__mult__eq__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.10 864 (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]. 1.70/2.10 865 (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)) -> ti(X_a,A_1) = abs_abs(X_a,A_1))))) # label(fact_944_abs__of__pos) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 866 (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]. 1.70/2.11 867 (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))) -> (N != zero_zero(nat) -> 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]. 1.70/2.11 868 (all A_1 all B_1 all C (minus_minus(int,A_1,B_1) = ti(int,C) -> ti(int,A_1) = plus_plus(int,C,B_1))) # label(fact_719_Int2_Oaux1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 869 (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,minus_minus(X_a,A_2,B),E_1),C_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,A_2,E_1),C_1)),plus_plus(X_a,times_times(X_a,B,E_1),D))))))) # label(fact_295_less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 870 (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]. 1.70/2.11 871 (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)) -> plus_plus(X_a,number_number_of(X_a,V_1),one_one(X_a)) = number_number_of(X_a,succ(V_1)))))) # label(fact_480_semiring__number__of__add__1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 872 (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]. 1.70/2.11 873 (all B_1 all Q all R_1 all B_2 all Q_1 all R_2 (plus_plus(int,times_times(int,B_1,Q),R_1) = plus_plus(int,times_times(int,B_2,Q_1),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(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]. 1.70/2.11 874 (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]. 1.70/2.11 875 (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]. 1.70/2.11 876 (all A_2 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Ma),A_2)) <-> hBOOL(hAPP(int,bool,zcong(A_2,zero_zero(int)),Ma)))) # label(fact_802_zcong__zero__equiv__div) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 877 (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]. 1.70/2.11 878 (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]. 1.70/2.11 879 (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]. 1.70/2.11 880 (all X_a (semiring_1(X_a) -> ti(fun(nat,X_a),semiring_1_of_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]. 1.70/2.11 881 (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]. 1.70/2.11 882 (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]. 1.70/2.11 883 (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]. 1.70/2.11 884 (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),X),Z)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z),X))))) # label(fact_847_dvd_Oless__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 885 (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]. 1.70/2.11 886 (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))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_2),B)) | ti(X_a,C_1) = zero_zero(X_a))))) # label(fact_740_dvd__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 887 (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]. 1.70/2.11 888 (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]. 1.70/2.11 889 (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]. 1.70/2.11 890 (all A_1 (one_one(int) = hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls)))) -> abs_abs(int,A_1) = one_one(int))) # label(fact_968_power2__eq1__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 891 (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]. 1.70/2.11 892 (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]. 1.70/2.11 893 (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]. 1.70/2.11 894 (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]. 1.70/2.11 895 (all B_1_1 all B_2_1 all X_a (dvd(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_0_res) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 896 (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]. 1.70/2.11 897 (all Xa all Ya (zero_zero(real) = plus_plus(real,times_times(real,Xa,Xa),times_times(real,Ya,Ya)) <-> zero_zero(real) = Ya & zero_zero(real) = Xa)) # label(fact_660_real__two__squares__add__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 898 (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]. 1.70/2.11 899 (all Z_2 all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) -> nat_1(times_times(int,Z,Z_2)) = times_times(nat,nat_1(Z),nat_1(Z_2)))) # label(fact_685_nat__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 900 (all Na all Ma (times_times(nat,Na,Ma) = one_one(nat) <-> one_one(nat) = Ma & one_one(nat) = Na)) # label(fact_625_nat__mult__eq__one) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 901 (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]. 1.70/2.11 902 (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]. 1.70/2.11 903 (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]. 1.70/2.11 904 (all M all N plus_plus(nat,M,N) = plus_plus(nat,N,M)) # label(fact_122_nat__add__commute) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 905 (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]. 1.70/2.11 906 (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),times_times(X_a,A_2,C_1)),times_times(X_a,B,C_1))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),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)))))) # label(fact_410_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 907 (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]. 1.70/2.11 908 (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]. 1.70/2.11 909 (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)) = zero_zero(nat)) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls)) -> number_number_of(nat,times_times(int,V_1,V_2)) = 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]. 1.70/2.11 910 (all X all Y all Z plus_plus(int,X,plus_plus(int,Y,Z)) = plus_plus(int,Y,plus_plus(int,X,Z))) # label(fact_46_zadd__left__commute) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 911 (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]. 1.70/2.11 912 (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]. 1.70/2.11 913 (all Z times_times(real,one_one(real),Z) = Z) # label(fact_632_real__mult__1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 914 (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]. 1.70/2.11 915 (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)))))) <-> zero_zero(X_a) != ti(X_a,A_2))))) # label(fact_95_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 916 (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]. 1.70/2.11 917 (all X_a (comm_semiring_1(X_a) -> (all A_1 all C plus_plus(X_a,C,A_1) = plus_plus(X_a,A_1,C)))) # label(fact_210_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 918 (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,bit1(N)))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,bit0(N)))))) # label(fact_914_number__of1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 919 (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]. 1.70/2.11 920 (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]. 1.70/2.11 921 (all B_1_1 all B_2_1 all X_a (power(X_a) -> times_times(X_a,B_1_1,B_2_1) = ti(X_a,times_times(X_a,B_1_1,B_2_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_1_res) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 922 (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]. 1.70/2.11 923 (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),Wa),Z_1)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z_1)))) # label(fact_484_zless__nat__conj) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 924 (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]. 1.70/2.11 925 (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]. 1.70/2.11 926 (all B_1_1 all B_2_1 all X_a (mult_zero(X_a) -> times_times(X_a,B_1_1,B_2_1) = ti(X_a,times_times(X_a,B_1_1,B_2_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_4_res) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 927 (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]. 1.70/2.11 928 (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]. 1.70/2.11 929 (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]. 1.70/2.11 930 (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]. 1.70/2.11 931 (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]. 1.70/2.11 932 (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]. 1.70/2.11 933 (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]. 1.70/2.11 934 (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]. 1.70/2.11 935 (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]. 1.70/2.11 936 (all X_a (ring_char_0(X_a) & number_ring(X_a) -> (all Xa all Ya (number_number_of(X_a,Xa) = number_number_of(X_a,Ya) <-> ti(int,Xa) = ti(int,Ya))))) # label(fact_40_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 937 (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]. 1.70/2.11 938 (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]. 1.70/2.11 939 (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]. 1.70/2.11 940 (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]. 1.70/2.11 941 (all X_a (power(X_a) & zero_neq_one(X_a) & no_zero_divisors(X_a) & mult_zero(X_a) -> (all A_2 all Wa (hAPP(nat,X_a,power_power(X_a,A_2),number_number_of(nat,Wa)) = zero_zero(X_a) <-> zero_zero(X_a) = ti(X_a,A_2) & zero_zero(nat) != number_number_of(nat,Wa))))) # label(fact_86_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 942 (all B_1_1 all B_2_1 zcong(B_1_1,B_2_1) = ti(fun(int,bool),zcong(B_1_1,B_2_1))) # label(tsy_c_IntPrimes_Ozcong_res) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 943 (all Ma (one_one(nat) = Ma <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),one_one(nat))))) # label(fact_813_nat__dvd__1__iff__1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 944 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K)),min)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),min)))) # label(fact_751_rel__simps_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 945 (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]. 1.70/2.11 946 (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]. 1.70/2.11 947 (all X_a (semiri456707255roduct(X_a) -> (all Wa all Ya all Xa all Z_1 (ti(X_a,Xa) = ti(X_a,Wa) | ti(X_a,Ya) = ti(X_a,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)))))) # label(fact_341_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 948 (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]. 1.70/2.11 949 (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)) <-> Xa != Ya & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)))) # label(fact_869_dvd_Oless__le) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 950 (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]. 1.70/2.11 951 (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]. 1.70/2.11 952 (all B_1_1 all B_2_1 all X_a (semiring(X_a) -> times_times(X_a,B_1_1,B_2_1) = times_times(X_a,B_1_1,ti(X_a,B_2_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_2_arg2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 953 (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]. 1.70/2.11 954 (all X_a (comm_semiring_1(X_a) -> (all A_1 plus_plus(X_a,zero_zero(X_a),A_1) = ti(X_a,A_1)))) # label(fact_218_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 955 (all X_a (group_add(X_a) -> (all A_1 zero_zero(X_a) = minus_minus(X_a,A_1,A_1)))) # label(fact_322_diff__self) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 956 (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]. 1.70/2.11 957 (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]. 1.70/2.11 958 (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]. 1.70/2.11 959 (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]. 1.70/2.11 960 (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]. 1.70/2.11 961 (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]. 1.70/2.11 962 (all Ma all K all Na (Ma = Na <-> plus_plus(nat,Na,K) = plus_plus(nat,Ma,K))) # label(fact_126_nat__add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 963 (all N plus_plus(nat,zero_zero(nat),N) = N) # label(fact_156_plus__nat_Oadd__0) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 964 (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]. 1.70/2.11 965 (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]. 1.70/2.11 966 (all X_a (comm_semiring_1(X_a) -> (all Lx all Rx all Ry times_times(X_a,Rx,times_times(X_a,Lx,Ry)) = times_times(X_a,Lx,times_times(X_a,Rx,Ry))))) # label(fact_240_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 967 (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]. 1.70/2.11 968 (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]. 1.70/2.11 969 (all X_a (group_add(X_a) -> (all A_1 all B_1 minus_minus(X_a,plus_plus(X_a,A_1,B_1),B_1) = ti(X_a,A_1)))) # label(fact_326_add__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 970 (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]. 1.70/2.11 971 (all X_a (number_ring(X_a) -> (all Z times_times(X_a,number_number_of(X_a,bit0(bit1(pls))),Z) = plus_plus(X_a,Z,Z)))) # label(fact_537_mult__2) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 972 (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(plus_plus(int,X,Y)) = plus_plus(nat,nat_1(X),nat_1(Y))))) # label(fact_396_Nat__Transfer_Otransfer__nat__int__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 973 (all B_1_1 bit1(ti(int,B_1_1)) = bit1(B_1_1)) # label(tsy_c_Int_OBit1_arg1) # label(hypothesis) # label(non_clause). [assumption]. 1.70/2.11 974 (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) <-> ti(X_a,Xa) = zero_zero(X_a) & zero_zero(X_a) = ti(X_a,Ya))))))) # label(fact_435_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 975 (all X_a (number_ring(X_a) -> (all A_1 ti(X_a,A_1) = times_times(X_a,A_1,number_number_of(X_a,bit1(pls)))))) # label(fact_497_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 976 (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]. 1.70/2.11 977 (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]. 1.70/2.11 978 (all K all L_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K)),bit0(L_1))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L_1)))) # label(fact_375_rel__simps_I31_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 979 (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]. 1.70/2.11 980 (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]. 1.70/2.11 981 (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]. 1.70/2.11 982 (all N all M M = minus_minus(nat,plus_plus(nat,N,M),N)) # label(fact_366_diff__add__inverse) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 983 (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]. 1.70/2.11 984 (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]. 1.70/2.11 985 (all X_a (comm_monoid_mult(X_a) -> (all A_1 ti(X_a,A_1) = times_times(X_a,one_one(X_a),A_1)))) # label(fact_350_mult__1) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 986 (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]. 1.70/2.11 987 (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]. 1.70/2.11 988 (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]. 1.70/2.11 989 (all Ma all 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))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)))) # label(fact_678_zle__int) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 990 (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]. 1.70/2.11 991 (all N N = times_times(nat,N,one_one(nat))) # label(fact_627_nat__mult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 992 (all X_a (comm_semiring_1(X_a) -> (all X all Y all Q hAPP(nat,X_a,power_power(X_a,times_times(X_a,X,Y)),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))))) # label(fact_364_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 993 (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]. 1.70/2.11 994 (all P_1 all Xa all Ya ((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(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Xa),Ya)) -> hBOOL(hAPP(int,bool,P_1,zero_zero(int)))) <-> 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]. 1.70/2.11 995 (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]. 1.70/2.11 996 (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]. 1.70/2.11 997 (all X_a (number_ring(X_a) -> (all N 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)) = one_one(X_a)))) # label(fact_791_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 998 (all X_a (comm_monoid_add(X_a) -> (all A_1 ti(X_a,A_1) = plus_plus(X_a,zero_zero(X_a),A_1)))) # label(fact_217_add__0) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 999 (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),bit1(Wa)),zero_zero(int))))) # label(fact_56_bin__less__0__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.70/2.11 1000 (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]. 1.70/2.11 1001 (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]. 1.70/2.11 1002 (all P_1 ((exists X_1 (hBOOL(hAPP(nat,bool,P_1,nat_1(X_1))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)))) <-> (exists X1 hBOOL(hAPP(nat,bool,P_1,X1))))) # label(fact_293_ex__nat) # label(axiom) # label(non_clause). [assumption]. 1.70/2.14 1003 (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]. 1.70/2.14 1004 (all A_1 all P_2 ((hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_2)) -> legendre(A_1,P_2) = zero_zero(int)) & (-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))))) # label(fact_877_Legendre__def) # label(axiom) # label(non_clause). [assumption]. 1.70/2.14 1005 (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,A_2,E_1),C_1) = plus_plus(X_a,times_times(X_a,B,E_1),D) <-> 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]. 1.70/2.14 1006 (all P_2 (hBOOL(hAPP(int,bool,zprime,P_2)) -> (number_number_of(int,bit0(bit1(pls))) != ti(int,P_2) -> (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]. 1.70/2.14 1007 (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]. 1.70/2.14 1008 (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)),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),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]. 1.70/2.14 1009 (all X_a (comm_monoid_mult(X_a) -> (all A_1 all B_1 all 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)) = hAPP(nat,X_a,power_power(X_a,times_times(X_a,A_1,B_1)),N)))) # label(fact_363_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.70/2.14 1010 (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]. 1.70/2.14 1011 -(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) # label(non_clause). [assumption]. 1.70/2.14 1.70/2.14 ============================== end of process non-clausal formulas === 1.70/2.14 1.70/2.14 ============================== PROCESS INITIAL CLAUSES =============== 1.70/2.14 1.70/2.14 ============================== PREDICATE ELIMINATION ================= 1.70/2.14 1012 -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(162)]. 1.70/2.14 1013 monoid_add(nat) # label(arity_Nat_Onat___Groups_Omonoid__add) # label(axiom). [assumption]. 1.70/2.14 Derived: plus_plus(nat,A,ti(nat,B)) = plus_plus(nat,A,B). [resolve(1012,a,1013,a)]. 1.70/2.14 1014 -monoid_add(A) | ti(A,B) = plus_plus(A,B,zero_zero(A)) # label(fact_215_add__0__right) # label(axiom). [clausify(293)]. 1.70/2.14 Derived: ti(nat,A) = plus_plus(nat,A,zero_zero(nat)). [resolve(1014,a,1013,a)]. 1.70/2.14 1015 -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(498)]. 1.70/2.14 Derived: ti(nat,plus_plus(nat,A,B)) = plus_plus(nat,A,B). [resolve(1015,a,1013,a)]. 2.08/2.34 1016 monoid_add(real) # label(arity_RealDef_Oreal___Groups_Omonoid__add) # label(axiom). [assumption]. 2.08/2.34 Derived: plus_plus(real,A,ti(real,B)) = plus_plus(real,A,B). [resolve(1016,a,1012,a)]. 2.08/2.34 Derived: ti(real,A) = plus_plus(real,A,zero_zero(real)). [resolve(1016,a,1014,a)]. 2.08/2.34 Derived: ti(real,plus_plus(real,A,B)) = plus_plus(real,A,B). [resolve(1016,a,1015,a)]. 2.08/2.34 1017 -monoid_add(A) | ti(A,B) = plus_plus(A,zero_zero(A),B) # label(fact_219_add__0__left) # label(axiom). [clausify(821)]. 2.08/2.34 Derived: ti(nat,A) = plus_plus(nat,zero_zero(nat),A). [resolve(1017,a,1013,a)]. 2.08/2.34 Derived: ti(real,A) = plus_plus(real,zero_zero(real),A). [resolve(1017,a,1016,a)]. 2.08/2.34 1018 -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(856)]. 2.08/2.34 Derived: plus_plus(nat,ti(nat,A),B) = plus_plus(nat,A,B). [resolve(1018,a,1013,a)]. 2.08/2.34 Derived: plus_plus(real,ti(real,A),B) = plus_plus(real,A,B). [resolve(1018,a,1016,a)]. 2.08/2.34 1019 monoid_add(int) # label(arity_Int_Oint___Groups_Omonoid__add) # label(axiom). [assumption]. 2.08/2.34 Derived: plus_plus(int,A,ti(int,B)) = plus_plus(int,A,B). [resolve(1019,a,1012,a)]. 2.08/2.34 Derived: ti(int,A) = plus_plus(int,A,zero_zero(int)). [resolve(1019,a,1014,a)]. 2.08/2.34 Derived: ti(int,plus_plus(int,A,B)) = plus_plus(int,A,B). [resolve(1019,a,1015,a)]. 2.08/2.34 Derived: ti(int,A) = plus_plus(int,zero_zero(int),A). [resolve(1019,a,1017,a)]. 2.08/2.34 Derived: plus_plus(int,ti(int,A),B) = plus_plus(int,A,B). [resolve(1019,a,1018,a)]. 2.08/2.34 1020 -ring_n68954251visors(A) | zero_zero(A) != ti(A,B) | zero_zero(A) = times_times(A,C,B) # label(fact_330_mult__eq__0__iff) # label(axiom). [clausify(47)]. 2.08/2.34 1021 ring_n68954251visors(real) # label(arity_RealDef_Oreal___Rings_Oring__no__zero__divisors) # label(axiom). [assumption]. 2.08/2.34 Derived: zero_zero(real) != ti(real,A) | zero_zero(real) = times_times(real,B,A). [resolve(1020,a,1021,a)]. 2.08/2.34 1022 -ring_n68954251visors(A) | zero_zero(A) != ti(A,B) | zero_zero(A) = times_times(A,B,C) # label(fact_330_mult__eq__0__iff) # label(axiom). [clausify(47)]. 2.08/2.34 Derived: zero_zero(real) != ti(real,A) | zero_zero(real) = times_times(real,A,B). [resolve(1022,a,1021,a)]. 2.08/2.34 1023 -ring_n68954251visors(A) | zero_zero(A) = ti(A,B) | zero_zero(A) = ti(A,C) | zero_zero(A) != times_times(A,C,B) # label(fact_330_mult__eq__0__iff) # label(axiom). [clausify(47)]. 2.08/2.34 Derived: zero_zero(real) = ti(real,A) | zero_zero(real) = ti(real,B) | zero_zero(real) != times_times(real,B,A). [resolve(1023,a,1021,a)]. 2.08/2.34 1024 ring_n68954251visors(int) # label(arity_Int_Oint___Rings_Oring__no__zero__divisors) # label(axiom). [assumption]. 2.08/2.34 Derived: zero_zero(int) != ti(int,A) | zero_zero(int) = times_times(int,B,A). [resolve(1024,a,1020,a)]. 2.08/2.34 Derived: zero_zero(int) != ti(int,A) | zero_zero(int) = times_times(int,A,B). [resolve(1024,a,1022,a)]. 2.08/2.34 Derived: zero_zero(int) = ti(int,A) | zero_zero(int) = ti(int,B) | zero_zero(int) != times_times(int,B,A). [resolve(1024,a,1023,a)]. 2.08/2.34 1025 ordere142940540dd_abs(int) # label(arity_Int_Oint___Groups_Oordered__ab__group__add__abs) # label(axiom). [assumption]. 2.08/2.34 1026 -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(2)]. 2.08/2.34 1027 -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(62)]. 2.08/2.34 1028 -ordere142940540dd_abs(A) | abs_abs(A,plus_plus(A,abs_abs(A,B),abs_abs(A,C))) = plus_plus(A,abs_abs(A,B),abs_abs(A,C)) # label(fact_932_abs__add__abs) # label(axiom). [clausify(98)]. 2.08/2.34 1029 -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(126)]. 2.08/2.34 1030 -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(145)]. 2.08/2.37 1031 -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(249)]. 2.08/2.37 1032 -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(282)]. 2.08/2.37 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(1025,a,1026,a)]. 2.08/2.37 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(1025,a,1027,a)]. 2.08/2.37 Derived: abs_abs(int,plus_plus(int,abs_abs(int,A),abs_abs(int,B))) = plus_plus(int,abs_abs(int,A),abs_abs(int,B)). [resolve(1025,a,1028,a)]. 2.08/2.37 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),abs_abs(int,A)),zero_zero(int))). [resolve(1025,a,1029,a)]. 2.08/2.37 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(1025,a,1030,a)]. 2.08/2.37 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(1025,a,1031,a)]. 2.08/2.37 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),abs_abs(int,A))). [resolve(1025,a,1032,a)]. 2.08/2.37 1033 -ordere142940540dd_abs(A) | abs_abs(A,ti(A,B)) = abs_abs(A,B) # label(tsy_c_Groups_Oabs__class_Oabs_arg1) # label(axiom). [clausify(325)]. 2.08/2.37 Derived: abs_abs(int,ti(int,A)) = abs_abs(int,A). [resolve(1033,a,1025,a)]. 2.08/2.37 1034 -ordere142940540dd_abs(A) | abs_abs(A,abs_abs(A,B)) = abs_abs(A,B) # label(fact_937_abs__idempotent) # label(axiom). [clausify(369)]. 2.08/2.37 Derived: abs_abs(int,abs_abs(int,A)) = abs_abs(int,A). [resolve(1034,a,1025,a)]. 2.08/2.37 1035 -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(407)]. 2.08/2.37 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(1035,a,1025,a)]. 2.08/2.37 1036 -ordere142940540dd_abs(A) | zero_zero(A) = ti(A,B) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),abs_abs(A,B))) # label(fact_945_zero__less__abs__iff) # label(axiom). [clausify(429)]. 2.08/2.37 Derived: zero_zero(int) = ti(int,A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),abs_abs(int,A))). [resolve(1036,a,1025,a)]. 2.08/2.37 1037 -ordere142940540dd_abs(A) | zero_zero(A) != ti(A,B) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),abs_abs(A,B))) # label(fact_945_zero__less__abs__iff) # label(axiom). [clausify(429)]. 2.08/2.37 Derived: zero_zero(int) != ti(int,A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),abs_abs(int,A))). [resolve(1037,a,1025,a)]. 2.08/2.37 1038 -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(430)]. 2.08/2.37 Derived: abs_abs(int,minus_minus(int,A,B)) = abs_abs(int,minus_minus(int,B,A)). [resolve(1038,a,1025,a)]. 2.08/2.37 1039 ordere142940540dd_abs(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__group__add__abs) # label(axiom). [assumption]. 2.08/2.37 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(1039,a,1026,a)]. 2.08/2.37 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(1039,a,1027,a)]. 2.17/2.39 Derived: abs_abs(real,plus_plus(real,abs_abs(real,A),abs_abs(real,B))) = plus_plus(real,abs_abs(real,A),abs_abs(real,B)). [resolve(1039,a,1028,a)]. 2.17/2.39 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),abs_abs(real,A)),zero_zero(real))). [resolve(1039,a,1029,a)]. 2.17/2.39 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(1039,a,1030,a)]. 2.17/2.39 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(1039,a,1031,a)]. 2.17/2.39 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),abs_abs(real,A))). [resolve(1039,a,1032,a)]. 2.17/2.39 Derived: abs_abs(real,ti(real,A)) = abs_abs(real,A). [resolve(1039,a,1033,a)]. 2.17/2.39 Derived: abs_abs(real,abs_abs(real,A)) = abs_abs(real,A). [resolve(1039,a,1034,a)]. 2.17/2.39 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(1039,a,1035,a)]. 2.17/2.39 Derived: zero_zero(real) = ti(real,A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),abs_abs(real,A))). [resolve(1039,a,1036,a)]. 2.17/2.39 Derived: zero_zero(real) != ti(real,A) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),abs_abs(real,A))). [resolve(1039,a,1037,a)]. 2.17/2.39 Derived: abs_abs(real,minus_minus(real,A,B)) = abs_abs(real,minus_minus(real,B,A)). [resolve(1039,a,1038,a)]. 2.17/2.39 1040 -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(668)]. 2.17/2.39 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),abs_abs(int,A))). [resolve(1040,a,1025,a)]. 2.17/2.39 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),abs_abs(real,A))). [resolve(1040,a,1039,a)]. 2.17/2.39 1041 -ordere142940540dd_abs(A) | ti(A,abs_abs(A,B)) = abs_abs(A,B) # label(tsy_c_Groups_Oabs__class_Oabs_res) # label(axiom). [clausify(714)]. 2.17/2.39 Derived: ti(int,abs_abs(int,A)) = abs_abs(int,A). [resolve(1041,a,1025,a)]. 2.17/2.39 Derived: ti(real,abs_abs(real,A)) = abs_abs(real,A). [resolve(1041,a,1039,a)]. 2.17/2.39 1042 -ordere142940540dd_abs(A) | zero_zero(A) != ti(A,B) | zero_zero(A) = abs_abs(A,B) # label(fact_931_abs__eq__0) # label(axiom). [clausify(739)]. 2.17/2.39 Derived: zero_zero(int) != ti(int,A) | zero_zero(int) = abs_abs(int,A). [resolve(1042,a,1025,a)]. 2.17/2.39 Derived: zero_zero(real) != ti(real,A) | zero_zero(real) = abs_abs(real,A). [resolve(1042,a,1039,a)]. 2.17/2.39 1043 -ordere142940540dd_abs(A) | zero_zero(A) = ti(A,B) | zero_zero(A) != abs_abs(A,B) # label(fact_931_abs__eq__0) # label(axiom). [clausify(739)]. 2.17/2.39 Derived: zero_zero(int) = ti(int,A) | zero_zero(int) != abs_abs(int,A). [resolve(1043,a,1025,a)]. 2.17/2.39 Derived: zero_zero(real) = ti(real,A) | zero_zero(real) != abs_abs(real,A). [resolve(1043,a,1039,a)]. 2.17/2.39 1044 -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(743)]. 2.17/2.39 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(1044,a,1025,a)]. 2.17/2.39 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(1044,a,1039,a)]. 2.17/2.39 1045 -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(801)]. 2.17/2.39 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(1045,a,1025,a)]. 2.17/2.39 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(1045,a,1039,a)]. 2.26/2.49 1046 -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(865)]. 2.26/2.49 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(1046,a,1025,a)]. 2.26/2.49 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(1046,a,1039,a)]. 2.26/2.49 1047 -ordere142940540dd_abs(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),abs_abs(A,B)),zero_zero(A))) | zero_zero(A) = ti(A,B) # label(fact_948_abs__le__zero__iff) # label(axiom). [clausify(920)]. 2.26/2.49 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),abs_abs(int,A)),zero_zero(int))) | zero_zero(int) = ti(int,A). [resolve(1047,a,1025,a)]. 2.26/2.49 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),abs_abs(real,A)),zero_zero(real))) | zero_zero(real) = ti(real,A). [resolve(1047,a,1039,a)]. 2.26/2.49 1048 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),abs_abs(A,B)),zero_zero(A))) | zero_zero(A) != ti(A,B) # label(fact_948_abs__le__zero__iff) # label(axiom). [clausify(920)]. 2.26/2.49 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),abs_abs(int,A)),zero_zero(int))) | zero_zero(int) != ti(int,A). [resolve(1048,a,1025,a)]. 2.26/2.49 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),abs_abs(real,A)),zero_zero(real))) | zero_zero(real) != ti(real,A). [resolve(1048,a,1039,a)]. 2.26/2.49 1049 -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(950)]. 2.26/2.49 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(1049,a,1025,a)]. 2.26/2.49 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(1049,a,1039,a)]. 2.26/2.49 1050 -ordere142940540dd_abs(A) | zero_zero(A) = abs_abs(A,zero_zero(A)) # label(fact_930_abs__zero) # label(axiom). [clausify(980)]. 2.26/2.49 Derived: zero_zero(int) = abs_abs(int,zero_zero(int)). [resolve(1050,a,1025,a)]. 2.26/2.49 Derived: zero_zero(real) = abs_abs(real,zero_zero(real)). [resolve(1050,a,1039,a)]. 2.26/2.49 1051 linordered_semidom(nat) # label(arity_Nat_Onat___Rings_Olinordered__semidom) # label(axiom). [assumption]. 2.26/2.49 1052 -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(5)]. 2.26/2.49 1053 -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_167_of__nat__less__iff) # label(axiom). [clausify(9)]. 2.26/2.49 1054 -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_167_of__nat__less__iff) # label(axiom). [clausify(9)]. 2.26/2.49 1055 -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(24)]. 2.26/2.49 1056 -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(27)]. 2.26/2.49 1057 -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(72)]. 2.26/2.49 1058 -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(76)]. 2.26/2.49 1059 -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(82)]. 2.26/2.49 1060 -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(119)]. 2.26/2.49 1061 -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(127)]. 2.26/2.49 1062 -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(127)]. 2.26/2.49 1063 -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(185)]. 2.26/2.49 1064 -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(215)]. 2.26/2.49 1065 -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(270)]. 2.26/2.49 1066 -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(302)]. 2.26/2.49 1067 -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(333)]. 2.26/2.49 1068 -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(341)]. 2.26/2.49 1069 -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(371)]. 2.26/2.49 1070 -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(383)]. 2.26/2.49 1071 -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(395)]. 2.26/2.49 1072 -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(395)]. 2.26/2.49 1073 -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(411)]. 2.26/2.49 1074 -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(424)]. 2.26/2.49 1075 -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(431)]. 2.26/2.49 1076 -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(437)]. 2.26/2.49 1077 -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(524)]. 2.26/2.49 1078 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(nat,A,semiring_1_of_nat(A),B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) # label(fact_107_of__nat__0__less__iff) # label(axiom). [clausify(609)]. 2.26/2.49 1079 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(nat,A,semiring_1_of_nat(A),B))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) # label(fact_107_of__nat__0__less__iff) # label(axiom). [clausify(609)]. 2.26/2.49 1080 -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(611)]. 2.26/2.49 1081 -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(612)]. 2.26/2.49 1082 -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(654)]. 2.26/2.49 1083 -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(654)]. 2.26/2.50 1084 -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(692)]. 2.26/2.50 1085 -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(692)]. 2.26/2.50 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,semiring_1_of_nat(nat),A)),hAPP(nat,nat,semiring_1_of_nat(nat),B))). [resolve(1051,a,1053,a)]. 2.26/2.50 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,semiring_1_of_nat(nat),A)),hAPP(nat,nat,semiring_1_of_nat(nat),B))). [resolve(1051,a,1054,a)]. 2.26/2.50 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(1051,a,1055,a)]. 2.26/2.50 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),hAPP(nat,nat,power_power(nat,A),B))). [resolve(1051,a,1056,a)]. 2.26/2.50 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(1051,a,1057,a)]. 2.26/2.50 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(1051,a,1058,a)]. 2.26/2.50 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(1051,a,1059,a)]. 2.26/2.50 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(1051,a,1060,a)]. 2.26/2.50 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(1051,a,1061,a)]. 2.26/2.50 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(1051,a,1062,a)]. 2.26/2.50 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(1051,a,1064,a)]. 2.26/2.50 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(1051,a,1065,a)]. 2.28/2.50 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),zero_zero(nat))). [resolve(1051,a,1066,a)]. 2.28/2.50 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(1051,a,1067,a)]. 2.28/2.50 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(1051,a,1069,a)]. 2.28/2.50 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(1051,a,1071,a)]. 2.28/2.50 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(1051,a,1072,a)]. 2.28/2.50 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),one_one(nat))). [resolve(1051,a,1073,a)]. 2.28/2.50 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(1051,a,1075,a)]. 2.28/2.50 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(1051,a,1076,a)]. 2.28/2.50 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(1051,a,1077,a)]. 2.28/2.50 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,semiring_1_of_nat(nat),A))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)). [resolve(1051,a,1078,a)]. 2.28/2.50 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,semiring_1_of_nat(nat),A))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)). [resolve(1051,a,1079,a)]. 2.28/2.50 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(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(1051,a,1080,a)]. 2.28/2.50 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,power_power(nat,C),A)),hAPP(nat,nat,power_power(nat,C),B))). [resolve(1051,a,1081,a)]. 2.28/2.50 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(1051,a,1082,a)]. 2.28/2.50 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(1051,a,1084,a)]. 2.28/2.50 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(1051,a,1085,a)]. 2.28/2.50 1086 -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(715)]. 2.28/2.51 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),plus_plus(nat,A,one_one(nat)))). [resolve(1086,a,1051,a)]. 2.28/2.51 1087 -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(732)]. 2.28/2.51 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,power_power(nat,A),B))). [resolve(1087,a,1051,a)]. 2.28/2.51 1088 linordered_semidom(int) # label(arity_Int_Oint___Rings_Olinordered__semidom) # label(axiom). [assumption]. 2.28/2.51 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),zero_zero(int))). [resolve(1088,a,1052,a)]. 2.28/2.51 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(1088,a,1055,a)]. 2.28/2.51 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(1088,a,1057,a)]. 2.28/2.51 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(1088,a,1058,a)]. 2.28/2.51 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(1088,a,1059,a)]. 2.28/2.51 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(1088,a,1060,a)]. 2.28/2.51 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(1088,a,1061,a)]. 2.28/2.51 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(1088,a,1062,a)]. 2.28/2.51 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),A)),zero_zero(int))). [resolve(1088,a,1063,a)]. 2.28/2.51 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(1088,a,1065,a)]. 2.28/2.51 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),zero_zero(int))). [resolve(1088,a,1066,a)]. 2.28/2.51 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(1088,a,1067,a)]. 2.28/2.51 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(1088,a,1069,a)]. 2.28/2.51 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,1071,a)]. 2.28/2.51 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)]. 2.28/2.51 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),one_one(int))). [resolve(1088,a,1073,a)]. 2.28/2.51 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(1088,a,1075,a)]. 2.28/2.51 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(1088,a,1076,a)]. 2.28/2.51 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(1088,a,1077,a)]. 2.28/2.51 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),A))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)). [resolve(1088,a,1078,a)]. 2.28/2.51 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),A))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)). [resolve(1088,a,1079,a)]. 2.28/2.51 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(1088,a,1080,a)]. 2.28/2.51 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,power_power(int,C),A)),hAPP(nat,int,power_power(int,C),B))). [resolve(1088,a,1081,a)]. 2.28/2.51 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(1088,a,1082,a)]. 2.28/2.51 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),plus_plus(int,A,one_one(int)))). [resolve(1088,a,1086,a)]. 2.28/2.51 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(1088,a,1087,a)]. 2.28/2.51 1089 -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(745)]. 2.28/2.51 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(1089,a,1051,a)]. 2.29/2.53 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(1089,a,1088,a)]. 2.29/2.53 1090 -linordered_semidom(A) | hAPP(nat,A,power_power(A,B),C) != hAPP(nat,A,power_power(A,D),C) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C)) | ti(A,B) = ti(A,D) # label(fact_526_power__eq__imp__eq__base) # label(axiom). [clausify(749)]. 2.29/2.53 Derived: hAPP(nat,nat,power_power(nat,A),B) != hAPP(nat,nat,power_power(nat,C),B) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | ti(nat,A) = ti(nat,C). [resolve(1090,a,1051,a)]. 2.29/2.53 Derived: hAPP(nat,int,power_power(int,A),B) != hAPP(nat,int,power_power(int,C),B) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),C)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | ti(int,A) = ti(int,C). [resolve(1090,a,1088,a)]. 2.29/2.53 1091 -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(808)]. 2.29/2.53 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),one_one(nat))). [resolve(1091,a,1051,a)]. 2.29/2.53 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),one_one(int))). [resolve(1091,a,1088,a)]. 2.29/2.53 1092 -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(834)]. 2.29/2.53 1093 -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(851)]. 2.29/2.53 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(1093,a,1088,a)]. 2.29/2.53 1094 -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(874)]. 2.29/2.53 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(1094,a,1051,a)]. 2.29/2.53 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(1094,a,1088,a)]. 2.29/2.53 1095 -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(896)]. 2.29/2.54 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(1095,a,1051,a)]. 2.29/2.54 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(1095,a,1088,a)]. 2.29/2.54 1096 -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(937)]. 2.29/2.54 1097 linordered_semidom(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semidom) # label(axiom). [assumption]. 2.29/2.54 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),zero_zero(real))). [resolve(1097,a,1052,a)]. 2.29/2.54 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),hAPP(nat,real,semiring_1_of_nat(real),A)),hAPP(nat,real,semiring_1_of_nat(real),B))). [resolve(1097,a,1053,a)]. 2.29/2.54 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),hAPP(nat,real,semiring_1_of_nat(real),A)),hAPP(nat,real,semiring_1_of_nat(real),B))). [resolve(1097,a,1054,a)]. 2.29/2.54 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(1097,a,1055,a)]. 2.29/2.54 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(1097,a,1056,a)]. 2.29/2.54 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(1097,a,1057,a)]. 2.29/2.54 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(1097,a,1058,a)]. 2.29/2.54 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(1097,a,1059,a)]. 2.29/2.54 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(1097,a,1060,a)]. 2.29/2.54 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(1097,a,1061,a)]. 2.29/2.54 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(1097,a,1062,a)]. 2.29/2.54 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(1097,a,1063,a)]. 2.29/2.54 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(1097,a,1064,a)]. 2.29/2.54 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(1097,a,1065,a)]. 2.29/2.54 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),zero_zero(real))). [resolve(1097,a,1066,a)]. 2.29/2.54 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(1097,a,1067,a)]. 2.29/2.54 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(1097,a,1069,a)]. 2.29/2.54 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(1097,a,1071,a)]. 2.29/2.54 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(1097,a,1072,a)]. 2.29/2.54 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),one_one(real))). [resolve(1097,a,1073,a)]. 2.29/2.54 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(1097,a,1075,a)]. 2.29/2.54 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(1097,a,1076,a)]. 2.29/2.54 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(1097,a,1077,a)]. 2.29/2.54 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,semiring_1_of_nat(real),A))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)). [resolve(1097,a,1078,a)]. 2.29/2.54 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,semiring_1_of_nat(real),A))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)). [resolve(1097,a,1079,a)]. 2.29/2.54 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(1097,a,1080,a)]. 2.29/2.54 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,power_power(real,C),A)),hAPP(nat,real,power_power(real,C),B))). [resolve(1097,a,1081,a)]. 2.29/2.55 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(1097,a,1082,a)]. 2.29/2.55 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(1097,a,1084,a)]. 2.29/2.55 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(1097,a,1085,a)]. 2.29/2.55 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),plus_plus(real,A,one_one(real)))). [resolve(1097,a,1086,a)]. 2.29/2.55 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(1097,a,1087,a)]. 2.29/2.55 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(1097,a,1089,a)]. 2.29/2.55 Derived: hAPP(nat,real,power_power(real,A),B) != hAPP(nat,real,power_power(real,C),B) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | ti(real,A) = ti(real,C). [resolve(1097,a,1090,a)]. 2.29/2.55 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),one_one(real))). [resolve(1097,a,1091,a)]. 2.29/2.55 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(1097,a,1094,a)]. 2.29/2.55 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(1097,a,1095,a)]. 2.29/2.55 1098 -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(977)]. 2.29/2.55 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(1098,a,1051,a)]. 2.29/2.55 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(1098,a,1088,a)]. 2.29/2.55 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(1098,a,1097,a)]. 2.29/2.59 1099 comm_semiring_1(real) # label(arity_RealDef_Oreal___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 2.29/2.59 1100 -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(6)]. 2.29/2.59 1101 -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(43)]. 2.29/2.59 1102 -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(69)]. 2.29/2.60 1103 -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(93)]. 2.29/2.60 1104 -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(99)]. 2.29/2.60 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),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(1099,a,1100,a)]. 2.29/2.60 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(1099,a,1101,a)]. 2.29/2.60 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),A),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),plus_plus(real,B,C))). [resolve(1099,a,1103,a)]. 2.29/2.60 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(1099,a,1104,a)]. 2.29/2.60 1105 -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(149)]. 2.29/2.60 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),A),times_times(real,C,B))). [resolve(1105,a,1099,a)]. 2.29/2.60 1106 -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(207)]. 2.29/2.60 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(1106,a,1099,a)]. 2.29/2.60 1107 -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(237)]. 2.29/2.60 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(1107,a,1099,a)]. 2.29/2.60 1108 -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(258)]. 2.29/2.60 Derived: times_times(real,times_times(real,A,B),times_times(real,C,D)) = times_times(real,C,times_times(real,times_times(real,A,B),D)). [resolve(1108,a,1099,a)]. 2.29/2.60 1109 -comm_semiring_1(A) | ti(A,B) = times_times(A,one_one(A),B) # label(fact_351_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom). [clausify(268)]. 2.29/2.60 Derived: ti(real,A) = times_times(real,one_one(real),A). [resolve(1109,a,1099,a)]. 2.29/2.60 1110 -comm_semiring_1(A) | zero_zero(A) = times_times(A,B,zero_zero(A)) # label(fact_333_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom). [clausify(269)]. 2.29/2.60 Derived: zero_zero(real) = times_times(real,A,zero_zero(real)). [resolve(1110,a,1099,a)]. 2.29/2.60 1111 -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(286)]. 2.29/2.60 1112 -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(303)]. 2.29/2.60 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),times_times(real,A,B))). [resolve(1112,a,1099,a)]. 2.29/2.60 1113 -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(314)]. 2.29/2.60 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(1113,a,1099,a)]. 2.29/2.60 1114 comm_semiring_1(int) # label(arity_Int_Oint___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 2.29/2.60 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),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(1114,a,1100,a)]. 2.29/2.60 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(1114,a,1101,a)]. 2.29/2.60 Derived: times_times(int,plus_plus(int,A,B),C) = plus_plus(int,times_times(int,A,C),times_times(int,B,C)). [resolve(1114,a,1102,a)]. 2.29/2.60 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),A),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),plus_plus(int,B,C))). [resolve(1114,a,1103,a)]. 2.29/2.60 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(1114,a,1104,a)]. 2.29/2.60 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),A),times_times(int,C,B))). [resolve(1114,a,1105,a)]. 2.29/2.60 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(1114,a,1106,a)]. 2.29/2.60 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(1114,a,1107,a)]. 2.29/2.60 Derived: times_times(int,times_times(int,A,B),times_times(int,C,D)) = times_times(int,C,times_times(int,times_times(int,A,B),D)). [resolve(1114,a,1108,a)]. 2.29/2.60 Derived: ti(int,A) = times_times(int,one_one(int),A). [resolve(1114,a,1109,a)]. 2.29/2.60 Derived: zero_zero(int) = times_times(int,A,zero_zero(int)). [resolve(1114,a,1110,a)]. 2.29/2.60 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),times_times(int,A,B))). [resolve(1114,a,1112,a)]. 2.29/2.60 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(1114,a,1113,a)]. 2.39/2.61 1115 -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(332)]. 2.39/2.61 1116 -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(337)]. 2.39/2.61 Derived: times_times(real,A,A) = hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))). [resolve(1116,a,1099,a)]. 2.39/2.61 Derived: times_times(int,A,A) = hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))). [resolve(1116,a,1114,a)]. 2.39/2.61 1117 -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(338)]. 2.39/2.61 Derived: times_times(real,plus_plus(real,A,one_one(real)),B) = plus_plus(real,times_times(real,A,B),B). [resolve(1117,a,1099,a)]. 2.39/2.61 Derived: times_times(int,plus_plus(int,A,one_one(int)),B) = plus_plus(int,times_times(int,A,B),B). [resolve(1117,a,1114,a)]. 2.39/2.61 1118 -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(346)]. 2.39/2.61 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(1118,a,1099,a)]. 2.39/2.61 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(1118,a,1114,a)]. 2.39/2.61 1119 -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(349)]. 2.39/2.61 Derived: times_times(real,A,plus_plus(real,B,C)) = plus_plus(real,times_times(real,A,B),times_times(real,A,C)). [resolve(1119,a,1099,a)]. 2.39/2.61 Derived: times_times(int,A,plus_plus(int,B,C)) = plus_plus(int,times_times(int,A,B),times_times(int,A,C)). [resolve(1119,a,1114,a)]. 2.39/2.61 1120 -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(361)]. 2.39/2.61 Derived: plus_plus(real,plus_plus(real,A,B),C) = plus_plus(real,A,plus_plus(real,B,C)). [resolve(1120,a,1099,a)]. 2.39/2.61 1121 comm_semiring_1(nat) # label(arity_Nat_Onat___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 2.39/2.61 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(1121,a,1100,a)]. 2.39/2.61 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(1121,a,1101,a)]. 2.39/2.61 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(1121,a,1103,a)]. 2.39/2.61 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(1121,a,1104,a)]. 2.39/2.61 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(1121,a,1105,a)]. 2.39/2.61 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(1121,a,1106,a)]. 2.39/2.63 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(1121,a,1107,a)]. 2.39/2.63 Derived: times_times(nat,times_times(nat,A,B),times_times(nat,C,D)) = times_times(nat,C,times_times(nat,times_times(nat,A,B),D)). [resolve(1121,a,1108,a)]. 2.39/2.63 Derived: ti(nat,A) = times_times(nat,one_one(nat),A). [resolve(1121,a,1109,a)]. 2.39/2.63 Derived: zero_zero(nat) = times_times(nat,A,zero_zero(nat)). [resolve(1121,a,1110,a)]. 2.39/2.63 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),times_times(nat,A,B))). [resolve(1121,a,1112,a)]. 2.39/2.63 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(1121,a,1113,a)]. 2.39/2.63 Derived: times_times(nat,A,A) = hAPP(nat,nat,power_power(nat,A),number_number_of(nat,bit0(bit1(pls)))). [resolve(1121,a,1116,a)]. 2.39/2.63 Derived: times_times(nat,plus_plus(nat,A,one_one(nat)),B) = plus_plus(nat,times_times(nat,A,B),B). [resolve(1121,a,1117,a)]. 2.39/2.63 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(1121,a,1118,a)]. 2.39/2.63 Derived: plus_plus(nat,plus_plus(nat,A,B),C) = plus_plus(nat,A,plus_plus(nat,B,C)). [resolve(1121,a,1120,a)]. 2.39/2.63 1122 -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(372)]. 2.39/2.63 Derived: times_times(real,plus_plus(real,A,one_one(real)),B) = plus_plus(real,B,times_times(real,A,B)). [resolve(1122,a,1099,a)]. 2.39/2.63 Derived: times_times(int,plus_plus(int,A,one_one(int)),B) = plus_plus(int,B,times_times(int,A,B)). [resolve(1122,a,1114,a)]. 2.39/2.63 Derived: times_times(nat,plus_plus(nat,A,one_one(nat)),B) = plus_plus(nat,B,times_times(nat,A,B)). [resolve(1122,a,1121,a)]. 2.39/2.63 1123 -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(392)]. 2.39/2.63 1124 -comm_semiring_1(A) | zero_zero(A) = times_times(A,zero_zero(A),B) # label(fact_334_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom). [clausify(459)]. 2.39/2.63 Derived: zero_zero(real) = times_times(real,zero_zero(real),A). [resolve(1124,a,1099,a)]. 2.39/2.63 Derived: zero_zero(int) = times_times(int,zero_zero(int),A). [resolve(1124,a,1114,a)]. 2.39/2.63 Derived: zero_zero(nat) = times_times(nat,zero_zero(nat),A). [resolve(1124,a,1121,a)]. 2.39/2.63 1125 -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(519)]. 2.39/2.63 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),zero_zero(real))). [resolve(1125,a,1099,a)]. 2.39/2.63 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),zero_zero(int))). [resolve(1125,a,1114,a)]. 2.39/2.63 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),zero_zero(nat))). [resolve(1125,a,1121,a)]. 2.39/2.63 1126 -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(534)]. 2.39/2.63 Derived: ti(real,A) = hAPP(nat,real,power_power(real,A),one_one(nat)). [resolve(1126,a,1099,a)]. 2.39/2.63 Derived: ti(int,A) = hAPP(nat,int,power_power(int,A),one_one(nat)). [resolve(1126,a,1114,a)]. 2.39/2.63 Derived: ti(nat,A) = hAPP(nat,nat,power_power(nat,A),one_one(nat)). [resolve(1126,a,1121,a)]. 2.39/2.63 1127 -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(537)]. 2.39/2.63 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(1127,a,1099,a)]. 2.39/2.64 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(1127,a,1114,a)]. 2.39/2.64 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(1127,a,1121,a)]. 2.39/2.64 1128 -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(542)]. 2.39/2.64 Derived: times_times(nat,times_times(nat,A,B),C) = times_times(nat,A,times_times(nat,B,C)). [resolve(1128,a,1121,a)]. 2.39/2.64 1129 -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(549)]. 2.39/2.64 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(1129,a,1099,a)]. 2.39/2.64 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(1129,a,1114,a)]. 2.39/2.64 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(1129,a,1121,a)]. 2.39/2.64 1130 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),zero_zero(A)),B)) | zero_zero(A) = ti(A,B) # label(fact_775_dvd__0__left) # label(axiom). [clausify(556)]. 2.39/2.64 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),zero_zero(real)),A)) | zero_zero(real) = ti(real,A). [resolve(1130,a,1099,a)]. 2.39/2.64 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),zero_zero(int)),A)) | zero_zero(int) = ti(int,A). [resolve(1130,a,1114,a)]. 2.39/2.64 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A)) | zero_zero(nat) = ti(nat,A). [resolve(1130,a,1121,a)]. 2.39/2.64 1131 -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(584)]. 2.39/2.64 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),times_times(real,B,A))). [resolve(1131,a,1099,a)]. 2.39/2.64 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),times_times(int,B,A))). [resolve(1131,a,1114,a)]. 2.39/2.64 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),times_times(nat,B,A))). [resolve(1131,a,1121,a)]. 2.39/2.64 1132 -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(593)]. 2.39/2.64 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),A)). [resolve(1132,a,1099,a)]. 2.39/2.64 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),A)). [resolve(1132,a,1114,a)]. 2.39/2.64 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),A)). [resolve(1132,a,1121,a)]. 2.39/2.64 1133 -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(634)]. 2.39/2.64 Derived: plus_plus(real,A,plus_plus(real,B,C)) = plus_plus(real,B,plus_plus(real,A,C)). [resolve(1133,a,1099,a)]. 2.39/2.64 Derived: plus_plus(nat,A,plus_plus(nat,B,C)) = plus_plus(nat,B,plus_plus(nat,A,C)). [resolve(1133,a,1121,a)]. 2.39/2.64 1134 -comm_semiring_1(A) | one_one(A) != ti(A,B) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(nat,A,power_power(A,B),C))) # label(fact_789_dvd__power) # label(axiom). [clausify(665)]. 2.39/2.66 Derived: one_one(real) != ti(real,A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(nat,real,power_power(real,A),B))). [resolve(1134,a,1099,a)]. 2.39/2.66 Derived: one_one(int) != ti(int,A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(nat,int,power_power(int,A),B))). [resolve(1134,a,1114,a)]. 2.39/2.66 Derived: one_one(nat) != ti(nat,A) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,power_power(nat,A),B))). [resolve(1134,a,1121,a)]. 2.39/2.66 1135 -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(665)]. 2.39/2.66 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(1135,a,1099,a)]. 2.39/2.66 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(1135,a,1114,a)]. 2.39/2.66 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(1135,a,1121,a)]. 2.39/2.66 1136 -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(673)]. 2.39/2.66 Derived: plus_plus(real,plus_plus(real,A,B),C) = plus_plus(real,plus_plus(real,A,C),B). [resolve(1136,a,1099,a)]. 2.39/2.66 Derived: plus_plus(int,plus_plus(int,A,B),C) = plus_plus(int,plus_plus(int,A,C),B). [resolve(1136,a,1114,a)]. 2.39/2.66 Derived: plus_plus(nat,plus_plus(nat,A,B),C) = plus_plus(nat,plus_plus(nat,A,C),B). [resolve(1136,a,1121,a)]. 2.39/2.66 1137 -comm_semiring_1(A) | ti(A,B) = times_times(A,B,one_one(A)) # label(fact_348_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom). [clausify(678)]. 2.39/2.66 Derived: ti(real,A) = times_times(real,A,one_one(real)). [resolve(1137,a,1099,a)]. 2.39/2.66 Derived: ti(nat,A) = times_times(nat,A,one_one(nat)). [resolve(1137,a,1121,a)]. 2.39/2.66 1138 -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(722)]. 2.39/2.66 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(1138,a,1099,a)]. 2.39/2.66 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(1138,a,1114,a)]. 2.39/2.66 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(1138,a,1121,a)]. 2.39/2.66 1139 -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(724)]. 2.39/2.66 1140 -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(729)]. 2.39/2.66 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(1140,a,1099,a)]. 2.39/2.66 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(1140,a,1114,a)]. 2.39/2.66 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(1140,a,1121,a)]. 2.39/2.66 1141 -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(730)]. 2.39/2.69 Derived: times_times(real,times_times(real,A,B),C) = times_times(real,times_times(real,A,C),B). [resolve(1141,a,1099,a)]. 2.39/2.69 Derived: times_times(int,times_times(int,A,B),C) = times_times(int,times_times(int,A,C),B). [resolve(1141,a,1114,a)]. 2.39/2.69 Derived: times_times(nat,times_times(nat,A,B),C) = times_times(nat,times_times(nat,A,C),B). [resolve(1141,a,1121,a)]. 2.39/2.69 1142 -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(735)]. 2.39/2.69 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(1142,a,1099,a)]. 2.39/2.69 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(1142,a,1114,a)]. 2.39/2.69 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(1142,a,1121,a)]. 2.39/2.69 1143 -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(764)]. 2.39/2.69 Derived: times_times(real,plus_plus(real,one_one(real),one_one(real)),A) = plus_plus(real,A,A). [resolve(1143,a,1099,a)]. 2.39/2.69 Derived: times_times(int,plus_plus(int,one_one(int),one_one(int)),A) = plus_plus(int,A,A). [resolve(1143,a,1114,a)]. 2.39/2.69 Derived: times_times(nat,plus_plus(nat,one_one(nat),one_one(nat)),A) = plus_plus(nat,A,A). [resolve(1143,a,1121,a)]. 2.39/2.69 1144 -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(795)]. 2.39/2.69 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),one_one(real)),A)). [resolve(1144,a,1099,a)]. 2.39/2.69 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),one_one(int)),A)). [resolve(1144,a,1114,a)]. 2.39/2.69 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),one_one(nat)),A)). [resolve(1144,a,1121,a)]. 2.39/2.69 1145 -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(804)]. 2.39/2.69 Derived: times_times(real,plus_plus(real,A,B),C) = plus_plus(real,times_times(real,A,C),times_times(real,B,C)). [resolve(1145,a,1099,a)]. 2.39/2.69 Derived: times_times(nat,plus_plus(nat,A,B),C) = plus_plus(nat,times_times(nat,A,C),times_times(nat,B,C)). [resolve(1145,a,1121,a)]. 2.39/2.69 1146 -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(853)]. 2.39/2.69 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(1146,a,1099,a)]. 2.39/2.69 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(1146,a,1114,a)]. 2.39/2.69 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(1146,a,1121,a)]. 2.39/2.69 1147 -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(917)]. 2.48/2.76 Derived: plus_plus(real,A,B) = plus_plus(real,B,A). [resolve(1147,a,1099,a)]. 2.48/2.76 Derived: plus_plus(int,A,B) = plus_plus(int,B,A). [resolve(1147,a,1114,a)]. 2.48/2.76 Derived: plus_plus(nat,A,B) = plus_plus(nat,B,A). [resolve(1147,a,1121,a)]. 2.48/2.76 1148 -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(919)]. 2.48/2.76 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(1148,a,1099,a)]. 2.48/2.76 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(1148,a,1114,a)]. 2.48/2.76 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(1148,a,1121,a)]. 2.48/2.76 1149 -comm_semiring_1(A) | one_one(A) = hAPP(nat,A,power_power(A,B),zero_zero(nat)) # label(fact_193_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom). [clausify(922)]. 2.48/2.76 Derived: one_one(real) = hAPP(nat,real,power_power(real,A),zero_zero(nat)). [resolve(1149,a,1099,a)]. 2.48/2.76 Derived: one_one(int) = hAPP(nat,int,power_power(int,A),zero_zero(nat)). [resolve(1149,a,1114,a)]. 2.48/2.76 Derived: one_one(nat) = hAPP(nat,nat,power_power(nat,A),zero_zero(nat)). [resolve(1149,a,1121,a)]. 2.48/2.76 1150 -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(954)]. 2.48/2.76 1151 -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(966)]. 2.48/2.76 Derived: times_times(real,A,times_times(real,B,C)) = times_times(real,B,times_times(real,A,C)). [resolve(1151,a,1099,a)]. 2.48/2.76 Derived: times_times(int,A,times_times(int,B,C)) = times_times(int,B,times_times(int,A,C)). [resolve(1151,a,1114,a)]. 2.48/2.76 Derived: times_times(nat,A,times_times(nat,B,C)) = times_times(nat,B,times_times(nat,A,C)). [resolve(1151,a,1121,a)]. 2.48/2.76 1152 -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(984)]. 2.48/2.76 Derived: times_times(real,A,B) = times_times(real,B,A). [resolve(1152,a,1099,a)]. 2.48/2.76 Derived: times_times(int,A,B) = times_times(int,B,A). [resolve(1152,a,1114,a)]. 2.48/2.76 Derived: times_times(nat,A,B) = times_times(nat,B,A). [resolve(1152,a,1121,a)]. 2.48/2.76 1153 -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(992)]. 2.48/2.76 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(1153,a,1099,a)]. 2.48/2.76 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(1153,a,1114,a)]. 2.48/2.76 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(1153,a,1121,a)]. 2.48/2.76 1154 semiring_1(real) # label(arity_RealDef_Oreal___Rings_Osemiring__1) # label(axiom). [assumption]. 2.57/2.78 1155 -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(7)]. 2.57/2.78 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(1154,a,1155,a)]. 2.57/2.78 1156 -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(233)]. 2.57/2.78 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(1156,a,1154,a)]. 2.57/2.78 1157 -semiring_1(A) | one_one(A) = hAPP(nat,A,semiring_1_of_nat(A),one_one(nat)) # label(fact_171_of__nat__1) # label(axiom). [clausify(271)]. 2.57/2.78 Derived: one_one(real) = hAPP(nat,real,semiring_1_of_nat(real),one_one(nat)). [resolve(1157,a,1154,a)]. 2.57/2.78 1158 -semiring_1(A) | zero_zero(A) = hAPP(nat,A,power_power(A,zero_zero(A)),number_number_of(nat,bit0(bit1(pls)))) # label(fact_4_zero__power2) # label(axiom). [clausify(321)]. 2.57/2.78 Derived: zero_zero(real) = hAPP(nat,real,power_power(real,zero_zero(real)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1158,a,1154,a)]. 2.57/2.78 1159 -semiring_1(A) | one_one(A) = hAPP(nat,A,power_power(A,one_one(A)),number_number_of(nat,bit0(bit1(pls)))) # label(fact_3_one__power2) # label(axiom). [clausify(402)]. 2.57/2.78 Derived: one_one(real) = hAPP(nat,real,power_power(real,one_one(real)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1159,a,1154,a)]. 2.57/2.78 1160 -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(405)]. 2.57/2.78 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(1160,a,1154,a)]. 2.57/2.78 1161 semiring_1(int) # label(arity_Int_Oint___Rings_Osemiring__1) # label(axiom). [assumption]. 2.57/2.78 Derived: hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,power_power(nat,A),B)) = hAPP(nat,int,power_power(int,hAPP(nat,int,semiring_1_of_nat(int),A)),B). [resolve(1161,a,1155,a)]. 2.57/2.78 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(1161,a,1156,a)]. 2.57/2.78 Derived: one_one(int) = hAPP(nat,int,semiring_1_of_nat(int),one_one(nat)). [resolve(1161,a,1157,a)]. 2.57/2.78 Derived: zero_zero(int) = hAPP(nat,int,power_power(int,zero_zero(int)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1161,a,1158,a)]. 2.57/2.78 Derived: one_one(int) = hAPP(nat,int,power_power(int,one_one(int)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1161,a,1159,a)]. 2.57/2.78 1162 semiring_1(nat) # label(arity_Nat_Onat___Rings_Osemiring__1) # label(axiom). [assumption]. 2.57/2.78 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(1162,a,1155,a)]. 2.57/2.78 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(1162,a,1156,a)]. 2.57/2.78 Derived: one_one(nat) = hAPP(nat,nat,semiring_1_of_nat(nat),one_one(nat)). [resolve(1162,a,1157,a)]. 2.84/3.07 Derived: zero_zero(nat) = hAPP(nat,nat,power_power(nat,zero_zero(nat)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1162,a,1158,a)]. 2.84/3.07 Derived: one_one(nat) = hAPP(nat,nat,power_power(nat,one_one(nat)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1162,a,1159,a)]. 2.84/3.07 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(1162,a,1160,a)]. 2.84/3.07 1163 -semiring_1(A) | zero_zero(A) = hAPP(nat,A,semiring_1_of_nat(A),zero_zero(nat)) # label(fact_188_of__nat__0) # label(axiom). [clausify(838)]. 2.84/3.07 Derived: zero_zero(real) = hAPP(nat,real,semiring_1_of_nat(real),zero_zero(nat)). [resolve(1163,a,1154,a)]. 2.84/3.07 Derived: zero_zero(int) = hAPP(nat,int,semiring_1_of_nat(int),zero_zero(nat)). [resolve(1163,a,1161,a)]. 2.84/3.07 Derived: zero_zero(nat) = hAPP(nat,nat,semiring_1_of_nat(nat),zero_zero(nat)). [resolve(1163,a,1162,a)]. 2.84/3.07 1164 -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(880)]. 2.84/3.07 Derived: semiring_1_of_nat(real) = ti(fun(nat,real),semiring_1_of_nat(real)). [resolve(1164,a,1154,a)]. 2.84/3.07 Derived: semiring_1_of_nat(int) = ti(fun(nat,int),semiring_1_of_nat(int)). [resolve(1164,a,1161,a)]. 2.84/3.07 Derived: semiring_1_of_nat(nat) = ti(fun(nat,nat),semiring_1_of_nat(nat)). [resolve(1164,a,1162,a)]. 2.84/3.07 1165 -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(930)]. 2.84/3.07 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(1165,a,1154,a)]. 2.84/3.07 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(1165,a,1161,a)]. 2.84/3.07 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(1165,a,1162,a)]. 2.84/3.07 1166 -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(403)]. 2.84/3.07 1167 ab_semigroup_add(int) # label(arity_Int_Oint___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 2.84/3.07 1168 ab_semigroup_add(real) # label(arity_RealDef_Oreal___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 2.84/3.07 1169 ab_semigroup_add(nat) # label(arity_Nat_Onat___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 2.84/3.07 1170 -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(481)]. 2.84/3.07 1171 -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(567)]. 2.84/3.07 1172 -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(996)]. 2.84/3.07 1173 -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(852)]. 2.84/3.07 1174 comm_ring_1(int) # label(arity_Int_Oint___Rings_Ocomm__ring__1) # label(axiom). [assumption]. 2.84/3.07 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(1173,a,1174,a)]. 2.84/3.07 1175 comm_ring_1(real) # label(arity_RealDef_Oreal___Rings_Ocomm__ring__1) # label(axiom). [assumption]. 2.84/3.07 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(1175,a,1173,a)]. 2.91/3.14 1176 linord581940658strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__ring__strict) # label(axiom). [assumption]. 2.91/3.14 1177 -linord581940658strict(A) | zero_zero(A) != ti(A,B) | zero_zero(A) != ti(A,C) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,times_times(A,B,B),times_times(A,C,C))),zero_zero(A))) # label(fact_311_sum__squares__le__zero__iff) # label(axiom). [clausify(15)]. 2.91/3.14 1178 -linord581940658strict(A) | zero_zero(A) = ti(A,B) | -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))) # label(fact_311_sum__squares__le__zero__iff) # label(axiom). [clausify(15)]. 2.91/3.14 1179 -linord581940658strict(A) | zero_zero(A) = ti(A,B) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,times_times(A,C,C),times_times(A,B,B))),zero_zero(A))) # label(fact_311_sum__squares__le__zero__iff) # label(axiom). [clausify(15)]. 2.91/3.14 Derived: zero_zero(real) != ti(real,A) | zero_zero(real) != ti(real,B) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),plus_plus(real,times_times(real,A,A),times_times(real,B,B))),zero_zero(real))). [resolve(1176,a,1177,a)]. 2.91/3.14 Derived: zero_zero(real) = ti(real,A) | -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))). [resolve(1176,a,1178,a)]. 2.91/3.14 Derived: zero_zero(real) = ti(real,A) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),plus_plus(real,times_times(real,B,B),times_times(real,A,A))),zero_zero(real))). [resolve(1176,a,1179,a)]. 2.91/3.14 1180 -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,B,D)),times_times(A,B,C))) # label(fact_418_mult__less__cancel__left__neg) # label(axiom). [clausify(101)]. 2.91/3.14 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,A,C)),times_times(real,A,B))). [resolve(1180,a,1176,a)]. 2.91/3.14 1181 -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,B,D)),times_times(A,B,C))) # label(fact_418_mult__less__cancel__left__neg) # label(axiom). [clausify(101)]. 2.91/3.14 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,A,C)),times_times(real,A,B))). [resolve(1181,a,1176,a)]. 2.91/3.14 1182 -linord581940658strict(A) | zero_zero(A) != ti(A,B) | zero_zero(A) != ti(A,C) | zero_zero(A) = plus_plus(A,times_times(A,B,B),times_times(A,C,C)) # label(fact_428_sum__squares__eq__zero__iff) # label(axiom). [clausify(131)]. 2.91/3.14 Derived: zero_zero(real) != ti(real,A) | zero_zero(real) != ti(real,B) | zero_zero(real) = plus_plus(real,times_times(real,A,A),times_times(real,B,B)). [resolve(1182,a,1176,a)]. 2.91/3.14 1183 -linord581940658strict(A) | zero_zero(A) = ti(A,B) | zero_zero(A) != plus_plus(A,times_times(A,B,B),times_times(A,C,C)) # label(fact_428_sum__squares__eq__zero__iff) # label(axiom). [clausify(131)]. 2.91/3.14 Derived: zero_zero(real) = ti(real,A) | zero_zero(real) != plus_plus(real,times_times(real,A,A),times_times(real,B,B)). [resolve(1183,a,1176,a)]. 2.91/3.14 1184 -linord581940658strict(A) | zero_zero(A) = ti(A,B) | zero_zero(A) != plus_plus(A,times_times(A,C,C),times_times(A,B,B)) # label(fact_428_sum__squares__eq__zero__iff) # label(axiom). [clausify(131)]. 2.91/3.14 Derived: zero_zero(real) = ti(real,A) | zero_zero(real) != plus_plus(real,times_times(real,B,B),times_times(real,A,A)). [resolve(1184,a,1176,a)]. 2.91/3.15 1185 -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(304)]. 2.91/3.15 1186 -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(304)]. 2.91/3.15 1187 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),times_times(A,B,C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) # label(fact_291_zero__le__mult__iff) # label(axiom). [clausify(351)]. 2.91/3.15 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),times_times(real,A,B))) | 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)),A)). [resolve(1187,a,1176,a)]. 2.91/3.15 1188 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),times_times(A,B,C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) # label(fact_291_zero__le__mult__iff) # label(axiom). [clausify(351)]. 2.91/3.15 1189 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),times_times(A,B,C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) # label(fact_291_zero__le__mult__iff) # label(axiom). [clausify(351)]. 2.91/3.15 1190 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),times_times(A,B,C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) # label(fact_291_zero__le__mult__iff) # label(axiom). [clausify(351)]. 2.91/3.15 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),times_times(real,A,B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)). [resolve(1190,a,1176,a)]. 2.91/3.15 1191 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),times_times(A,B,C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) # label(fact_291_zero__le__mult__iff) # label(axiom). [clausify(351)]. 2.91/3.15 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),times_times(real,A,B))) | -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(1191,a,1176,a)]. 2.91/3.15 1192 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),times_times(A,B,C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) # label(fact_291_zero__le__mult__iff) # label(axiom). [clausify(351)]. 2.91/3.15 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),times_times(real,A,B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)). [resolve(1192,a,1176,a)]. 2.91/3.15 1193 linord581940658strict(int) # label(arity_Int_Oint___Rings_Olinordered__ring__strict) # label(axiom). [assumption]. 2.91/3.16 Derived: zero_zero(int) != ti(int,A) | zero_zero(int) != ti(int,B) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,times_times(int,A,A),times_times(int,B,B))),zero_zero(int))). [resolve(1193,a,1177,a)]. 2.91/3.16 Derived: zero_zero(int) = ti(int,A) | -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))). [resolve(1193,a,1178,a)]. 2.91/3.16 Derived: zero_zero(int) = ti(int,A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,times_times(int,B,B),times_times(int,A,A))),zero_zero(int))). [resolve(1193,a,1179,a)]. 2.91/3.16 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),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(1193,a,1180,a)]. 2.91/3.16 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),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(1193,a,1181,a)]. 2.91/3.16 Derived: zero_zero(int) != ti(int,A) | zero_zero(int) != ti(int,B) | zero_zero(int) = plus_plus(int,times_times(int,A,A),times_times(int,B,B)). [resolve(1193,a,1182,a)]. 2.91/3.16 Derived: zero_zero(int) = ti(int,A) | zero_zero(int) != plus_plus(int,times_times(int,A,A),times_times(int,B,B)). [resolve(1193,a,1183,a)]. 2.91/3.16 Derived: zero_zero(int) = ti(int,A) | zero_zero(int) != plus_plus(int,times_times(int,B,B),times_times(int,A,A)). [resolve(1193,a,1184,a)]. 2.91/3.16 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(1193,a,1185,a)]. 2.91/3.16 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(1193,a,1186,a)]. 2.91/3.16 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),times_times(int,A,B))) | 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)),A)). [resolve(1193,a,1187,a)]. 2.91/3.16 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),times_times(int,A,B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)). [resolve(1193,a,1190,a)]. 2.91/3.16 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),times_times(int,A,B))) | -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(1193,a,1191,a)]. 2.91/3.16 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),times_times(int,A,B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)). [resolve(1193,a,1192,a)]. 2.91/3.16 1194 -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(398)]. 2.91/3.16 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),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(1194,a,1176,a)]. 2.91/3.17 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(1194,a,1193,a)]. 2.91/3.17 1195 -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(400)]. 2.91/3.17 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(1195,a,1176,a)]. 2.91/3.17 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(1195,a,1193,a)]. 2.91/3.17 1196 -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(400)]. 2.91/3.17 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(1196,a,1176,a)]. 2.91/3.17 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(1196,a,1193,a)]. 2.91/3.17 1197 -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,B,D)),times_times(A,B,C))) # label(fact_411_mult__less__cancel__left__disj) # label(axiom). [clausify(566)]. 2.91/3.17 1198 -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,C)),times_times(A,B,D))) # label(fact_411_mult__less__cancel__left__disj) # label(axiom). [clausify(566)]. 2.91/3.17 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),times_times(real,A,B)),times_times(real,A,C))). [resolve(1198,a,1176,a)]. 2.91/3.17 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),times_times(int,A,B)),times_times(int,A,C))). [resolve(1198,a,1193,a)]. 2.91/3.17 1199 -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,B,C)),times_times(A,B,D))) # label(fact_411_mult__less__cancel__left__disj) # label(axiom). [clausify(566)]. 2.91/3.17 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,A,B)),times_times(real,A,C))). [resolve(1199,a,1176,a)]. 2.91/3.17 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,A,B)),times_times(int,A,C))). [resolve(1199,a,1193,a)]. 2.91/3.18 1200 -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,B,C)),times_times(A,B,D))) # label(fact_411_mult__less__cancel__left__disj) # label(axiom). [clausify(566)]. 2.91/3.18 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,A,B)),times_times(real,A,C))). [resolve(1200,a,1176,a)]. 2.91/3.18 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,A,B)),times_times(int,A,C))). [resolve(1200,a,1193,a)]. 2.91/3.18 1201 -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,C)),times_times(A,D,B))) # label(fact_411_mult__less__cancel__left__disj) # label(axiom). [clausify(566)]. 2.91/3.18 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,B)),times_times(real,C,A))). [resolve(1201,a,1176,a)]. 2.91/3.18 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,B)),times_times(int,C,A))). [resolve(1201,a,1193,a)]. 2.91/3.18 1202 -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,C)),times_times(A,D,B))) # label(fact_411_mult__less__cancel__left__disj) # label(axiom). [clausify(566)]. 2.91/3.18 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,B)),times_times(real,C,A))). [resolve(1202,a,1176,a)]. 2.91/3.18 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,B)),times_times(int,C,A))). [resolve(1202,a,1193,a)]. 2.91/3.18 1203 -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,C)),times_times(A,B,D))) # label(fact_412_mult__less__cancel__left__pos) # label(axiom). [clausify(603)]. 2.91/3.18 1204 -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,C)),times_times(A,B,D))) # label(fact_412_mult__less__cancel__left__pos) # label(axiom). [clausify(603)]. 2.91/3.18 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,B)),times_times(real,A,C))). [resolve(1204,a,1176,a)]. 2.91/3.18 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,B)),times_times(int,A,C))). [resolve(1204,a,1193,a)]. 2.91/3.18 1205 -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(618)]. 2.91/3.18 1206 -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(618)]. 2.91/3.18 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(1206,a,1176,a)]. 2.91/3.18 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(1206,a,1193,a)]. 2.91/3.18 1207 -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(618)]. 2.91/3.18 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(1207,a,1176,a)]. 2.91/3.18 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(1207,a,1193,a)]. 2.91/3.18 1208 -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(618)]. 2.91/3.18 1209 -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(618)]. 2.91/3.18 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),zero_zero(real)),A)). [resolve(1209,a,1176,a)]. 2.91/3.18 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)),A)). [resolve(1209,a,1193,a)]. 2.91/3.18 1210 -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(618)]. 2.91/3.18 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)),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))). [resolve(1210,a,1176,a)]. 2.91/3.22 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)),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))). [resolve(1210,a,1193,a)]. 2.91/3.22 1211 -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(710)]. 2.91/3.22 1212 -linord581940658strict(A) | zero_zero(A) = ti(A,B) | 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(765)]. 2.91/3.22 Derived: zero_zero(real) = ti(real,A) | 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(1212,a,1176,a)]. 2.91/3.22 Derived: zero_zero(int) = ti(int,A) | 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(1212,a,1193,a)]. 2.91/3.22 1213 -linord581940658strict(A) | zero_zero(A) = ti(A,B) | 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(765)]. 2.91/3.22 Derived: zero_zero(real) = ti(real,A) | 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(1213,a,1176,a)]. 2.91/3.22 Derived: zero_zero(int) = ti(int,A) | 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(1213,a,1193,a)]. 2.91/3.22 1214 -linord581940658strict(A) | zero_zero(A) != ti(A,B) | zero_zero(A) != ti(A,C) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),plus_plus(A,times_times(A,C,C),times_times(A,B,B)))) # label(fact_486_sum__squares__gt__zero__iff) # label(axiom). [clausify(765)]. 2.91/3.22 Derived: zero_zero(real) != ti(real,A) | zero_zero(real) != ti(real,B) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),plus_plus(real,times_times(real,B,B),times_times(real,A,A)))). [resolve(1214,a,1176,a)]. 2.91/3.22 Derived: zero_zero(int) != ti(int,A) | zero_zero(int) != ti(int,B) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),plus_plus(int,times_times(int,B,B),times_times(int,A,A)))). [resolve(1214,a,1193,a)]. 2.91/3.22 1215 -linord581940658strict(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(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),zero_zero(A))) # label(fact_410_mult__less__cancel__right__disj) # label(axiom). [clausify(906)]. 2.91/3.22 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(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),zero_zero(real))). [resolve(1215,a,1176,a)]. 2.91/3.22 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(int),zero_zero(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))). [resolve(1215,a,1193,a)]. 2.91/3.22 1216 -linord581940658strict(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(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),B)) # label(fact_410_mult__less__cancel__right__disj) # label(axiom). [clausify(906)]. 3.01/3.23 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(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),A)). [resolve(1216,a,1176,a)]. 3.01/3.23 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(int),zero_zero(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),A)). [resolve(1216,a,1193,a)]. 3.01/3.23 1217 -linord581940658strict(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(A),B),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),zero_zero(A))) # label(fact_410_mult__less__cancel__right__disj) # label(axiom). [clausify(906)]. 3.01/3.23 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(real),A),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),zero_zero(real))). [resolve(1217,a,1176,a)]. 3.01/3.23 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(int),A),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))). [resolve(1217,a,1193,a)]. 3.01/3.23 1218 -linord581940658strict(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(A),B),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),B)) # label(fact_410_mult__less__cancel__right__disj) # label(axiom). [clausify(906)]. 3.01/3.23 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(real),A),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),A)). [resolve(1218,a,1176,a)]. 3.01/3.23 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(int),A),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),A)). [resolve(1218,a,1193,a)]. 3.01/3.23 1219 -linord581940658strict(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(A),zero_zero(A)),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),D)) # label(fact_410_mult__less__cancel__right__disj) # label(axiom). [clausify(906)]. 3.01/3.23 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(int),zero_zero(int)),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),C)). [resolve(1219,a,1193,a)]. 3.01/3.23 1220 -linord581940658strict(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(A),C),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),B)) # label(fact_410_mult__less__cancel__right__disj) # label(axiom). [clausify(906)]. 3.01/3.23 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(real),B),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),A)). [resolve(1220,a,1176,a)]. 3.01/3.23 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(int),B),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),A)). [resolve(1220,a,1193,a)]. 3.01/3.23 1221 -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(987)]. 3.20/3.43 1222 cancel_semigroup_add(int) # label(arity_Int_Oint___Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 3.20/3.43 1223 -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(19)]. 3.20/3.43 1224 -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(92)]. 3.20/3.43 1225 cancel_semigroup_add(nat) # label(arity_Nat_Onat___Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 3.20/3.43 1226 -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(533)]. 3.20/3.43 Derived: plus_plus(int,A,B) != plus_plus(int,C,B) | ti(int,A) = ti(int,C). [resolve(1226,a,1222,a)]. 3.20/3.43 Derived: plus_plus(nat,A,B) != plus_plus(nat,C,B) | ti(nat,A) = ti(nat,C). [resolve(1226,a,1225,a)]. 3.20/3.43 1227 -cancel_semigroup_add(A) | plus_plus(A,B,C) != plus_plus(A,D,C) | ti(A,B) = ti(A,D) # label(fact_203_add__right__cancel) # label(axiom). [clausify(672)]. 3.20/3.43 1228 -cancel_semigroup_add(A) | plus_plus(A,B,C) = plus_plus(A,D,C) | ti(A,B) != ti(A,D) # label(fact_203_add__right__cancel) # label(axiom). [clausify(672)]. 3.20/3.43 Derived: plus_plus(int,A,B) = plus_plus(int,C,B) | ti(int,A) != ti(int,C). [resolve(1228,a,1222,a)]. 3.20/3.43 Derived: plus_plus(nat,A,B) = plus_plus(nat,C,B) | ti(nat,A) != ti(nat,C). [resolve(1228,a,1225,a)]. 3.20/3.43 1229 -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(705)]. 3.20/3.43 Derived: ti(int,A) != ti(int,B) | plus_plus(int,C,A) = plus_plus(int,C,B). [resolve(1229,a,1222,a)]. 3.20/3.43 Derived: ti(nat,A) != ti(nat,B) | plus_plus(nat,C,A) = plus_plus(nat,C,B). [resolve(1229,a,1225,a)]. 3.20/3.43 1230 -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(705)]. 3.20/3.43 Derived: ti(int,A) = ti(int,B) | plus_plus(int,C,A) != plus_plus(int,C,B). [resolve(1230,a,1222,a)]. 3.20/3.43 Derived: ti(nat,A) = ti(nat,B) | plus_plus(nat,C,A) != plus_plus(nat,C,B). [resolve(1230,a,1225,a)]. 3.20/3.43 1231 -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(733)]. 3.20/3.43 1232 -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(736)]. 3.20/3.43 1233 cancel_semigroup_add(real) # label(arity_RealDef_Oreal___Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 3.20/3.43 Derived: plus_plus(real,A,B) != plus_plus(real,C,B) | ti(real,A) = ti(real,C). [resolve(1233,a,1226,a)]. 3.20/3.43 Derived: plus_plus(real,A,B) = plus_plus(real,C,B) | ti(real,A) != ti(real,C). [resolve(1233,a,1228,a)]. 3.20/3.43 Derived: ti(real,A) != ti(real,B) | plus_plus(real,C,A) = plus_plus(real,C,B). [resolve(1233,a,1229,a)]. 3.20/3.43 Derived: ti(real,A) = ti(real,B) | plus_plus(real,C,A) != plus_plus(real,C,B). [resolve(1233,a,1230,a)]. 3.20/3.43 1234 -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(100)]. 3.20/3.43 1235 ordere453448008miring(nat) # label(arity_Nat_Onat___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 3.20/3.43 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(1234,a,1235,a)]. 3.36/3.58 1236 -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(316)]. 3.36/3.58 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(1236,a,1235,a)]. 3.36/3.58 1237 ordere453448008miring(real) # label(arity_RealDef_Oreal___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 3.36/3.58 1238 ordere453448008miring(int) # label(arity_Int_Oint___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 3.36/3.58 1239 -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(809)]. 3.36/3.58 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),times_times(nat,A,B))). [resolve(1239,a,1235,a)]. 3.36/3.58 1240 -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(847)]. 3.36/3.58 1241 -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_269_split__mult__neg__le) # label(axiom). [clausify(847)]. 3.36/3.58 1242 -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(960)]. 3.36/3.58 1243 no_zero_divisors(int) # label(arity_Int_Oint___Rings_Ono__zero__divisors) # label(axiom). [assumption]. 3.36/3.58 1244 -no_zero_divisors(A) | ti(A,times_times(A,B,C)) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_3_res) # label(axiom). [clausify(29)]. 3.36/3.58 1245 -no_zero_divisors(A) | times_times(A,ti(A,B),C) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_3_arg1) # label(axiom). [clausify(354)]. 3.36/3.58 Derived: ti(int,times_times(int,A,B)) = times_times(int,A,B). [resolve(1243,a,1244,a)]. 3.36/3.58 Derived: times_times(int,ti(int,A),B) = times_times(int,A,B). [resolve(1243,a,1245,a)]. 3.36/3.58 1246 -mult_zero(A) | -no_zero_divisors(A) | -zero_neq_one(A) | -power(A) | zero_zero(A) != hAPP(nat,A,power_power(A,B),C) | zero_zero(A) = ti(A,B) # label(fact_185_power__eq__0__iff) # label(axiom). [clausify(484)]. 3.36/3.58 Derived: -mult_zero(int) | -zero_neq_one(int) | -power(int) | zero_zero(int) != hAPP(nat,int,power_power(int,A),B) | zero_zero(int) = ti(int,A). [resolve(1246,b,1243,a)]. 3.36/3.58 1247 -mult_zero(A) | -no_zero_divisors(A) | -zero_neq_one(A) | -power(A) | zero_zero(A) != hAPP(nat,A,power_power(A,B),C) | zero_zero(nat) != C # label(fact_185_power__eq__0__iff) # label(axiom). [clausify(484)]. 3.36/3.58 Derived: -mult_zero(int) | -zero_neq_one(int) | -power(int) | zero_zero(int) != hAPP(nat,int,power_power(int,A),B) | zero_zero(nat) != B. [resolve(1247,b,1243,a)]. 3.36/3.58 1248 -mult_zero(A) | -no_zero_divisors(A) | -zero_neq_one(A) | -power(A) | zero_zero(A) = hAPP(nat,A,power_power(A,B),C) | zero_zero(A) != ti(A,B) | zero_zero(nat) = C # label(fact_185_power__eq__0__iff) # label(axiom). [clausify(484)]. 3.36/3.62 Derived: -mult_zero(int) | -zero_neq_one(int) | -power(int) | zero_zero(int) = hAPP(nat,int,power_power(int,A),B) | zero_zero(int) != ti(int,A) | zero_zero(nat) = B. [resolve(1248,b,1243,a)]. 3.36/3.62 1249 -no_zero_divisors(A) | zero_zero(A) = ti(A,B) | zero_zero(A) = ti(A,C) | zero_zero(A) != times_times(A,B,C) # label(fact_331_no__zero__divisors) # label(axiom). [clausify(515)]. 3.36/3.62 1250 -no_zero_divisors(A) | zero_zero(A) != times_times(A,B,C) | zero_zero(A) = ti(A,B) | zero_zero(A) = ti(A,C) # label(fact_332_divisors__zero) # label(axiom). [clausify(516)]. 3.36/3.62 1251 -no_zero_divisors(A) | times_times(A,B,ti(A,C)) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_3_arg2) # label(axiom). [clausify(528)]. 3.36/3.62 Derived: times_times(int,A,ti(int,B)) = times_times(int,A,B). [resolve(1251,a,1243,a)]. 3.36/3.62 1252 no_zero_divisors(nat) # label(arity_Nat_Onat___Rings_Ono__zero__divisors) # label(axiom). [assumption]. 3.36/3.62 Derived: ti(nat,times_times(nat,A,B)) = times_times(nat,A,B). [resolve(1252,a,1244,a)]. 3.36/3.62 Derived: times_times(nat,ti(nat,A),B) = times_times(nat,A,B). [resolve(1252,a,1245,a)]. 3.36/3.62 Derived: -mult_zero(nat) | -zero_neq_one(nat) | -power(nat) | zero_zero(nat) != hAPP(nat,nat,power_power(nat,A),B) | zero_zero(nat) = ti(nat,A). [resolve(1252,a,1246,b)]. 3.36/3.62 Derived: -mult_zero(nat) | -zero_neq_one(nat) | -power(nat) | zero_zero(nat) != hAPP(nat,nat,power_power(nat,A),B) | zero_zero(nat) != B. [resolve(1252,a,1247,b)]. 3.36/3.62 Derived: -mult_zero(nat) | -zero_neq_one(nat) | -power(nat) | zero_zero(nat) = hAPP(nat,nat,power_power(nat,A),B) | zero_zero(nat) != ti(nat,A) | zero_zero(nat) = B. [resolve(1252,a,1248,b)]. 3.36/3.62 Derived: zero_zero(nat) = ti(nat,A) | zero_zero(nat) = ti(nat,B) | zero_zero(nat) != times_times(nat,A,B). [resolve(1252,a,1249,a)]. 3.36/3.62 Derived: times_times(nat,A,ti(nat,B)) = times_times(nat,A,B). [resolve(1252,a,1251,a)]. 3.36/3.62 1253 no_zero_divisors(real) # label(arity_RealDef_Oreal___Rings_Ono__zero__divisors) # label(axiom). [assumption]. 3.36/3.62 Derived: ti(real,times_times(real,A,B)) = times_times(real,A,B). [resolve(1253,a,1244,a)]. 3.36/3.62 Derived: times_times(real,ti(real,A),B) = times_times(real,A,B). [resolve(1253,a,1245,a)]. 3.36/3.62 Derived: -mult_zero(real) | -zero_neq_one(real) | -power(real) | zero_zero(real) != hAPP(nat,real,power_power(real,A),B) | zero_zero(real) = ti(real,A). [resolve(1253,a,1246,b)]. 3.36/3.62 Derived: -mult_zero(real) | -zero_neq_one(real) | -power(real) | zero_zero(real) != hAPP(nat,real,power_power(real,A),B) | zero_zero(nat) != B. [resolve(1253,a,1247,b)]. 3.36/3.62 Derived: -mult_zero(real) | -zero_neq_one(real) | -power(real) | zero_zero(real) = hAPP(nat,real,power_power(real,A),B) | zero_zero(real) != ti(real,A) | zero_zero(nat) = B. [resolve(1253,a,1248,b)]. 3.36/3.62 Derived: times_times(real,A,ti(real,B)) = times_times(real,A,B). [resolve(1253,a,1251,a)]. 3.36/3.62 1254 -power(A) | -zero_neq_one(A) | -no_zero_divisors(A) | -mult_zero(A) | zero_zero(A) != hAPP(nat,A,power_power(A,B),number_number_of(nat,C)) | zero_zero(A) = ti(A,B) # label(fact_86_power__eq__0__iff__number__of) # label(axiom). [clausify(941)]. 3.36/3.62 1255 -power(A) | -zero_neq_one(A) | -no_zero_divisors(A) | -mult_zero(A) | zero_zero(A) != hAPP(nat,A,power_power(A,B),number_number_of(nat,C)) | number_number_of(nat,C) != zero_zero(nat) # label(fact_86_power__eq__0__iff__number__of) # label(axiom). [clausify(941)]. 3.36/3.62 Derived: -power(int) | -zero_neq_one(int) | -mult_zero(int) | zero_zero(int) != hAPP(nat,int,power_power(int,A),number_number_of(nat,B)) | number_number_of(nat,B) != zero_zero(nat). [resolve(1255,c,1243,a)]. 3.36/3.62 Derived: -power(nat) | -zero_neq_one(nat) | -mult_zero(nat) | zero_zero(nat) != hAPP(nat,nat,power_power(nat,A),number_number_of(nat,B)) | number_number_of(nat,B) != zero_zero(nat). [resolve(1255,c,1252,a)]. 3.36/3.62 Derived: -power(real) | -zero_neq_one(real) | -mult_zero(real) | zero_zero(real) != hAPP(nat,real,power_power(real,A),number_number_of(nat,B)) | number_number_of(nat,B) != zero_zero(nat). [resolve(1255,c,1253,a)]. 3.36/3.62 1256 -power(A) | -zero_neq_one(A) | -no_zero_divisors(A) | -mult_zero(A) | zero_zero(A) = hAPP(nat,A,power_power(A,B),number_number_of(nat,C)) | zero_zero(A) != ti(A,B) | number_number_of(nat,C) = zero_zero(nat) # label(fact_86_power__eq__0__iff__number__of) # label(axiom). [clausify(941)]. 3.50/3.73 Derived: -power(int) | -zero_neq_one(int) | -mult_zero(int) | zero_zero(int) = hAPP(nat,int,power_power(int,A),number_number_of(nat,B)) | zero_zero(int) != ti(int,A) | number_number_of(nat,B) = zero_zero(nat). [resolve(1256,c,1243,a)]. 3.50/3.73 Derived: -power(nat) | -zero_neq_one(nat) | -mult_zero(nat) | zero_zero(nat) = hAPP(nat,nat,power_power(nat,A),number_number_of(nat,B)) | zero_zero(nat) != ti(nat,A) | number_number_of(nat,B) = zero_zero(nat). [resolve(1256,c,1252,a)]. 3.50/3.73 Derived: -power(real) | -zero_neq_one(real) | -mult_zero(real) | zero_zero(real) = hAPP(nat,real,power_power(real,A),number_number_of(nat,B)) | zero_zero(real) != ti(real,A) | number_number_of(nat,B) = zero_zero(nat). [resolve(1256,c,1253,a)]. 3.50/3.73 1257 monoid_mult(nat) # label(arity_Nat_Onat___Groups_Omonoid__mult) # label(axiom). [assumption]. 3.50/3.73 1258 -monoid_mult(A) | ti(A,B) = times_times(A,B,one_one(A)) # label(fact_349_mult__1__right) # label(axiom). [clausify(30)]. 3.50/3.73 1259 -monoid_mult(A) | ti(A,B) = hAPP(nat,A,power_power(A,B),one_one(nat)) # label(fact_157_power__one__right) # label(axiom). [clausify(34)]. 3.50/3.73 1260 -monoid_mult(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_459_power__add) # label(axiom). [clausify(63)]. 3.50/3.73 1261 -monoid_mult(A) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | times_times(A,hAPP(nat,A,power_power(A,C),minus_minus(nat,B,one_one(nat))),C) = hAPP(nat,A,power_power(A,C),B) # label(fact_405_realpow__minus__mult) # label(axiom). [clausify(96)]. 3.50/3.73 1262 -monoid_mult(A) | hAPP(nat,A,power_power(A,hAPP(nat,A,power_power(A,B),C)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,A,power_power(A,B),times_times(nat,number_number_of(nat,bit0(bit1(pls))),C)) # label(fact_690_power__even__eq) # label(axiom). [clausify(133)]. 3.50/3.73 1263 -monoid_mult(A) | hAPP(nat,A,power_power(A,hAPP(nat,A,power_power(A,B),C)),D) = hAPP(nat,A,power_power(A,B),times_times(nat,C,D)) # label(fact_600_power__mult) # label(axiom). [clausify(163)]. 3.50/3.73 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | times_times(nat,hAPP(nat,nat,power_power(nat,B),minus_minus(nat,A,one_one(nat))),B) = hAPP(nat,nat,power_power(nat,B),A). [resolve(1257,a,1261,a)]. 3.50/3.73 Derived: hAPP(nat,nat,power_power(nat,hAPP(nat,nat,power_power(nat,A),B)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,nat,power_power(nat,A),times_times(nat,number_number_of(nat,bit0(bit1(pls))),B)). [resolve(1257,a,1262,a)]. 3.50/3.73 1264 -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(208)]. 3.50/3.73 1265 -monoid_mult(A) | ti(A,B) = times_times(A,one_one(A),B) # label(fact_352_mult__1__left) # label(axiom). [clausify(427)]. 3.50/3.73 1266 -monoid_mult(A) | times_times(A,B,B) = hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))) # label(fact_540_power2__eq__square) # label(axiom). [clausify(531)]. 3.50/3.73 1267 -monoid_mult(A) | one_one(A) = hAPP(nat,A,power_power(A,one_one(A)),B) # label(fact_166_power__one) # label(axiom). [clausify(553)]. 3.50/3.73 Derived: one_one(nat) = hAPP(nat,nat,power_power(nat,one_one(nat)),A). [resolve(1267,a,1257,a)]. 3.50/3.73 1268 -monoid_mult(A) | times_times(A,hAPP(nat,A,power_power(A,B),C),B) = times_times(A,B,hAPP(nat,A,power_power(A,B),C)) # label(fact_362_power__commutes) # label(axiom). [clausify(607)]. 3.50/3.73 1269 -monoid_mult(A) | times_times(A,times_times(A,B,B),B) = hAPP(nat,A,power_power(A,B),number_number_of(nat,bit1(bit1(pls)))) # label(fact_527_power3__eq__cube) # label(axiom). [clausify(794)]. 3.50/3.73 Derived: times_times(nat,times_times(nat,A,A),A) = hAPP(nat,nat,power_power(nat,A),number_number_of(nat,bit1(bit1(pls)))). [resolve(1269,a,1257,a)]. 3.61/3.86 1270 monoid_mult(real) # label(arity_RealDef_Oreal___Groups_Omonoid__mult) # label(axiom). [assumption]. 3.61/3.86 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | times_times(real,hAPP(nat,real,power_power(real,B),minus_minus(nat,A,one_one(nat))),B) = hAPP(nat,real,power_power(real,B),A). [resolve(1270,a,1261,a)]. 3.61/3.86 Derived: hAPP(nat,real,power_power(real,hAPP(nat,real,power_power(real,A),B)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,real,power_power(real,A),times_times(nat,number_number_of(nat,bit0(bit1(pls))),B)). [resolve(1270,a,1262,a)]. 3.61/3.86 Derived: one_one(real) = hAPP(nat,real,power_power(real,one_one(real)),A). [resolve(1270,a,1267,a)]. 3.61/3.86 Derived: times_times(real,times_times(real,A,A),A) = hAPP(nat,real,power_power(real,A),number_number_of(nat,bit1(bit1(pls)))). [resolve(1270,a,1269,a)]. 3.61/3.86 1271 monoid_mult(int) # label(arity_Int_Oint___Groups_Omonoid__mult) # label(axiom). [assumption]. 3.61/3.86 Derived: ti(int,A) = times_times(int,A,one_one(int)). [resolve(1271,a,1258,a)]. 3.61/3.86 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | times_times(int,hAPP(nat,int,power_power(int,B),minus_minus(nat,A,one_one(nat))),B) = hAPP(nat,int,power_power(int,B),A). [resolve(1271,a,1261,a)]. 3.61/3.86 Derived: hAPP(nat,int,power_power(int,hAPP(nat,int,power_power(int,A),B)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,int,power_power(int,A),times_times(nat,number_number_of(nat,bit0(bit1(pls))),B)). [resolve(1271,a,1262,a)]. 3.61/3.86 Derived: one_one(int) = hAPP(nat,int,power_power(int,one_one(int)),A). [resolve(1271,a,1267,a)]. 3.61/3.86 Derived: times_times(int,times_times(int,A,A),A) = hAPP(nat,int,power_power(int,A),number_number_of(nat,bit1(bit1(pls)))). [resolve(1271,a,1269,a)]. 3.61/3.86 1272 order(int) # label(arity_Int_Oint___Orderings_Oorder) # label(axiom). [assumption]. 3.61/3.86 1273 -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(31)]. 3.61/3.86 1274 -order(A) | ti(bool,hAPP(A,bool,B,C)) = hAPP(A,bool,B,C) # label(tsy_c_hAPP_1_res) # label(axiom). [clausify(44)]. 3.61/3.86 1275 -order(A) | hAPP(A,bool,B,ti(A,C)) = hAPP(A,bool,B,C) # label(tsy_c_hAPP_1_arg2) # label(axiom). [clausify(113)]. 3.61/3.86 1276 -order(A) | ti(fun(A,fun(A,bool)),ord_less_eq(A)) = ord_less_eq(A) # label(tsy_c_Orderings_Oord__class_Oless__eq_res) # label(axiom). [clausify(199)]. 3.61/3.86 1277 -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(387)]. 3.61/3.86 1278 -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(462)]. 3.61/3.86 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | ti(int,B) = ti(int,A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)). [resolve(1272,a,1273,a)]. 3.61/3.86 Derived: ti(bool,hAPP(int,bool,A,B)) = hAPP(int,bool,A,B). [resolve(1272,a,1274,a)]. 3.61/3.86 Derived: hAPP(int,bool,A,ti(int,B)) = hAPP(int,bool,A,B). [resolve(1272,a,1275,a)]. 3.61/3.86 Derived: ti(fun(int,fun(int,bool)),ord_less_eq(int)) = ord_less_eq(int). [resolve(1272,a,1276,a)]. 3.61/3.86 Derived: hAPP(int,bool,ti(fun(int,bool),A),B) = hAPP(int,bool,A,B). [resolve(1272,a,1277,a)]. 3.61/3.86 Derived: hAPP(int,fun(int,bool),A,ti(int,B)) = hAPP(int,fun(int,bool),A,B). [resolve(1272,a,1278,a)]. 3.61/3.86 1279 -order(A) | ord_less(A) = ti(fun(A,fun(A,bool)),ord_less(A)) # label(tsy_c_Orderings_Oord__class_Oless_res) # label(axiom). [clausify(569)]. 3.61/3.86 Derived: ord_less(int) = ti(fun(int,fun(int,bool)),ord_less(int)). [resolve(1279,a,1272,a)]. 3.61/3.86 1280 order(real) # label(arity_RealDef_Oreal___Orderings_Oorder) # label(axiom). [assumption]. 3.61/3.86 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(1280,a,1273,a)]. 3.61/3.86 Derived: ti(bool,hAPP(real,bool,A,B)) = hAPP(real,bool,A,B). [resolve(1280,a,1274,a)]. 3.61/3.86 Derived: hAPP(real,bool,A,ti(real,B)) = hAPP(real,bool,A,B). [resolve(1280,a,1275,a)]. 3.66/3.95 Derived: ti(fun(real,fun(real,bool)),ord_less_eq(real)) = ord_less_eq(real). [resolve(1280,a,1276,a)]. 3.66/3.95 Derived: hAPP(real,bool,ti(fun(real,bool),A),B) = hAPP(real,bool,A,B). [resolve(1280,a,1277,a)]. 3.66/3.95 Derived: hAPP(real,fun(real,bool),A,ti(real,B)) = hAPP(real,fun(real,bool),A,B). [resolve(1280,a,1278,a)]. 3.66/3.95 Derived: ord_less(real) = ti(fun(real,fun(real,bool)),ord_less(real)). [resolve(1280,a,1279,a)]. 3.66/3.95 1281 -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(812)]. 3.66/3.95 Derived: ti(fun(int,bool),hAPP(int,fun(int,bool),A,B)) = hAPP(int,fun(int,bool),A,B). [resolve(1281,a,1272,a)]. 3.66/3.95 Derived: ti(fun(real,bool),hAPP(real,fun(real,bool),A,B)) = hAPP(real,fun(real,bool),A,B). [resolve(1281,a,1280,a)]. 3.66/3.95 1282 order(bool) # label(arity_HOL_Obool___Orderings_Oorder) # label(axiom). [assumption]. 3.66/3.95 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(1282,a,1273,a)]. 3.66/3.95 Derived: ti(bool,hAPP(bool,bool,A,B)) = hAPP(bool,bool,A,B). [resolve(1282,a,1274,a)]. 3.66/3.95 Derived: hAPP(bool,bool,A,ti(bool,B)) = hAPP(bool,bool,A,B). [resolve(1282,a,1275,a)]. 3.66/3.95 Derived: ti(fun(bool,fun(bool,bool)),ord_less_eq(bool)) = ord_less_eq(bool). [resolve(1282,a,1276,a)]. 3.66/3.95 Derived: hAPP(bool,bool,ti(fun(bool,bool),A),B) = hAPP(bool,bool,A,B). [resolve(1282,a,1277,a)]. 3.66/3.95 Derived: hAPP(bool,fun(bool,bool),A,ti(bool,B)) = hAPP(bool,fun(bool,bool),A,B). [resolve(1282,a,1278,a)]. 3.66/3.95 Derived: ord_less(bool) = ti(fun(bool,fun(bool,bool)),ord_less(bool)). [resolve(1282,a,1279,a)]. 3.66/3.95 Derived: ti(fun(bool,bool),hAPP(bool,fun(bool,bool),A,B)) = hAPP(bool,fun(bool,bool),A,B). [resolve(1282,a,1281,a)]. 3.66/3.95 1283 -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(902)]. 3.66/3.95 Derived: hAPP(int,fun(int,bool),ti(fun(int,fun(int,bool)),A),B) = hAPP(int,fun(int,bool),A,B). [resolve(1283,a,1272,a)]. 3.66/3.95 Derived: hAPP(real,fun(real,bool),ti(fun(real,fun(real,bool)),A),B) = hAPP(real,fun(real,bool),A,B). [resolve(1283,a,1280,a)]. 3.66/3.95 Derived: hAPP(bool,fun(bool,bool),ti(fun(bool,fun(bool,bool)),A),B) = hAPP(bool,fun(bool,bool),A,B). [resolve(1283,a,1282,a)]. 3.66/3.95 1284 order(nat) # label(arity_Nat_Onat___Orderings_Oorder) # label(axiom). [assumption]. 3.66/3.95 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(1284,a,1273,a)]. 3.66/3.95 Derived: ti(bool,hAPP(nat,bool,A,B)) = hAPP(nat,bool,A,B). [resolve(1284,a,1274,a)]. 3.66/3.95 Derived: hAPP(nat,bool,A,ti(nat,B)) = hAPP(nat,bool,A,B). [resolve(1284,a,1275,a)]. 3.66/3.95 Derived: ti(fun(nat,fun(nat,bool)),ord_less_eq(nat)) = ord_less_eq(nat). [resolve(1284,a,1276,a)]. 3.66/3.95 Derived: hAPP(nat,bool,ti(fun(nat,bool),A),B) = hAPP(nat,bool,A,B). [resolve(1284,a,1277,a)]. 3.66/3.95 Derived: hAPP(nat,fun(nat,bool),A,ti(nat,B)) = hAPP(nat,fun(nat,bool),A,B). [resolve(1284,a,1278,a)]. 3.66/3.95 Derived: ord_less(nat) = ti(fun(nat,fun(nat,bool)),ord_less(nat)). [resolve(1284,a,1279,a)]. 3.66/3.95 Derived: ti(fun(nat,bool),hAPP(nat,fun(nat,bool),A,B)) = hAPP(nat,fun(nat,bool),A,B). [resolve(1284,a,1281,a)]. 3.66/3.95 Derived: hAPP(nat,fun(nat,bool),ti(fun(nat,fun(nat,bool)),A),B) = hAPP(nat,fun(nat,bool),A,B). [resolve(1284,a,1283,a)]. 3.66/3.95 1285 number_semiring(real) # label(arity_RealDef_Oreal___Int_Onumber__semiring) # label(axiom). [assumption]. 3.66/3.95 1286 -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(35)]. 3.66/3.95 Derived: times_times(real,number_number_of(real,bit0(bit1(pls))),A) = plus_plus(real,A,A). [resolve(1285,a,1286,a)]. 3.66/3.95 1287 -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(134)]. 3.66/3.95 Derived: number_number_of(real,hAPP(nat,int,semiring_1_of_nat(int),A)) = hAPP(nat,real,semiring_1_of_nat(real),A). [resolve(1287,a,1285,a)]. 3.74/3.96 1288 -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(154)]. 3.74/3.96 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(1288,a,1285,a)]. 3.74/3.96 1289 -number_semiring(A) | number_number_of(A,bit1(pls)) = one_one(A) # label(fact_101_semiring__numeral__1__eq__1) # label(axiom). [clausify(229)]. 3.74/3.96 Derived: number_number_of(real,bit1(pls)) = one_one(real). [resolve(1289,a,1285,a)]. 3.74/3.96 1290 -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,times_times(int,B,C)) = times_times(A,number_number_of(A,B),number_number_of(A,C)) # label(fact_250_semiring__mult__number__of) # label(axiom). [clausify(248)]. 3.74/3.96 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,times_times(int,A,B)) = times_times(real,number_number_of(real,A),number_number_of(real,B)). [resolve(1290,a,1285,a)]. 3.74/3.96 1291 number_semiring(nat) # label(arity_Nat_Onat___Int_Onumber__semiring) # label(axiom). [assumption]. 3.74/3.96 Derived: number_number_of(nat,hAPP(nat,int,semiring_1_of_nat(int),A)) = hAPP(nat,nat,semiring_1_of_nat(nat),A). [resolve(1291,a,1287,a)]. 3.74/3.96 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(1291,a,1288,a)]. 3.74/3.96 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,times_times(int,A,B)) = times_times(nat,number_number_of(nat,A),number_number_of(nat,B)). [resolve(1291,a,1290,a)]. 3.74/3.96 1292 -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(436)]. 3.74/3.96 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(1292,a,1291,a)]. 3.74/3.96 1293 -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(493)]. 3.74/3.96 Derived: number_number_of(real,bit0(bit1(pls))) = plus_plus(real,one_one(real),one_one(real)). [resolve(1293,a,1285,a)]. 3.74/3.96 Derived: number_number_of(nat,bit0(bit1(pls))) = plus_plus(nat,one_one(nat),one_one(nat)). [resolve(1293,a,1291,a)]. 3.74/3.96 1294 number_semiring(int) # label(arity_Int_Oint___Int_Onumber__semiring) # label(axiom). [assumption]. 3.74/3.96 Derived: times_times(int,number_number_of(int,bit0(bit1(pls))),A) = plus_plus(int,A,A). [resolve(1294,a,1286,a)]. 3.74/3.96 Derived: number_number_of(int,hAPP(nat,int,semiring_1_of_nat(int),A)) = hAPP(nat,int,semiring_1_of_nat(int),A). [resolve(1294,a,1287,a)]. 3.74/3.96 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) | number_number_of(int,plus_plus(int,A,B)) = plus_plus(int,number_number_of(int,A),number_number_of(int,B)). [resolve(1294,a,1288,a)]. 3.83/4.08 Derived: number_number_of(int,bit1(pls)) = one_one(int). [resolve(1294,a,1289,a)]. 3.83/4.08 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) | number_number_of(int,times_times(int,A,B)) = times_times(int,number_number_of(int,A),number_number_of(int,B)). [resolve(1294,a,1290,a)]. 3.83/4.08 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(1294,a,1292,a)]. 3.83/4.08 Derived: number_number_of(int,bit0(bit1(pls))) = plus_plus(int,one_one(int),one_one(int)). [resolve(1294,a,1293,a)]. 3.83/4.08 1295 -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(572)]. 3.83/4.08 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(1295,a,1285,a)]. 3.83/4.08 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(1295,a,1291,a)]. 3.83/4.08 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(1295,a,1294,a)]. 3.83/4.08 1296 -number_semiring(A) | number_number_of(A,pls) = zero_zero(A) # label(fact_81_semiring__numeral__0__eq__0) # label(axiom). [clausify(815)]. 3.83/4.08 Derived: number_number_of(real,pls) = zero_zero(real). [resolve(1296,a,1285,a)]. 3.83/4.08 Derived: number_number_of(int,pls) = zero_zero(int). [resolve(1296,a,1294,a)]. 3.83/4.08 1297 -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(871)]. 3.83/4.08 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(1297,a,1285,a)]. 3.83/4.08 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(1297,a,1291,a)]. 3.83/4.08 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(1297,a,1294,a)]. 3.83/4.08 1298 -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(946)]. 3.83/4.08 Derived: times_times(real,A,number_number_of(real,bit0(bit1(pls)))) = plus_plus(real,A,A). [resolve(1298,a,1285,a)]. 3.83/4.08 Derived: times_times(nat,A,number_number_of(nat,bit0(bit1(pls)))) = plus_plus(nat,A,A). [resolve(1298,a,1291,a)]. 3.83/4.08 Derived: times_times(int,A,number_number_of(int,bit0(bit1(pls)))) = plus_plus(int,A,A). [resolve(1298,a,1294,a)]. 3.83/4.08 1299 linord20386208strict(int) # label(arity_Int_Oint___Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 3.83/4.08 1300 -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(40)]. 3.83/4.08 1301 -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(71)]. 3.83/4.09 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),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(1299,a,1300,a)]. 3.83/4.09 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(1299,a,1301,a)]. 3.83/4.09 1302 -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(251)]. 3.83/4.09 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(1302,a,1299,a)]. 3.83/4.09 1303 -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(260)]. 3.83/4.09 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),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(1303,a,1299,a)]. 3.83/4.09 1304 -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(319)]. 3.83/4.09 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(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(1304,a,1299,a)]. 3.83/4.09 1305 -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(350)]. 3.83/4.09 1306 linord20386208strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 3.83/4.09 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(1306,a,1300,a)]. 3.83/4.09 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(1306,a,1301,a)]. 3.83/4.09 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(1306,a,1302,a)]. 3.83/4.10 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),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(1306,a,1304,a)]. 3.83/4.10 1307 -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(444)]. 3.83/4.10 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(1307,a,1299,a)]. 3.83/4.10 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(1307,a,1306,a)]. 3.83/4.10 1308 -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(466)]. 3.83/4.10 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_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(1308,a,1299,a)]. 3.83/4.10 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_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(1308,a,1306,a)]. 3.83/4.10 1309 -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(503)]. 3.83/4.10 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_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(1309,a,1299,a)]. 3.83/4.10 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_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(1309,a,1306,a)]. 3.89/4.12 1310 -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(522)]. 3.89/4.12 1311 linord20386208strict(nat) # label(arity_Nat_Onat___Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 3.89/4.12 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(1311,a,1304,a)]. 3.89/4.12 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),times_times(nat,A,B)),times_times(nat,A,C))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)). [resolve(1311,a,1307,a)]. 3.89/4.12 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(1311,a,1308,a)]. 3.89/4.12 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(1311,a,1309,a)]. 3.89/4.12 1312 -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(696)]. 3.89/4.12 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(1312,a,1299,a)]. 3.89/4.12 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),times_times(real,A,B)),times_times(real,C,B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),C)). [resolve(1312,a,1306,a)]. 3.89/4.12 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(1312,a,1311,a)]. 3.89/4.12 1313 -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(756)]. 3.89/4.12 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(1313,a,1299,a)]. 3.89/4.14 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(1313,a,1306,a)]. 3.89/4.14 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(1313,a,1311,a)]. 3.89/4.14 1314 -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(859)]. 3.89/4.14 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),times_times(nat,C,A)),times_times(nat,C,B))). [resolve(1314,a,1311,a)]. 3.89/4.14 1315 -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(875)]. 3.89/4.14 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(1315,a,1299,a)]. 3.89/4.14 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(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(1315,a,1306,a)]. 3.89/4.14 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(1315,a,1311,a)]. 3.89/4.14 1316 -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(916)]. 3.89/4.14 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(1316,a,1299,a)]. 3.89/4.14 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(1316,a,1306,a)]. 3.89/4.14 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(1316,a,1311,a)]. 4.13/4.36 1317 -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(929)]. 4.13/4.36 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(1317,a,1306,a)]. 4.13/4.36 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),times_times(nat,A,B)),times_times(nat,A,C))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)). [resolve(1317,a,1311,a)]. 4.13/4.36 1318 comm_ring(int) # label(arity_Int_Oint___Rings_Ocomm__ring) # label(axiom). [assumption]. 4.13/4.36 1319 -comm_ring(A) | -dvd(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),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(49)]. 4.13/4.36 1320 -comm_ring(A) | -dvd(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),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(49)]. 4.13/4.36 Derived: -dvd(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),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(1318,a,1319,a)]. 4.13/4.36 Derived: -dvd(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),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(1318,a,1320,a)]. 4.13/4.36 1321 comm_ring(real) # label(arity_RealDef_Oreal___Rings_Ocomm__ring) # label(axiom). [assumption]. 4.13/4.36 Derived: -dvd(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),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(1321,a,1319,a)]. 4.13/4.36 Derived: -dvd(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),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(1321,a,1320,a)]. 4.13/4.36 1322 -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(877)]. 4.13/4.36 1323 -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(877)]. 4.13/4.36 1324 linordered_idom(int) # label(arity_Int_Oint___Rings_Olinordered__idom) # label(axiom). [assumption]. 4.13/4.36 1325 -linordered_idom(A) | abs_abs(A,hAPP(nat,A,semiring_1_of_nat(A),B)) = hAPP(nat,A,semiring_1_of_nat(A),B) # label(fact_935_abs__of__nat) # label(axiom). [clausify(50)]. 4.13/4.37 1326 -linordered_idom(A) | abs_abs(A,B) != abs_abs(A,C) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) # label(fact_919_dvd__if__abs__eq) # label(axiom). [clausify(54)]. 4.13/4.37 Derived: abs_abs(int,A) != abs_abs(int,B) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)). [resolve(1324,a,1326,a)]. 4.13/4.37 1327 -linordered_idom(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))) | zero_zero(A) = ti(A,B) # label(fact_553_sum__power2__le__zero__iff) # label(axiom). [clausify(104)]. 4.13/4.37 Derived: -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))) | zero_zero(int) = ti(int,A). [resolve(1327,a,1324,a)]. 4.13/4.37 1328 -linordered_idom(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))) | zero_zero(A) = ti(A,C) # label(fact_553_sum__power2__le__zero__iff) # label(axiom). [clausify(104)]. 4.13/4.37 Derived: -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))) | zero_zero(int) = ti(int,B). [resolve(1328,a,1324,a)]. 4.13/4.37 1329 -linordered_idom(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))) | zero_zero(A) != ti(A,B) | zero_zero(A) != ti(A,C) # label(fact_553_sum__power2__le__zero__iff) # label(axiom). [clausify(104)]. 4.13/4.37 Derived: 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))) | zero_zero(int) != ti(int,A) | zero_zero(int) != ti(int,B). [resolve(1329,a,1324,a)]. 4.13/4.37 1330 -number_ring(A) | -linordered_idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),number_number_of(A,B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) # label(fact_389_le__special_I1_J) # label(axiom). [clausify(117)]. 4.13/4.37 Derived: -number_ring(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)). [resolve(1330,b,1324,a)]. 4.13/4.37 1331 -number_ring(A) | -linordered_idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),number_number_of(A,B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) # label(fact_389_le__special_I1_J) # label(axiom). [clausify(117)]. 4.13/4.37 Derived: -number_ring(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)). [resolve(1331,b,1324,a)]. 4.13/4.37 1332 -number_ring(A) | -linordered_idom(A) | one_one(A) = abs_abs(A,number_number_of(A,min)) # label(fact_929_abs__minus__one) # label(axiom). [clausify(157)]. 4.13/4.37 Derived: -number_ring(int) | one_one(int) = abs_abs(int,number_number_of(int,min)). [resolve(1332,b,1324,a)]. 4.13/4.37 1333 -linordered_idom(A) | -number_ring(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(259)]. 4.13/4.37 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(1333,a,1324,a)]. 4.13/4.37 1334 -linordered_idom(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))))) | zero_zero(A) = ti(A,C) # label(fact_2_sum__power2__eq__zero__iff) # label(axiom). [clausify(264)]. 4.13/4.37 Derived: 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))))) | zero_zero(int) = ti(int,B). [resolve(1334,a,1324,a)]. 4.13/4.37 1335 -linordered_idom(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))))) | zero_zero(A) = ti(A,B) # label(fact_2_sum__power2__eq__zero__iff) # label(axiom). [clausify(264)]. 4.13/4.37 Derived: 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))))) | zero_zero(int) = ti(int,A). [resolve(1335,a,1324,a)]. 4.13/4.37 1336 -linordered_idom(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))))) | zero_zero(A) != ti(A,C) | zero_zero(A) != ti(A,B) # label(fact_2_sum__power2__eq__zero__iff) # label(axiom). [clausify(264)]. 4.13/4.37 Derived: 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))))) | zero_zero(int) != ti(int,B) | zero_zero(int) != ti(int,A). [resolve(1336,a,1324,a)]. 4.13/4.37 1337 -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(277)]. 4.13/4.37 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(1337,a,1324,a)]. 4.13/4.37 1338 -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(277)]. 4.13/4.37 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(1338,a,1324,a)]. 4.13/4.37 1339 -number_ring(A) | -linordered_idom(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,B)),number_number_of(A,C))) # label(fact_257_le__number__of) # label(axiom). [clausify(284)]. 4.13/4.37 1340 -number_ring(A) | -linordered_idom(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,B)),number_number_of(A,C))) # label(fact_257_le__number__of) # label(axiom). [clausify(284)]. 4.13/4.37 1341 -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(320)]. 4.13/4.37 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(1341,a,1324,a)]. 4.13/4.37 1342 -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(330)]. 4.13/4.38 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(1342,a,1324,a)]. 4.13/4.38 1343 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),pls)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,B)),zero_zero(A))) # label(fact_390_le__special_I3_J) # label(axiom). [clausify(352)]. 4.13/4.38 Derived: -number_ring(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),pls)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,A)),zero_zero(int))). [resolve(1343,a,1324,a)]. 4.13/4.38 1344 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),pls)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,B)),zero_zero(A))) # label(fact_390_le__special_I3_J) # label(axiom). [clausify(352)]. 4.13/4.38 Derived: -number_ring(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),pls)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,A)),zero_zero(int))). [resolve(1344,a,1324,a)]. 4.13/4.38 1345 linordered_idom(real) # label(arity_RealDef_Oreal___Rings_Olinordered__idom) # label(axiom). [assumption]. 4.13/4.38 Derived: abs_abs(real,hAPP(nat,real,semiring_1_of_nat(real),A)) = hAPP(nat,real,semiring_1_of_nat(real),A). [resolve(1345,a,1325,a)]. 4.13/4.38 Derived: abs_abs(real,A) != abs_abs(real,B) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)). [resolve(1345,a,1326,a)]. 4.13/4.38 Derived: -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))) | zero_zero(real) = ti(real,A). [resolve(1345,a,1327,a)]. 4.13/4.38 Derived: -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))) | zero_zero(real) = ti(real,B). [resolve(1345,a,1328,a)]. 4.13/4.38 Derived: 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))) | zero_zero(real) != ti(real,A) | zero_zero(real) != ti(real,B). [resolve(1345,a,1329,a)]. 4.13/4.38 Derived: -number_ring(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),number_number_of(real,A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)). [resolve(1345,a,1330,b)]. 4.13/4.38 Derived: -number_ring(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),number_number_of(real,A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)). [resolve(1345,a,1331,b)]. 4.13/4.38 Derived: -number_ring(real) | one_one(real) = abs_abs(real,number_number_of(real,min)). [resolve(1345,a,1332,b)]. 4.13/4.38 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(1345,a,1333,a)]. 4.13/4.38 Derived: 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))))) | zero_zero(real) = ti(real,B). [resolve(1345,a,1334,a)]. 4.13/4.38 Derived: 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))))) | zero_zero(real) = ti(real,A). [resolve(1345,a,1335,a)]. 4.13/4.38 Derived: 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))))) | zero_zero(real) != ti(real,B) | zero_zero(real) != ti(real,A). [resolve(1345,a,1336,a)]. 4.13/4.38 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(1345,a,1337,a)]. 4.13/4.38 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(1345,a,1338,a)]. 4.13/4.38 Derived: -number_ring(real) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),number_number_of(real,B))). [resolve(1345,a,1339,b)]. 4.13/4.38 Derived: -number_ring(real) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),number_number_of(real,B))). [resolve(1345,a,1340,b)]. 4.13/4.38 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),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(1345,a,1341,a)]. 4.13/4.38 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(1345,a,1342,a)]. 4.13/4.38 Derived: -number_ring(real) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),pls)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),zero_zero(real))). [resolve(1345,a,1343,a)]. 4.13/4.38 Derived: -number_ring(real) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),pls)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),zero_zero(real))). [resolve(1345,a,1344,a)]. 4.13/4.38 1346 -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(397)]. 4.13/4.38 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(1346,a,1345,a)]. 4.13/4.38 1347 -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(406)]. 4.13/4.38 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(1347,a,1345,a)]. 4.13/4.38 1348 -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(461)]. 4.13/4.38 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(1348,a,1324,a)]. 4.13/4.38 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(1348,a,1345,a)]. 4.13/4.38 1349 -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(461)]. 4.13/4.38 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(1349,a,1324,a)]. 4.13/4.39 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(1349,a,1345,a)]. 4.13/4.39 1350 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),pls)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,B)),zero_zero(A))) # label(fact_49_less__special_I3_J) # label(axiom). [clausify(469)]. 4.13/4.39 Derived: -number_ring(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,A)),zero_zero(int))). [resolve(1350,a,1324,a)]. 4.13/4.39 Derived: -number_ring(real) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,A)),zero_zero(real))). [resolve(1350,a,1345,a)]. 4.13/4.39 1351 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),pls)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,B)),zero_zero(A))) # label(fact_49_less__special_I3_J) # label(axiom). [clausify(469)]. 4.13/4.39 Derived: -number_ring(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,A)),zero_zero(int))). [resolve(1351,a,1324,a)]. 4.13/4.39 Derived: -number_ring(real) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,A)),zero_zero(real))). [resolve(1351,a,1345,a)]. 4.13/4.39 1352 -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(472)]. 4.13/4.39 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(1352,a,1324,a)]. 4.13/4.39 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(1352,a,1345,a)]. 4.13/4.39 1353 -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(478)]. 4.13/4.39 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(1353,a,1324,a)]. 4.13/4.39 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(1353,a,1345,a)]. 4.13/4.39 1354 -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(478)]. 4.13/4.39 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(1354,a,1324,a)]. 4.13/4.39 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(1354,a,1345,a)]. 4.13/4.39 1355 -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(535)]. 4.13/4.40 Derived: times_times(int,abs_abs(int,A),abs_abs(int,A)) = times_times(int,A,A). [resolve(1355,a,1324,a)]. 4.13/4.40 Derived: times_times(real,abs_abs(real,A),abs_abs(real,A)) = times_times(real,A,A). [resolve(1355,a,1345,a)]. 4.13/4.40 1356 -number_ring(A) | -linordered_idom(A) | one_one(A) = abs_abs(A,hAPP(nat,A,power_power(A,number_number_of(A,min)),B)) # label(fact_958_abs__power__minus__one) # label(axiom). [clausify(545)]. 4.13/4.40 Derived: -number_ring(int) | one_one(int) = abs_abs(int,hAPP(nat,int,power_power(int,number_number_of(int,min)),A)). [resolve(1356,b,1324,a)]. 4.13/4.40 Derived: -number_ring(real) | one_one(real) = abs_abs(real,hAPP(nat,real,power_power(real,number_number_of(real,min)),A)). [resolve(1356,b,1345,a)]. 4.13/4.40 1357 -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(564)]. 4.13/4.40 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(1357,a,1324,a)]. 4.13/4.40 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(1357,a,1345,a)]. 4.13/4.40 1358 -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(581)]. 4.13/4.40 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(1358,a,1324,a)]. 4.13/4.40 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(1358,a,1345,a)]. 4.13/4.40 1359 -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(616)]. 4.13/4.40 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(1359,a,1324,a)]. 4.13/4.40 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(1359,a,1345,a)]. 4.13/4.40 1360 -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(616)]. 4.13/4.40 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(1360,a,1324,a)]. 4.13/4.40 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(1360,a,1345,a)]. 4.13/4.40 1361 -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(630)]. 4.13/4.40 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(1361,a,1324,a)]. 4.13/4.40 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(1361,a,1345,a)]. 4.13/4.40 1362 -number_ring(A) | -linordered_idom(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),number_number_of(A,B))) # label(fact_403_le__special_I2_J) # label(axiom). [clausify(631)]. 4.13/4.40 Derived: -number_ring(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),number_number_of(int,A))). [resolve(1362,b,1324,a)]. 4.13/4.40 Derived: -number_ring(real) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),number_number_of(real,A))). [resolve(1362,b,1345,a)]. 4.13/4.40 1363 -number_ring(A) | -linordered_idom(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),number_number_of(A,B))) # label(fact_403_le__special_I2_J) # label(axiom). [clausify(631)]. 4.13/4.40 Derived: -number_ring(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),number_number_of(int,A))). [resolve(1363,b,1324,a)]. 4.13/4.40 Derived: -number_ring(real) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),number_number_of(real,A))). [resolve(1363,b,1345,a)]. 4.13/4.40 1364 -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(652)]. 4.13/4.40 Derived: -number_ring(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,A)),number_number_of(int,B))). [resolve(1364,a,1324,a)]. 4.13/4.40 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(1364,a,1345,a)]. 4.13/4.40 1365 -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(652)]. 4.13/4.40 Derived: -number_ring(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,A)),number_number_of(int,B))). [resolve(1365,a,1324,a)]. 4.13/4.40 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(1365,a,1345,a)]. 4.13/4.40 1366 -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(677)]. 4.13/4.40 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)) | -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(1366,a,1324,a)]. 4.19/4.41 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),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(1366,a,1345,a)]. 4.19/4.41 1367 -linordered_idom(A) | abs_abs(A,hAPP(nat,A,power_power(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_964_abs__power2) # label(axiom). [clausify(689)]. 4.19/4.41 Derived: abs_abs(int,hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls))))) = hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))). [resolve(1367,a,1324,a)]. 4.19/4.41 Derived: abs_abs(real,hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls))))) = hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))). [resolve(1367,a,1345,a)]. 4.19/4.41 1368 -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(690)]. 4.19/4.41 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(1368,a,1324,a)]. 4.19/4.41 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(1368,a,1345,a)]. 4.19/4.41 1369 -linordered_idom(A) | zero_zero(A) = ti(A,B) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),plus_plus(A,hAPP(nat,A,power_power(A,C),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_97_sum__power2__gt__zero__iff) # label(axiom). [clausify(701)]. 4.19/4.41 Derived: zero_zero(int) = ti(int,A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),plus_plus(int,hAPP(nat,int,power_power(int,B),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls))))))). [resolve(1369,a,1324,a)]. 4.19/4.41 Derived: zero_zero(real) = ti(real,A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),plus_plus(real,hAPP(nat,real,power_power(real,B),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls))))))). [resolve(1369,a,1345,a)]. 4.19/4.41 1370 -linordered_idom(A) | zero_zero(A) = ti(A,B) | 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))))))) # label(fact_97_sum__power2__gt__zero__iff) # label(axiom). [clausify(701)]. 4.19/4.41 Derived: zero_zero(int) = ti(int,A) | 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))))))). [resolve(1370,a,1324,a)]. 4.19/4.41 Derived: zero_zero(real) = ti(real,A) | 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))))))). [resolve(1370,a,1345,a)]. 4.19/4.41 1371 -linordered_idom(A) | zero_zero(A) != ti(A,B) | zero_zero(A) != ti(A,C) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),plus_plus(A,hAPP(nat,A,power_power(A,C),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_97_sum__power2__gt__zero__iff) # label(axiom). [clausify(701)]. 4.19/4.42 Derived: zero_zero(int) != ti(int,A) | zero_zero(int) != ti(int,B) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),plus_plus(int,hAPP(nat,int,power_power(int,B),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls))))))). [resolve(1371,a,1324,a)]. 4.19/4.42 Derived: zero_zero(real) != ti(real,A) | zero_zero(real) != ti(real,B) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),plus_plus(real,hAPP(nat,real,power_power(real,B),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls))))))). [resolve(1371,a,1345,a)]. 4.19/4.42 1372 -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(720)]. 4.19/4.42 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(1372,b,1324,a)]. 4.19/4.42 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(1372,b,1345,a)]. 4.19/4.42 1373 -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(720)]. 4.19/4.42 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(1373,b,1324,a)]. 4.19/4.42 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(1373,b,1345,a)]. 4.19/4.42 1374 -linordered_idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),abs_abs(A,C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) # label(fact_921_dvd__abs__iff) # label(axiom). [clausify(725)]. 4.19/4.42 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),abs_abs(int,B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)). [resolve(1374,a,1324,a)]. 4.19/4.42 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),abs_abs(real,B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)). [resolve(1374,a,1345,a)]. 4.19/4.42 1375 -linordered_idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),abs_abs(A,C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) # label(fact_921_dvd__abs__iff) # label(axiom). [clausify(725)]. 4.19/4.42 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),abs_abs(int,B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)). [resolve(1375,a,1324,a)]. 4.19/4.42 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),abs_abs(real,B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)). [resolve(1375,a,1345,a)]. 4.19/4.42 1376 -linordered_idom(A) | abs_abs(A,hAPP(nat,A,power_power(A,B),C)) = hAPP(nat,A,power_power(A,abs_abs(A,B)),C) # label(fact_934_power__abs) # label(axiom). [clausify(740)]. 4.19/4.42 Derived: abs_abs(int,hAPP(nat,int,power_power(int,A),B)) = hAPP(nat,int,power_power(int,abs_abs(int,A)),B). [resolve(1376,a,1324,a)]. 4.19/4.42 Derived: abs_abs(real,hAPP(nat,real,power_power(real,A),B)) = hAPP(nat,real,power_power(real,abs_abs(real,A)),B). [resolve(1376,a,1345,a)]. 4.19/4.42 1377 -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(750)]. 4.19/4.42 Derived: times_times(int,abs_abs(int,A),abs_abs(int,B)) = abs_abs(int,times_times(int,A,B)). [resolve(1377,a,1324,a)]. 4.19/4.43 Derived: times_times(real,abs_abs(real,A),abs_abs(real,B)) = abs_abs(real,times_times(real,A,B)). [resolve(1377,a,1345,a)]. 4.19/4.43 1378 -number_ring(A) | -linordered_idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,B)),one_one(A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),bit1(pls))) # label(fact_404_le__special_I4_J) # label(axiom). [clausify(824)]. 4.19/4.43 Derived: -number_ring(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,A)),one_one(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),bit1(pls))). [resolve(1378,b,1324,a)]. 4.19/4.43 Derived: -number_ring(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),one_one(real))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),bit1(pls))). [resolve(1378,b,1345,a)]. 4.19/4.43 1379 -number_ring(A) | -linordered_idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,B)),one_one(A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),bit1(pls))) # label(fact_404_le__special_I4_J) # label(axiom). [clausify(824)]. 4.19/4.43 Derived: -number_ring(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,A)),one_one(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),bit1(pls))). [resolve(1379,b,1324,a)]. 4.19/4.43 Derived: -number_ring(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),one_one(real))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),bit1(pls))). [resolve(1379,b,1345,a)]. 4.19/4.43 1380 -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))) | zero_zero(A) = ti(A,B) # label(fact_699_even__power__le__0__imp__0) # label(axiom). [clausify(908)]. 4.19/4.43 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))) | zero_zero(int) = ti(int,A). [resolve(1380,a,1324,a)]. 4.19/4.43 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))) | zero_zero(real) = ti(real,A). [resolve(1380,a,1345,a)]. 4.19/4.43 1381 -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(912)]. 4.19/4.43 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(1381,a,1324,a)]. 4.19/4.43 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(1381,a,1345,a)]. 4.19/4.43 1382 -linordered_idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))))) | zero_zero(A) != ti(A,B) # label(fact_95_zero__less__power2) # label(axiom). [clausify(915)]. 4.19/4.43 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))))) | zero_zero(int) != ti(int,A). [resolve(1382,a,1324,a)]. 4.19/4.43 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))))) | zero_zero(real) != ti(real,A). [resolve(1382,a,1345,a)]. 4.19/4.43 1383 -linordered_idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))))) | zero_zero(A) = ti(A,B) # label(fact_95_zero__less__power2) # label(axiom). [clausify(915)]. 4.19/4.43 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))))) | zero_zero(int) = ti(int,A). [resolve(1383,a,1324,a)]. 4.26/4.54 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))))) | zero_zero(real) = ti(real,A). [resolve(1383,a,1345,a)]. 4.26/4.54 1384 -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(928)]. 4.26/4.54 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(1384,a,1324,a)]. 4.26/4.54 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(1384,a,1345,a)]. 4.26/4.54 1385 -linordered_idom(A) | one_one(A) = abs_abs(A,one_one(A)) # label(fact_933_abs__one) # label(axiom). [clausify(995)]. 4.26/4.54 Derived: one_one(int) = abs_abs(int,one_one(int)). [resolve(1385,a,1324,a)]. 4.26/4.54 Derived: one_one(real) = abs_abs(real,one_one(real)). [resolve(1385,a,1345,a)]. 4.26/4.54 1386 -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(91)]. 4.26/4.54 1387 ordere216010020id_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 4.26/4.54 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(1386,a,1387,a)]. 4.26/4.54 1388 -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(118)]. 4.26/4.54 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(1388,a,1387,a)]. 4.26/4.54 1389 -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(156)]. 4.26/4.54 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(1389,a,1387,a)]. 4.26/4.54 1390 -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(492)]. 4.26/4.54 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),plus_plus(real,A,B))). [resolve(1390,a,1387,a)]. 4.26/4.54 1391 -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(562)]. 4.26/4.55 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(1391,a,1387,a)]. 4.26/4.55 1392 -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(563)]. 4.26/4.55 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(1392,a,1387,a)]. 4.26/4.55 1393 ordere216010020id_add(int) # label(arity_Int_Oint___Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 4.26/4.55 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(1393,a,1386,a)]. 4.26/4.55 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(1393,a,1388,a)]. 4.26/4.55 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)),plus_plus(int,A,B))). [resolve(1393,a,1389,a)]. 4.26/4.55 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),plus_plus(int,A,B))). [resolve(1393,a,1390,a)]. 4.26/4.55 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(1393,a,1391,a)]. 4.26/4.55 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(1393,a,1392,a)]. 4.26/4.55 1394 -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(601)]. 4.26/4.55 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(1394,a,1387,a)]. 4.26/4.55 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(1394,a,1393,a)]. 4.26/4.55 1395 -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(660)]. 4.35/4.58 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(1395,a,1387,a)]. 4.35/4.58 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(1395,a,1393,a)]. 4.35/4.58 1396 -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(703)]. 4.35/4.58 1397 -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(712)]. 4.35/4.58 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(1397,a,1387,a)]. 4.35/4.58 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(1397,a,1393,a)]. 4.35/4.58 1398 ordere216010020id_add(nat) # label(arity_Nat_Onat___Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 4.35/4.58 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),plus_plus(nat,A,B))). [resolve(1398,a,1389,a)]. 4.35/4.58 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),plus_plus(nat,A,B))). [resolve(1398,a,1390,a)]. 4.35/4.58 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(1398,a,1391,a)]. 4.35/4.58 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(1398,a,1392,a)]. 4.35/4.58 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),ord_less_eq(nat),zero_zero(nat)),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),plus_plus(nat,A,B))). [resolve(1398,a,1394,a)]. 4.35/4.58 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),plus_plus(nat,A,C))). [resolve(1398,a,1395,a)]. 4.35/4.58 1399 -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(894)]. 4.35/4.59 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(1399,a,1387,a)]. 4.35/4.59 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(1399,a,1393,a)]. 4.35/4.59 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(1399,a,1398,a)]. 4.35/4.59 1400 -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(970)]. 4.35/4.59 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(1400,a,1387,a)]. 4.35/4.59 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(1400,a,1393,a)]. 4.35/4.59 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(1400,a,1398,a)]. 4.35/4.59 1401 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | zero_zero(A) != plus_plus(A,B,C) | zero_zero(A) = ti(A,B) # label(fact_435_add__nonneg__eq__0__iff) # label(axiom). [clausify(974)]. 4.35/4.59 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | zero_zero(real) != plus_plus(real,A,B) | zero_zero(real) = ti(real,A). [resolve(1401,a,1387,a)]. 4.35/4.59 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | zero_zero(int) != plus_plus(int,A,B) | zero_zero(int) = ti(int,A). [resolve(1401,a,1393,a)]. 4.35/4.59 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | zero_zero(nat) != plus_plus(nat,A,B) | zero_zero(nat) = ti(nat,A). [resolve(1401,a,1398,a)]. 4.35/4.59 1402 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | zero_zero(A) != plus_plus(A,B,C) | zero_zero(A) = ti(A,C) # label(fact_435_add__nonneg__eq__0__iff) # label(axiom). [clausify(974)]. 4.35/4.59 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | zero_zero(real) != plus_plus(real,A,B) | zero_zero(real) = ti(real,B). [resolve(1402,a,1387,a)]. 4.35/4.59 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | zero_zero(int) != plus_plus(int,A,B) | zero_zero(int) = ti(int,B). [resolve(1402,a,1393,a)]. 4.35/4.59 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | zero_zero(nat) != plus_plus(nat,A,B) | zero_zero(nat) = ti(nat,B). [resolve(1402,a,1398,a)]. 4.73/4.95 1403 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | zero_zero(A) = plus_plus(A,B,C) | zero_zero(A) != ti(A,B) | zero_zero(A) != ti(A,C) # label(fact_435_add__nonneg__eq__0__iff) # label(axiom). [clausify(974)]. 4.73/4.95 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | zero_zero(real) = plus_plus(real,A,B) | zero_zero(real) != ti(real,A) | zero_zero(real) != ti(real,B). [resolve(1403,a,1387,a)]. 4.73/4.95 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | zero_zero(int) = plus_plus(int,A,B) | zero_zero(int) != ti(int,A) | zero_zero(int) != ti(int,B). [resolve(1403,a,1393,a)]. 4.73/4.95 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | zero_zero(nat) = plus_plus(nat,A,B) | zero_zero(nat) != ti(nat,A) | zero_zero(nat) != ti(nat,B). [resolve(1403,a,1398,a)]. 4.73/4.95 1404 -comm_monoid_mult(A) | ti(A,B) = times_times(A,B,one_one(A)) # label(fact_347_mult_Ocomm__neutral) # label(axiom). [clausify(342)]. 4.73/4.95 1405 comm_monoid_mult(int) # label(arity_Int_Oint___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 4.73/4.95 1406 comm_monoid_mult(real) # label(arity_RealDef_Oreal___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 4.73/4.95 1407 comm_monoid_mult(nat) # label(arity_Nat_Onat___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 4.73/4.95 1408 -comm_monoid_mult(A) | ti(A,B) = times_times(A,one_one(A),B) # label(fact_350_mult__1) # label(axiom). [clausify(985)]. 4.73/4.95 1409 -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(1009)]. 4.73/4.95 1410 -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(299)]. 4.73/4.95 1411 linordered_semiring(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring) # label(axiom). [assumption]. 4.73/4.95 1412 linordered_semiring(nat) # label(arity_Nat_Onat___Rings_Olinordered__semiring) # label(axiom). [assumption]. 4.73/4.95 1413 linordered_semiring(int) # label(arity_Int_Oint___Rings_Olinordered__semiring) # label(axiom). [assumption]. 4.73/4.95 1414 -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(1001)]. 4.73/4.95 1415 linord626643107strict(int) # label(arity_Int_Oint___Rings_Olinordered__semiring__1__strict) # label(axiom). [assumption]. 4.73/4.95 1416 -linord626643107strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),F)) | one_one(A) != plus_plus(A,E,F) | 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(61)]. 4.73/4.95 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),D)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),E)) | one_one(int) != plus_plus(int,D,E) | 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(1415,a,1416,a)]. 5.01/5.25 1417 linord626643107strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring__1__strict) # label(axiom). [assumption]. 5.01/5.25 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),D)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),E)) | one_one(real) != plus_plus(real,D,E) | 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(1417,a,1416,a)]. 5.01/5.25 1418 semiring(nat) # label(arity_Nat_Onat___Rings_Osemiring) # label(axiom). [assumption]. 5.01/5.25 1419 -semiring(A) | -number(A) | times_times(A,plus_plus(A,B,C),number_number_of(A,D)) = plus_plus(A,times_times(A,B,number_number_of(A,D)),times_times(A,C,number_number_of(A,D))) # label(fact_444_left__distrib__number__of) # label(axiom). [clausify(66)]. 5.01/5.25 1420 -semiring(A) | -number(A) | times_times(A,number_number_of(A,B),plus_plus(A,C,D)) = plus_plus(A,times_times(A,number_number_of(A,B),C),times_times(A,number_number_of(A,B),D)) # label(fact_443_right__distrib__number__of) # label(axiom). [clausify(141)]. 5.01/5.25 1421 semiring(int) # label(arity_Int_Oint___Rings_Osemiring) # label(axiom). [assumption]. 5.01/5.25 1422 -semiring(A) | ti(A,times_times(A,B,C)) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_2_res) # label(axiom). [clausify(393)]. 5.01/5.25 1423 semiring(real) # label(arity_RealDef_Oreal___Rings_Osemiring) # label(axiom). [assumption]. 5.01/5.25 1424 -semiring(A) | plus_plus(A,times_times(A,plus_plus(A,B,C),D),E) = plus_plus(A,times_times(A,B,D),plus_plus(A,times_times(A,C,D),E)) # label(fact_340_combine__common__factor) # label(axiom). [clausify(933)]. 5.01/5.25 Derived: plus_plus(nat,times_times(nat,plus_plus(nat,A,B),C),D) = plus_plus(nat,times_times(nat,A,C),plus_plus(nat,times_times(nat,B,C),D)). [resolve(1424,a,1418,a)]. 5.01/5.25 Derived: plus_plus(int,times_times(int,plus_plus(int,A,B),C),D) = plus_plus(int,times_times(int,A,C),plus_plus(int,times_times(int,B,C),D)). [resolve(1424,a,1421,a)]. 5.01/5.25 Derived: plus_plus(real,times_times(real,plus_plus(real,A,B),C),D) = plus_plus(real,times_times(real,A,C),plus_plus(real,times_times(real,B,C),D)). [resolve(1424,a,1423,a)]. 5.01/5.25 1425 -semiring(A) | times_times(A,B,ti(A,C)) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_2_arg2) # label(axiom). [clausify(952)]. 5.01/5.25 1426 -semiring(A) | times_times(A,ti(A,B),C) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_2_arg1) # label(axiom). [clausify(993)]. 5.01/5.25 1427 semiring_char_0(nat) # label(arity_Nat_Onat___Nat_Osemiring__char__0) # label(axiom). [assumption]. 5.01/5.25 1428 -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(67)]. 5.01/5.25 1429 -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(67)]. 5.01/5.25 Derived: hAPP(nat,nat,semiring_1_of_nat(nat),A) != hAPP(nat,nat,semiring_1_of_nat(nat),B) | A = B. [resolve(1427,a,1428,a)]. 5.01/5.25 Derived: hAPP(nat,nat,semiring_1_of_nat(nat),A) = hAPP(nat,nat,semiring_1_of_nat(nat),B) | A != B. [resolve(1427,a,1429,a)]. 5.01/5.25 1430 semiring_char_0(int) # label(arity_Int_Oint___Nat_Osemiring__char__0) # label(axiom). [assumption]. 5.01/5.25 1431 semiring_char_0(real) # label(arity_RealDef_Oreal___Nat_Osemiring__char__0) # label(axiom). [assumption]. 5.01/5.25 Derived: hAPP(nat,real,semiring_1_of_nat(real),A) != hAPP(nat,real,semiring_1_of_nat(real),B) | A = B. [resolve(1431,a,1428,a)]. 5.01/5.25 Derived: hAPP(nat,real,semiring_1_of_nat(real),A) = hAPP(nat,real,semiring_1_of_nat(real),B) | A != B. [resolve(1431,a,1429,a)]. 5.10/5.37 1432 -power(A) | -semiring_0(A) | number_number_of(nat,B) = zero_zero(nat) | zero_zero(A) = hAPP(nat,A,power_power(A,zero_zero(A)),number_number_of(nat,B)) # label(fact_11_power__0__left__number__of) # label(axiom). [clausify(147)]. 5.10/5.37 1433 semiring_0(int) # label(arity_Int_Oint___Rings_Osemiring__0) # label(axiom). [assumption]. 5.10/5.37 Derived: -power(int) | number_number_of(nat,A) = zero_zero(nat) | zero_zero(int) = hAPP(nat,int,power_power(int,zero_zero(int)),number_number_of(nat,A)). [resolve(1432,b,1433,a)]. 5.10/5.37 1434 -power(A) | -semiring_0(A) | number_number_of(nat,B) != zero_zero(nat) | one_one(A) = hAPP(nat,A,power_power(A,zero_zero(A)),number_number_of(nat,B)) # label(fact_11_power__0__left__number__of) # label(axiom). [clausify(147)]. 5.10/5.37 Derived: -power(int) | number_number_of(nat,A) != zero_zero(nat) | one_one(int) = hAPP(nat,int,power_power(int,zero_zero(int)),number_number_of(nat,A)). [resolve(1434,b,1433,a)]. 5.10/5.37 1435 semiring_0(real) # label(arity_RealDef_Oreal___Rings_Osemiring__0) # label(axiom). [assumption]. 5.10/5.37 Derived: -power(real) | number_number_of(nat,A) = zero_zero(nat) | zero_zero(real) = hAPP(nat,real,power_power(real,zero_zero(real)),number_number_of(nat,A)). [resolve(1435,a,1432,b)]. 5.10/5.37 Derived: -power(real) | number_number_of(nat,A) != zero_zero(nat) | one_one(real) = hAPP(nat,real,power_power(real,zero_zero(real)),number_number_of(nat,A)). [resolve(1435,a,1434,b)]. 5.10/5.37 1436 -dvd(A) | -semiring_0(A) | -hBOOL(hAPP(A,bool,B,times_times(A,C,D))) | hBOOL(hAPP(A,bool,B,f8(A,B,C))) # label(fact_908_unity__coeff__ex) # label(axiom). [clausify(273)]. 5.10/5.37 Derived: -dvd(int) | -hBOOL(hAPP(int,bool,A,times_times(int,B,C))) | hBOOL(hAPP(int,bool,A,f8(int,A,B))). [resolve(1436,b,1433,a)]. 5.10/5.37 Derived: -dvd(real) | -hBOOL(hAPP(real,bool,A,times_times(real,B,C))) | hBOOL(hAPP(real,bool,A,f8(real,A,B))). [resolve(1436,b,1435,a)]. 5.10/5.37 1437 -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,f8(A,B,C),zero_zero(A)))) # label(fact_908_unity__coeff__ex) # label(axiom). [clausify(273)]. 5.10/5.37 Derived: -dvd(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,f8(int,A,B),zero_zero(int)))). [resolve(1437,b,1433,a)]. 5.10/5.37 Derived: -dvd(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,f8(real,A,B),zero_zero(real)))). [resolve(1437,b,1435,a)]. 5.10/5.37 1438 -dvd(A) | -semiring_0(A) | hBOOL(hAPP(A,bool,B,times_times(A,C,f9(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(273)]. 5.10/5.37 Derived: -dvd(int) | hBOOL(hAPP(int,bool,A,times_times(int,B,f9(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(1438,b,1433,a)]. 5.10/5.37 Derived: -dvd(real) | hBOOL(hAPP(real,bool,A,times_times(real,B,f9(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(1438,b,1435,a)]. 5.10/5.37 1439 -power(A) | -semiring_0(A) | zero_zero(nat) = B | zero_zero(A) = hAPP(nat,A,power_power(A,zero_zero(A)),B) # label(fact_109_power__0__left) # label(axiom). [clausify(451)]. 5.10/5.37 Derived: -power(int) | zero_zero(nat) = A | zero_zero(int) = hAPP(nat,int,power_power(int,zero_zero(int)),A). [resolve(1439,b,1433,a)]. 5.10/5.37 Derived: -power(real) | zero_zero(nat) = A | zero_zero(real) = hAPP(nat,real,power_power(real,zero_zero(real)),A). [resolve(1439,b,1435,a)]. 5.10/5.37 1440 -power(A) | -semiring_0(A) | zero_zero(nat) != B | one_one(A) = hAPP(nat,A,power_power(A,zero_zero(A)),B) # label(fact_109_power__0__left) # label(axiom). [clausify(451)]. 5.10/5.37 1441 semiring_0(nat) # label(arity_Nat_Onat___Rings_Osemiring__0) # label(axiom). [assumption]. 5.10/5.37 Derived: -power(nat) | number_number_of(nat,A) = zero_zero(nat) | zero_zero(nat) = hAPP(nat,nat,power_power(nat,zero_zero(nat)),number_number_of(nat,A)). [resolve(1441,a,1432,b)]. 5.50/5.77 Derived: -power(nat) | number_number_of(nat,A) != zero_zero(nat) | one_one(nat) = hAPP(nat,nat,power_power(nat,zero_zero(nat)),number_number_of(nat,A)). [resolve(1441,a,1434,b)]. 5.50/5.77 Derived: -dvd(nat) | -hBOOL(hAPP(nat,bool,A,times_times(nat,B,C))) | hBOOL(hAPP(nat,bool,A,f8(nat,A,B))). [resolve(1441,a,1436,b)]. 5.50/5.77 Derived: -dvd(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,f8(nat,A,B),zero_zero(nat)))). [resolve(1441,a,1437,b)]. 5.50/5.77 Derived: -dvd(nat) | hBOOL(hAPP(nat,bool,A,times_times(nat,B,f9(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(1441,a,1438,b)]. 5.50/5.77 Derived: -power(nat) | zero_zero(nat) = A | zero_zero(nat) = hAPP(nat,nat,power_power(nat,zero_zero(nat)),A). [resolve(1441,a,1439,b)]. 5.50/5.77 Derived: -power(nat) | zero_zero(nat) != A | one_one(nat) = hAPP(nat,nat,power_power(nat,zero_zero(nat)),A). [resolve(1441,a,1440,b)]. 5.50/5.78 1442 -comm_monoid_add(A) | ti(A,B) = plus_plus(A,B,zero_zero(A)) # label(fact_213_add_Ocomm__neutral) # label(axiom). [clausify(711)]. 5.50/5.78 1443 comm_monoid_add(nat) # label(arity_Nat_Onat___Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 5.50/5.78 1444 comm_monoid_add(real) # label(arity_RealDef_Oreal___Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 5.50/5.78 1445 comm_monoid_add(int) # label(arity_Int_Oint___Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 5.50/5.78 1446 -comm_monoid_add(A) | ti(A,B) = plus_plus(A,zero_zero(A),B) # label(fact_217_add__0) # label(axiom). [clausify(998)]. 5.50/5.78 1447 ab_semigroup_mult(real) # label(arity_RealDef_Oreal___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 5.50/5.78 1448 -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(81)]. 5.50/5.78 1449 -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(490)]. 5.50/5.78 1450 -ab_semigroup_mult(A) | ti(A,times_times(A,B,C)) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_5_res) # label(axiom). [clausify(585)]. 5.50/5.78 1451 ab_semigroup_mult(nat) # label(arity_Nat_Onat___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 5.50/5.78 1452 -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(773)]. 5.50/5.78 Derived: times_times(real,times_times(real,A,B),C) = times_times(real,A,times_times(real,B,C)). [resolve(1452,a,1447,a)]. 5.50/5.78 1453 ab_semigroup_mult(int) # label(arity_Int_Oint___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 5.50/5.78 Derived: times_times(int,times_times(int,A,B),C) = times_times(int,A,times_times(int,B,C)). [resolve(1453,a,1452,a)]. 5.50/5.78 1454 -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,C),abs_abs(A,B)) = abs_abs(A,times_times(A,C,B)) # label(fact_925_abs__eq__mult) # label(axiom). [clausify(589)]. 5.50/5.78 1455 ordered_ring_abs(int) # label(arity_Int_Oint___Rings_Oordered__ring__abs) # label(axiom). [assumption]. 5.50/5.78 1456 -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,C),abs_abs(A,B)) = abs_abs(A,times_times(A,C,B)) # label(fact_925_abs__eq__mult) # label(axiom). [clausify(589)]. 5.50/5.78 1457 -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,C),abs_abs(A,B)) = abs_abs(A,times_times(A,C,B)) # label(fact_925_abs__eq__mult) # label(axiom). [clausify(589)]. 5.50/5.78 1458 -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,C),abs_abs(A,B)) = abs_abs(A,times_times(A,C,B)) # label(fact_925_abs__eq__mult) # label(axiom). [clausify(589)]. 5.69/5.90 1459 ordered_ring_abs(real) # label(arity_RealDef_Oreal___Rings_Oordered__ring__abs) # label(axiom). [assumption]. 5.69/5.90 1460 -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),D),plus_plus(A,times_times(A,minus_minus(A,E,B),C),F))) # label(fact_287_le__add__iff2) # label(axiom). [clausify(123)]. 5.69/5.90 1461 ordered_ring(real) # label(arity_RealDef_Oreal___Rings_Oordered__ring) # label(axiom). [assumption]. 5.69/5.90 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),C),plus_plus(real,times_times(real,minus_minus(real,D,A),B),E))). [resolve(1460,a,1461,a)]. 5.69/5.90 1462 -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),D),plus_plus(A,times_times(A,minus_minus(A,E,B),C),F))) # label(fact_287_le__add__iff2) # label(axiom). [clausify(123)]. 5.69/5.90 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),C),plus_plus(real,times_times(real,minus_minus(real,D,A),B),E))). [resolve(1462,a,1461,a)]. 5.69/5.90 1463 -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(200)]. 5.69/5.90 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(1463,a,1461,a)]. 5.69/5.90 1464 -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(287)]. 5.69/5.90 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(1464,a,1461,a)]. 5.69/5.90 1465 -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(287)]. 5.69/5.90 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(1465,a,1461,a)]. 5.69/5.90 1466 -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(450)]. 5.69/5.90 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(1466,a,1461,a)]. 5.70/5.94 1467 -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(450)]. 5.70/5.94 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(1467,a,1461,a)]. 5.70/5.94 1468 -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(694)]. 5.70/5.94 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(1468,a,1461,a)]. 5.70/5.94 1469 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,times_times(A,minus_minus(A,B,C),D),E)),F)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,times_times(A,B,D),E)),plus_plus(A,times_times(A,C,D),F))) # label(fact_295_less__add__iff1) # label(axiom). [clausify(869)]. 5.70/5.94 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),plus_plus(real,times_times(real,minus_minus(real,A,B),C),D)),E)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),plus_plus(real,times_times(real,A,C),D)),plus_plus(real,times_times(real,B,C),E))). [resolve(1469,a,1461,a)]. 5.70/5.94 1470 -ordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,times_times(A,minus_minus(A,B,C),D),E)),F)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,times_times(A,B,D),E)),plus_plus(A,times_times(A,C,D),F))) # label(fact_295_less__add__iff1) # label(axiom). [clausify(869)]. 5.70/5.94 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),plus_plus(real,times_times(real,minus_minus(real,A,B),C),D)),E)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),plus_plus(real,times_times(real,A,C),D)),plus_plus(real,times_times(real,B,C),E))). [resolve(1470,a,1461,a)]. 5.70/5.94 1471 ordered_ring(int) # label(arity_Int_Oint___Rings_Oordered__ring) # label(axiom). [assumption]. 5.70/5.94 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),C),plus_plus(int,times_times(int,minus_minus(int,D,A),B),E))). [resolve(1471,a,1460,a)]. 5.70/5.94 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),C),plus_plus(int,times_times(int,minus_minus(int,D,A),B),E))). [resolve(1471,a,1462,a)]. 5.70/5.94 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(1471,a,1463,a)]. 5.70/5.94 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(1471,a,1464,a)]. 5.70/5.94 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(1471,a,1465,a)]. 5.80/6.05 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(1471,a,1466,a)]. 5.80/6.05 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(1471,a,1467,a)]. 5.80/6.05 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(1471,a,1468,a)]. 5.80/6.05 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,times_times(int,minus_minus(int,A,B),C),D)),E)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,times_times(int,A,C),D)),plus_plus(int,times_times(int,B,C),E))). [resolve(1471,a,1469,a)]. 5.80/6.05 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,times_times(int,minus_minus(int,A,B),C),D)),E)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,times_times(int,A,C),D)),plus_plus(int,times_times(int,B,C),E))). [resolve(1471,a,1470,a)]. 5.80/6.05 1472 -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(1000)]. 5.80/6.05 1473 -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(1008)]. 5.80/6.05 1474 -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,C,B))) # label(fact_270_split__mult__pos__le) # label(axiom). [clausify(1008)]. 5.80/6.05 1475 ordere223160158up_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 5.80/6.05 1476 -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(95)]. 5.80/6.05 1477 -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(359)]. 5.80/6.05 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(1475,a,1476,a)]. 5.80/6.05 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),plus_plus(real,A,C)),plus_plus(real,B,C))). [resolve(1475,a,1477,a)]. 5.80/6.05 1478 -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(592)]. 5.88/6.16 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(1478,a,1475,a)]. 5.88/6.16 1479 ordere223160158up_add(int) # label(arity_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 5.88/6.16 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(1479,a,1476,a)]. 5.88/6.16 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(1479,a,1477,a)]. 5.88/6.16 1480 -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(757)]. 5.88/6.16 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(1480,a,1475,a)]. 5.88/6.16 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(1480,a,1479,a)]. 5.88/6.16 1481 -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(891)]. 5.88/6.16 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(1481,a,1475,a)]. 5.88/6.16 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(1481,a,1479,a)]. 5.88/6.16 1482 ordere223160158up_add(nat) # label(arity_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 5.88/6.16 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(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(1482,a,1476,a)]. 5.88/6.16 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(1482,a,1478,a)]. 5.88/6.16 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),plus_plus(nat,A,C)),plus_plus(nat,B,D))). [resolve(1482,a,1480,a)]. 5.88/6.16 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(1482,a,1481,a)]. 5.88/6.16 1483 -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(197)]. 5.88/6.16 1484 ordere779506340up_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 5.88/6.16 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(1483,a,1484,a)]. 6.03/6.31 1485 -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(252)]. 6.03/6.31 1486 ordere779506340up_add(int) # label(arity_Int_Oint___Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 6.03/6.31 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(1486,a,1483,a)]. 6.03/6.31 1487 -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(848)]. 6.03/6.31 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(1487,a,1484,a)]. 6.03/6.31 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(1487,a,1486,a)]. 6.03/6.31 1488 ordere779506340up_add(nat) # label(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 6.03/6.31 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(1488,a,1483,a)]. 6.03/6.31 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(1488,a,1485,a)]. 6.03/6.31 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(1488,a,1487,a)]. 6.03/6.31 1489 -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(470)]. 6.03/6.31 1490 group_add(real) # label(arity_RealDef_Oreal___Groups_Ogroup__add) # label(axiom). [assumption]. 6.03/6.31 Derived: minus_minus(real,ti(real,A),B) = minus_minus(real,A,B). [resolve(1489,a,1490,a)]. 6.03/6.31 1491 -group_add(A) | ti(A,B) = minus_minus(A,B,zero_zero(A)) # label(fact_323_diff__0__right) # label(axiom). [clausify(499)]. 6.03/6.31 Derived: ti(real,A) = minus_minus(real,A,zero_zero(real)). [resolve(1491,a,1490,a)]. 6.03/6.31 1492 -group_add(A) | ti(A,B) = plus_plus(A,minus_minus(A,B,C),C) # label(fact_327_diff__add__cancel) # label(axiom). [clausify(536)]. 6.03/6.31 Derived: ti(real,A) = plus_plus(real,minus_minus(real,A,B),B). [resolve(1492,a,1490,a)]. 6.03/6.31 1493 -group_add(A) | ti(A,minus_minus(A,B,C)) = minus_minus(A,B,C) # label(tsy_c_Groups_Ominus__class_Ominus_0_res) # label(axiom). [clausify(586)]. 6.03/6.31 Derived: ti(real,minus_minus(real,A,B)) = minus_minus(real,A,B). [resolve(1493,a,1490,a)]. 6.03/6.31 1494 group_add(int) # label(arity_Int_Oint___Groups_Ogroup__add) # label(axiom). [assumption]. 6.03/6.31 Derived: minus_minus(int,ti(int,A),B) = minus_minus(int,A,B). [resolve(1494,a,1489,a)]. 6.03/6.31 Derived: ti(int,A) = minus_minus(int,A,zero_zero(int)). [resolve(1494,a,1491,a)]. 6.03/6.31 Derived: ti(int,A) = plus_plus(int,minus_minus(int,A,B),B). [resolve(1494,a,1492,a)]. 6.03/6.31 Derived: ti(int,minus_minus(int,A,B)) = minus_minus(int,A,B). [resolve(1494,a,1493,a)]. 6.03/6.31 1495 -group_add(A) | ti(A,B) != ti(A,C) | zero_zero(A) = minus_minus(A,C,B) # label(fact_320_right__minus__eq) # label(axiom). [clausify(687)]. 6.03/6.31 Derived: ti(real,A) != ti(real,B) | zero_zero(real) = minus_minus(real,B,A). [resolve(1495,a,1490,a)]. 6.19/6.44 Derived: ti(int,A) != ti(int,B) | zero_zero(int) = minus_minus(int,B,A). [resolve(1495,a,1494,a)]. 6.19/6.44 1496 -group_add(A) | ti(A,B) = ti(A,C) | zero_zero(A) != minus_minus(A,C,B) # label(fact_320_right__minus__eq) # label(axiom). [clausify(687)]. 6.19/6.44 Derived: ti(real,A) = ti(real,B) | zero_zero(real) != minus_minus(real,B,A). [resolve(1496,a,1490,a)]. 6.19/6.44 Derived: ti(int,A) = ti(int,B) | zero_zero(int) != minus_minus(int,B,A). [resolve(1496,a,1494,a)]. 6.19/6.44 1497 -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(787)]. 6.19/6.44 Derived: minus_minus(real,A,ti(real,B)) = minus_minus(real,A,B). [resolve(1497,a,1490,a)]. 6.19/6.44 Derived: minus_minus(int,A,ti(int,B)) = minus_minus(int,A,B). [resolve(1497,a,1494,a)]. 6.19/6.44 1498 -group_add(A) | zero_zero(A) = minus_minus(A,B,B) # label(fact_322_diff__self) # label(axiom). [clausify(955)]. 6.19/6.44 Derived: zero_zero(real) = minus_minus(real,A,A). [resolve(1498,a,1490,a)]. 6.19/6.44 Derived: zero_zero(int) = minus_minus(int,A,A). [resolve(1498,a,1494,a)]. 6.19/6.44 1499 -group_add(A) | ti(A,B) = minus_minus(A,plus_plus(A,B,C),C) # label(fact_326_add__diff__cancel) # label(axiom). [clausify(969)]. 6.19/6.44 Derived: ti(real,A) = minus_minus(real,plus_plus(real,A,B),B). [resolve(1499,a,1490,a)]. 6.19/6.44 Derived: ti(int,A) = minus_minus(int,plus_plus(int,A,B),B). [resolve(1499,a,1494,a)]. 6.19/6.44 1500 -semiri456707255roduct(A) | zero_zero(A) = ti(A,B) | ti(A,C) = ti(A,D) | ti(A,E) != ti(A,F) | plus_plus(A,E,times_times(A,B,C)) != plus_plus(A,F,times_times(A,B,D)) # label(fact_429_add__scale__eq__noteq) # label(axiom). [clausify(146)]. 6.19/6.44 1501 semiri456707255roduct(int) # label(arity_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc) # label(axiom). [assumption]. 6.19/6.44 Derived: zero_zero(int) = ti(int,A) | ti(int,B) = ti(int,C) | ti(int,D) != ti(int,E) | plus_plus(int,D,times_times(int,A,B)) != plus_plus(int,E,times_times(int,A,C)). [resolve(1500,a,1501,a)]. 6.19/6.44 1502 -semiri456707255roduct(A) | ti(A,B) = ti(A,C) | ti(A,D) = ti(A,E) | plus_plus(A,times_times(A,E,B),times_times(A,D,C)) != plus_plus(A,times_times(A,E,C),times_times(A,D,B)) # label(fact_344_crossproduct__noteq) # label(axiom). [clausify(191)]. 6.19/6.44 Derived: ti(int,A) = ti(int,B) | ti(int,C) = ti(int,D) | plus_plus(int,times_times(int,D,A),times_times(int,C,B)) != plus_plus(int,times_times(int,D,B),times_times(int,C,A)). [resolve(1502,a,1501,a)]. 6.19/6.44 1503 -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(191)]. 6.19/6.44 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(1503,a,1501,a)]. 6.19/6.44 1504 -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(191)]. 6.19/6.44 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(1504,a,1501,a)]. 6.19/6.44 1505 semiri456707255roduct(nat) # label(arity_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc) # label(axiom). [assumption]. 6.19/6.44 Derived: zero_zero(nat) = ti(nat,A) | ti(nat,B) = ti(nat,C) | ti(nat,D) != ti(nat,E) | plus_plus(nat,D,times_times(nat,A,B)) != plus_plus(nat,E,times_times(nat,A,C)). [resolve(1505,a,1500,a)]. 6.19/6.44 Derived: ti(nat,A) = ti(nat,B) | ti(nat,C) = ti(nat,D) | plus_plus(nat,times_times(nat,D,A),times_times(nat,C,B)) != plus_plus(nat,times_times(nat,D,B),times_times(nat,C,A)). [resolve(1505,a,1502,a)]. 6.19/6.44 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(1505,a,1503,a)]. 6.19/6.44 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(1505,a,1504,a)]. 6.33/6.58 1506 -semiri456707255roduct(A) | ti(A,B) != plus_plus(A,B,C) | zero_zero(A) = ti(A,C) # label(fact_212_add__0__iff) # label(axiom). [clausify(548)]. 6.33/6.58 Derived: ti(int,A) != plus_plus(int,A,B) | zero_zero(int) = ti(int,B). [resolve(1506,a,1501,a)]. 6.33/6.58 Derived: ti(nat,A) != plus_plus(nat,A,B) | zero_zero(nat) = ti(nat,B). [resolve(1506,a,1505,a)]. 6.33/6.58 1507 -semiri456707255roduct(A) | ti(A,B) = plus_plus(A,B,C) | zero_zero(A) != ti(A,C) # label(fact_212_add__0__iff) # label(axiom). [clausify(548)]. 6.33/6.58 Derived: ti(int,A) = plus_plus(int,A,B) | zero_zero(int) != ti(int,B). [resolve(1507,a,1501,a)]. 6.33/6.58 Derived: ti(nat,A) = plus_plus(nat,A,B) | zero_zero(nat) != ti(nat,B). [resolve(1507,a,1505,a)]. 6.33/6.58 1508 semiri456707255roduct(real) # label(arity_RealDef_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossp) # label(axiom). [assumption]. 6.33/6.58 Derived: zero_zero(real) = ti(real,A) | ti(real,B) = ti(real,C) | ti(real,D) != ti(real,E) | plus_plus(real,D,times_times(real,A,B)) != plus_plus(real,E,times_times(real,A,C)). [resolve(1508,a,1500,a)]. 6.33/6.58 Derived: ti(real,A) = ti(real,B) | ti(real,C) = ti(real,D) | plus_plus(real,times_times(real,D,A),times_times(real,C,B)) != plus_plus(real,times_times(real,D,B),times_times(real,C,A)). [resolve(1508,a,1502,a)]. 6.33/6.58 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(1508,a,1503,a)]. 6.33/6.58 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(1508,a,1504,a)]. 6.33/6.58 Derived: ti(real,A) != plus_plus(real,A,B) | zero_zero(real) = ti(real,B). [resolve(1508,a,1506,a)]. 6.33/6.58 Derived: ti(real,A) = plus_plus(real,A,B) | zero_zero(real) != ti(real,B). [resolve(1508,a,1507,a)]. 6.33/6.58 1509 -semiri456707255roduct(A) | ti(A,B) != ti(A,C) | plus_plus(A,times_times(A,B,D),times_times(A,C,E)) = plus_plus(A,times_times(A,B,E),times_times(A,C,D)) # label(fact_341_crossproduct__eq) # label(axiom). [clausify(947)]. 6.33/6.58 Derived: ti(int,A) != ti(int,B) | plus_plus(int,times_times(int,A,C),times_times(int,B,D)) = plus_plus(int,times_times(int,A,D),times_times(int,B,C)). [resolve(1509,a,1501,a)]. 6.33/6.58 Derived: ti(nat,A) != ti(nat,B) | plus_plus(nat,times_times(nat,A,C),times_times(nat,B,D)) = plus_plus(nat,times_times(nat,A,D),times_times(nat,B,C)). [resolve(1509,a,1505,a)]. 6.33/6.58 Derived: ti(real,A) != ti(real,B) | plus_plus(real,times_times(real,A,C),times_times(real,B,D)) = plus_plus(real,times_times(real,A,D),times_times(real,B,C)). [resolve(1509,a,1508,a)]. 6.33/6.58 1510 -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_341_crossproduct__eq) # label(axiom). [clausify(947)]. 6.33/6.58 1511 -semiri456707255roduct(A) | ti(A,B) = ti(A,C) | ti(A,D) = ti(A,E) | plus_plus(A,times_times(A,B,D),times_times(A,C,E)) != plus_plus(A,times_times(A,B,E),times_times(A,C,D)) # label(fact_341_crossproduct__eq) # label(axiom). [clausify(947)]. 6.33/6.58 Derived: ti(int,A) = ti(int,B) | ti(int,C) = ti(int,D) | plus_plus(int,times_times(int,A,C),times_times(int,B,D)) != plus_plus(int,times_times(int,A,D),times_times(int,B,C)). [resolve(1511,a,1501,a)]. 6.33/6.58 Derived: ti(nat,A) = ti(nat,B) | ti(nat,C) = ti(nat,D) | plus_plus(nat,times_times(nat,A,C),times_times(nat,B,D)) != plus_plus(nat,times_times(nat,A,D),times_times(nat,B,C)). [resolve(1511,a,1505,a)]. 6.33/6.58 Derived: ti(real,A) = ti(real,B) | ti(real,C) = ti(real,D) | plus_plus(real,times_times(real,A,C),times_times(real,B,D)) != plus_plus(real,times_times(real,A,D),times_times(real,B,C)). [resolve(1511,a,1508,a)]. 6.33/6.58 1512 ordered_semiring(nat) # label(arity_Nat_Onat___Rings_Oordered__semiring) # label(axiom). [assumption]. 6.33/6.58 1513 -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(106)]. 6.33/6.60 1514 -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(410)]. 6.33/6.60 1515 ordered_semiring(int) # label(arity_Int_Oint___Rings_Oordered__semiring) # label(axiom). [assumption]. 6.33/6.60 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(1515,a,1513,a)]. 6.33/6.60 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(1515,a,1514,a)]. 6.33/6.60 1516 -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(691)]. 6.33/6.60 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),times_times(nat,C,A)),times_times(nat,C,B))). [resolve(1516,a,1512,a)]. 6.33/6.60 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(1516,a,1515,a)]. 6.33/6.60 1517 ordered_semiring(real) # label(arity_RealDef_Oreal___Rings_Oordered__semiring) # label(axiom). [assumption]. 6.33/6.60 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(1517,a,1513,a)]. 6.33/6.60 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(1517,a,1514,a)]. 6.33/6.60 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(1517,a,1516,a)]. 6.33/6.60 1518 -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(845)]. 6.33/6.60 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(1518,a,1515,a)]. 6.73/6.96 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),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(1518,a,1517,a)]. 6.73/6.96 1519 -linorder(A) | -number(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(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,C)),number_number_of(A,B))) # label(fact_448_le__number__of__eq__not__less) # label(axiom). [clausify(577)]. 6.73/6.96 1520 linorder(int) # label(arity_Int_Oint___Orderings_Olinorder) # label(axiom). [assumption]. 6.73/6.96 1521 linorder(real) # label(arity_RealDef_Oreal___Orderings_Olinorder) # label(axiom). [assumption]. 6.73/6.96 Derived: -number(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,A)),number_number_of(int,B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,B)),number_number_of(int,A))). [resolve(1519,a,1520,a)]. 6.73/6.96 Derived: -number(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(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,B)),number_number_of(real,A))). [resolve(1519,a,1521,a)]. 6.73/6.96 1522 -linorder(A) | -number(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(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,C)),number_number_of(A,B))) # label(fact_448_le__number__of__eq__not__less) # label(axiom). [clausify(577)]. 6.73/6.96 Derived: -number(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,A)),number_number_of(int,B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,B)),number_number_of(int,A))). [resolve(1522,a,1520,a)]. 6.73/6.96 Derived: -number(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(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,B)),number_number_of(real,A))). [resolve(1522,a,1521,a)]. 6.73/6.96 1523 linorder(nat) # label(arity_Nat_Onat___Orderings_Olinorder) # label(axiom). [assumption]. 6.73/6.96 Derived: -number(nat) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),number_number_of(nat,A)),number_number_of(nat,B))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),number_number_of(nat,B)),number_number_of(nat,A))). [resolve(1523,a,1519,a)]. 6.73/6.96 Derived: -number(nat) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),number_number_of(nat,A)),number_number_of(nat,B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),number_number_of(nat,B)),number_number_of(nat,A))). [resolve(1523,a,1522,a)]. 6.73/6.96 1524 zero(nat) # label(arity_Nat_Onat___Groups_Ozero) # label(axiom). [assumption]. 6.73/6.96 1525 -zero(A) | zero_zero(A) = ti(A,zero_zero(A)) # label(tsy_c_Groups_Ozero__class_Ozero_res) # label(axiom). [clausify(109)]. 6.73/6.96 Derived: zero_zero(nat) = ti(nat,zero_zero(nat)). [resolve(1524,a,1525,a)]. 6.73/6.96 1526 zero(real) # label(arity_RealDef_Oreal___Groups_Ozero) # label(axiom). [assumption]. 6.73/6.96 Derived: zero_zero(real) = ti(real,zero_zero(real)). [resolve(1526,a,1525,a)]. 6.73/6.96 1527 zero(int) # label(arity_Int_Oint___Groups_Ozero) # label(axiom). [assumption]. 6.73/6.96 Derived: zero_zero(int) = ti(int,zero_zero(int)). [resolve(1527,a,1525,a)]. 6.73/6.96 1528 linord219039673up_add(int) # label(arity_Int_Oint___Groups_Olinordered__ab__group__add) # label(axiom). [assumption]. 6.73/6.97 1529 -linord219039673up_add(A) | zero_zero(A) != ti(A,B) | zero_zero(A) = plus_plus(A,B,B) # label(fact_65_double__eq__0__iff) # label(axiom). [clausify(128)]. 6.73/6.97 1530 -linord219039673up_add(A) | zero_zero(A) = ti(A,B) | zero_zero(A) != plus_plus(A,B,B) # label(fact_65_double__eq__0__iff) # label(axiom). [clausify(128)]. 6.73/6.97 Derived: zero_zero(int) != ti(int,A) | zero_zero(int) = plus_plus(int,A,A). [resolve(1528,a,1529,a)]. 6.73/6.97 Derived: zero_zero(int) = ti(int,A) | zero_zero(int) != plus_plus(int,A,A). [resolve(1528,a,1530,a)]. 6.73/6.97 1531 -linord219039673up_add(A) | zero_zero(A) != ti(A,B) | zero_zero(A) = plus_plus(A,B,B) # label(fact_216_double__zero__sym) # label(axiom). [clausify(274)]. 6.73/6.97 1532 -linord219039673up_add(A) | zero_zero(A) = ti(A,B) | zero_zero(A) != plus_plus(A,B,B) # label(fact_216_double__zero__sym) # label(axiom). [clausify(274)]. 6.73/6.97 1533 linord219039673up_add(real) # label(arity_RealDef_Oreal___Groups_Olinordered__ab__group__add) # label(axiom). [assumption]. 6.73/6.97 Derived: zero_zero(real) != ti(real,A) | zero_zero(real) = plus_plus(real,A,A). [resolve(1533,a,1529,a)]. 6.73/6.97 Derived: zero_zero(real) = ti(real,A) | zero_zero(real) != plus_plus(real,A,A). [resolve(1533,a,1530,a)]. 6.73/6.97 1534 -linord219039673up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),plus_plus(A,B,B))) # label(fact_432_zero__le__double__add__iff__zero__le__single__add) # label(axiom). [clausify(339)]. 6.73/6.97 1535 -linord219039673up_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),plus_plus(A,B,B))) # label(fact_432_zero__le__double__add__iff__zero__le__single__add) # label(axiom). [clausify(339)]. 6.73/6.97 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)),plus_plus(int,A,A))). [resolve(1535,a,1528,a)]. 6.73/6.97 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)),plus_plus(real,A,A))). [resolve(1535,a,1533,a)]. 6.73/6.97 1536 -linord219039673up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,B,B)),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) # label(fact_433_double__add__le__zero__iff__single__add__le__zero) # label(axiom). [clausify(399)]. 6.73/6.97 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,A,A)),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))). [resolve(1536,a,1528,a)]. 6.73/6.97 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),plus_plus(real,A,A)),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))). [resolve(1536,a,1533,a)]. 6.73/6.97 1537 -linord219039673up_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,B,B)),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) # label(fact_433_double__add__le__zero__iff__single__add__le__zero) # label(axiom). [clausify(399)]. 6.73/6.97 1538 -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(512)]. 6.73/6.97 1539 -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(512)]. 6.73/6.97 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(1539,a,1528,a)]. 7.25/7.48 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(1539,a,1533,a)]. 7.25/7.48 1540 -linord219039673up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,B,B)),zero_zero(A))) # label(fact_195_double__add__less__zero__iff__single__add__less__zero) # label(axiom). [clausify(610)]. 7.25/7.48 1541 -linord219039673up_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,B,B)),zero_zero(A))) # label(fact_195_double__add__less__zero__iff__single__add__less__zero) # label(axiom). [clausify(610)]. 7.25/7.48 1542 cancel146912293up_add(real) # label(arity_RealDef_Oreal___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 7.25/7.48 1543 -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(136)]. 7.25/7.48 1544 cancel146912293up_add(int) # label(arity_Int_Oint___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 7.25/7.48 1545 cancel146912293up_add(nat) # label(arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 7.25/7.48 1546 ab_group_add(int) # label(arity_Int_Oint___Groups_Oab__group__add) # label(axiom). [assumption]. 7.25/7.48 1547 -ab_group_add(A) | ti(A,B) != ti(A,C) | zero_zero(A) = minus_minus(A,C,B) # label(fact_321_eq__iff__diff__eq__0) # label(axiom). [clausify(148)]. 7.25/7.48 1548 -ab_group_add(A) | ti(A,B) = ti(A,C) | zero_zero(A) != minus_minus(A,C,B) # label(fact_321_eq__iff__diff__eq__0) # label(axiom). [clausify(148)]. 7.25/7.48 1549 -ab_group_add(A) | minus_minus(A,B,C) != minus_minus(A,D,E) | ti(A,B) != ti(A,C) | ti(A,E) = ti(A,D) # label(fact_248_diff__eq__diff__eq) # label(axiom). [clausify(313)]. 7.25/7.48 1550 -ab_group_add(A) | minus_minus(A,B,C) != minus_minus(A,D,E) | ti(A,B) = ti(A,C) | ti(A,E) != ti(A,D) # label(fact_248_diff__eq__diff__eq) # label(axiom). [clausify(313)]. 7.25/7.48 1551 -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(477)]. 7.25/7.48 Derived: minus_minus(int,A,B) != minus_minus(int,C,D) | ti(int,A) != ti(int,B) | ti(int,D) = ti(int,C). [resolve(1546,a,1549,a)]. 7.25/7.48 Derived: minus_minus(int,A,B) != minus_minus(int,C,D) | ti(int,A) = ti(int,B) | ti(int,D) != ti(int,C). [resolve(1546,a,1550,a)]. 7.25/7.48 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(1546,a,1551,a)]. 7.25/7.48 1552 ab_group_add(real) # label(arity_RealDef_Oreal___Groups_Oab__group__add) # label(axiom). [assumption]. 7.25/7.48 Derived: minus_minus(real,A,B) != minus_minus(real,C,D) | ti(real,A) != ti(real,B) | ti(real,D) = ti(real,C). [resolve(1552,a,1549,a)]. 7.25/7.48 Derived: minus_minus(real,A,B) != minus_minus(real,C,D) | ti(real,A) = ti(real,B) | ti(real,D) != ti(real,C). [resolve(1552,a,1550,a)]. 7.25/7.48 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(1552,a,1551,a)]. 7.25/7.48 1553 comm_semiring(real) # label(arity_RealDef_Oreal___Rings_Ocomm__semiring) # label(axiom). [assumption]. 7.25/7.48 1554 -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(155)]. 7.25/7.48 1555 comm_semiring(int) # label(arity_Int_Oint___Rings_Ocomm__semiring) # label(axiom). [assumption]. 7.25/7.48 1556 comm_semiring(nat) # label(arity_Nat_Onat___Rings_Ocomm__semiring) # label(axiom). [assumption]. 7.25/7.48 1557 ring(real) # label(arity_RealDef_Oreal___Rings_Oring) # label(axiom). [assumption]. 7.25/7.48 1558 -ring(A) | -number(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(167)]. 7.36/7.61 Derived: -number(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(1557,a,1558,a)]. 7.36/7.61 1559 -number(A) | -ring(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(311)]. 7.36/7.61 Derived: -number(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(1559,b,1557,a)]. 7.36/7.61 1560 ring(int) # label(arity_Int_Oint___Rings_Oring) # label(axiom). [assumption]. 7.36/7.61 1561 -ring(A) | plus_plus(A,times_times(A,B,C),D) != plus_plus(A,times_times(A,E,C),F) | ti(A,D) = plus_plus(A,times_times(A,minus_minus(A,E,B),C),F) # label(fact_286_eq__add__iff1) # label(axiom). [clausify(676)]. 7.36/7.61 Derived: plus_plus(real,times_times(real,A,B),C) != plus_plus(real,times_times(real,D,B),E) | ti(real,C) = plus_plus(real,times_times(real,minus_minus(real,D,A),B),E). [resolve(1561,a,1557,a)]. 7.36/7.61 Derived: plus_plus(int,times_times(int,A,B),C) != plus_plus(int,times_times(int,D,B),E) | ti(int,C) = plus_plus(int,times_times(int,minus_minus(int,D,A),B),E). [resolve(1561,a,1560,a)]. 7.36/7.61 1562 -ring(A) | plus_plus(A,times_times(A,B,C),D) = plus_plus(A,times_times(A,E,C),F) | ti(A,D) != plus_plus(A,times_times(A,minus_minus(A,E,B),C),F) # label(fact_286_eq__add__iff1) # label(axiom). [clausify(676)]. 7.36/7.61 Derived: plus_plus(real,times_times(real,A,B),C) = plus_plus(real,times_times(real,D,B),E) | ti(real,C) != plus_plus(real,times_times(real,minus_minus(real,D,A),B),E). [resolve(1562,a,1557,a)]. 7.36/7.61 Derived: plus_plus(int,times_times(int,A,B),C) = plus_plus(int,times_times(int,D,B),E) | ti(int,C) != plus_plus(int,times_times(int,minus_minus(int,D,A),B),E). [resolve(1562,a,1560,a)]. 7.36/7.61 1563 -ring(A) | minus_minus(A,times_times(A,B,C),times_times(A,D,E)) = plus_plus(A,times_times(A,B,minus_minus(A,C,E)),times_times(A,minus_minus(A,B,D),E)) # label(fact_289_mult__diff__mult) # label(axiom). [clausify(719)]. 7.36/7.61 Derived: minus_minus(real,times_times(real,A,B),times_times(real,C,D)) = plus_plus(real,times_times(real,A,minus_minus(real,B,D)),times_times(real,minus_minus(real,A,C),D)). [resolve(1563,a,1557,a)]. 7.36/7.61 Derived: minus_minus(int,times_times(int,A,B),times_times(int,C,D)) = plus_plus(int,times_times(int,A,minus_minus(int,B,D)),times_times(int,minus_minus(int,A,C),D)). [resolve(1563,a,1560,a)]. 7.36/7.61 1564 -ring(A) | plus_plus(A,times_times(A,B,C),D) != plus_plus(A,times_times(A,E,C),F) | ti(A,F) = plus_plus(A,times_times(A,minus_minus(A,B,E),C),D) # label(fact_288_eq__add__iff2) # label(axiom). [clausify(1005)]. 7.36/7.61 Derived: plus_plus(real,times_times(real,A,B),C) != plus_plus(real,times_times(real,D,B),E) | ti(real,E) = plus_plus(real,times_times(real,minus_minus(real,A,D),B),C). [resolve(1564,a,1557,a)]. 7.36/7.61 Derived: plus_plus(int,times_times(int,A,B),C) != plus_plus(int,times_times(int,D,B),E) | ti(int,E) = plus_plus(int,times_times(int,minus_minus(int,A,D),B),C). [resolve(1564,a,1560,a)]. 7.36/7.61 1565 -ring(A) | plus_plus(A,times_times(A,B,C),D) = plus_plus(A,times_times(A,E,C),F) | ti(A,F) != plus_plus(A,times_times(A,minus_minus(A,B,E),C),D) # label(fact_288_eq__add__iff2) # label(axiom). [clausify(1005)]. 7.36/7.61 Derived: plus_plus(real,times_times(real,A,B),C) = plus_plus(real,times_times(real,D,B),E) | ti(real,E) != plus_plus(real,times_times(real,minus_minus(real,A,D),B),C). [resolve(1565,a,1557,a)]. 7.36/7.61 Derived: plus_plus(int,times_times(int,A,B),C) = plus_plus(int,times_times(int,D,B),E) | ti(int,E) != plus_plus(int,times_times(int,minus_minus(int,A,D),B),C). [resolve(1565,a,1560,a)]. 7.36/7.61 1566 ordered_ab_group_add(int) # label(arity_Int_Oint___Groups_Oordered__ab__group__add) # label(axiom). [assumption]. 7.36/7.61 1567 -ordered_ab_group_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),minus_minus(A,B,C)),zero_zero(A))) # label(fact_238_le__iff__diff__le__0) # label(axiom). [clausify(175)]. 7.36/7.64 1568 -ordered_ab_group_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),minus_minus(A,B,C)),zero_zero(A))) # label(fact_238_le__iff__diff__le__0) # label(axiom). [clausify(175)]. 7.36/7.64 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),minus_minus(int,A,B)),zero_zero(int))). [resolve(1566,a,1567,a)]. 7.36/7.64 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),minus_minus(int,A,B)),zero_zero(int))). [resolve(1566,a,1568,a)]. 7.36/7.64 1569 ordered_ab_group_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__group__add) # label(axiom). [assumption]. 7.36/7.64 1570 -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),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),E)) # label(fact_249_diff__eq__diff__less__eq) # label(axiom). [clausify(695)]. 7.36/7.64 Derived: minus_minus(int,A,B) != minus_minus(int,C,D) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),D)). [resolve(1570,a,1566,a)]. 7.36/7.64 Derived: minus_minus(real,A,B) != minus_minus(real,C,D) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),D)). [resolve(1570,a,1569,a)]. 7.36/7.64 1571 -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),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),E)) # label(fact_249_diff__eq__diff__less__eq) # label(axiom). [clausify(695)]. 7.36/7.64 Derived: minus_minus(int,A,B) != minus_minus(int,C,D) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),D)). [resolve(1571,a,1566,a)]. 7.36/7.64 Derived: minus_minus(real,A,B) != minus_minus(real,C,D) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),D)). [resolve(1571,a,1569,a)]. 7.36/7.64 1572 -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),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) # label(fact_324_diff__eq__diff__less) # label(axiom). [clausify(796)]. 7.36/7.64 Derived: minus_minus(int,A,B) != minus_minus(int,C,D) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),D)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)). [resolve(1572,a,1566,a)]. 7.36/7.64 Derived: minus_minus(real,A,B) != minus_minus(real,C,D) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)). [resolve(1572,a,1569,a)]. 7.36/7.64 1573 -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),D),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) # label(fact_324_diff__eq__diff__less) # label(axiom). [clausify(796)]. 7.36/7.64 Derived: minus_minus(int,A,B) != minus_minus(int,C,D) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),D)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)). [resolve(1573,a,1566,a)]. 7.36/7.64 Derived: minus_minus(real,A,B) != minus_minus(real,C,D) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),D)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)). [resolve(1573,a,1569,a)]. 7.36/7.64 1574 -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(883)]. 7.36/7.64 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),minus_minus(int,A,B)),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)). [resolve(1574,a,1566,a)]. 7.67/7.88 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(1574,a,1569,a)]. 7.67/7.88 1575 -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(883)]. 7.67/7.88 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),minus_minus(int,A,B)),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)). [resolve(1575,a,1566,a)]. 7.67/7.88 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(1575,a,1569,a)]. 7.67/7.88 1576 ring_1(int) # label(arity_Int_Oint___Rings_Oring__1) # label(axiom). [assumption]. 7.67/7.88 1577 -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(180)]. 7.67/7.88 1578 -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(232)]. 7.67/7.88 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(1576,a,1578,a)]. 7.67/7.88 1579 ring_1(real) # label(arity_RealDef_Oreal___Rings_Oring__1) # label(axiom). [assumption]. 7.67/7.88 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(1579,a,1577,a)]. 7.67/7.88 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(1579,a,1578,a)]. 7.67/7.88 1580 ordere236663937imp_le(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 7.67/7.88 1581 -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,D,B)),plus_plus(A,D,C))) # label(fact_225_add__less__cancel__left) # label(axiom). [clausify(219)]. 7.67/7.88 1582 -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,D,B)),plus_plus(A,D,C))) # label(fact_225_add__less__cancel__left) # label(axiom). [clausify(219)]. 7.67/7.88 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(1580,a,1582,a)]. 7.67/7.88 1583 ordere236663937imp_le(int) # label(arity_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 7.67/7.88 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(1583,a,1582,a)]. 7.67/7.88 1584 -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(432)]. 7.67/7.88 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),plus_plus(real,A,B)),plus_plus(real,C,B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),C)). [resolve(1584,a,1580,a)]. 7.67/7.88 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,A,B)),plus_plus(int,C,B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),C)). [resolve(1584,a,1583,a)]. 7.80/8.02 1585 -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(571)]. 7.80/8.02 1586 -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_359_add__le__cancel__right) # label(axiom). [clausify(573)]. 7.80/8.02 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),plus_plus(real,A,B)),plus_plus(real,C,B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),C)). [resolve(1586,a,1580,a)]. 7.80/8.02 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,A,B)),plus_plus(int,C,B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),C)). [resolve(1586,a,1583,a)]. 7.80/8.02 1587 -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_359_add__le__cancel__right) # label(axiom). [clausify(573)]. 7.80/8.02 1588 -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_226_add__less__cancel__right) # label(axiom). [clausify(578)]. 7.80/8.02 1589 -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_226_add__less__cancel__right) # label(axiom). [clausify(578)]. 7.80/8.02 1590 ordere236663937imp_le(nat) # label(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 7.80/8.02 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,A,B)),plus_plus(nat,C,B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),C)). [resolve(1590,a,1584,a)]. 7.80/8.02 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),plus_plus(nat,A,B)),plus_plus(nat,C,B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),C)). [resolve(1590,a,1586,a)]. 7.80/8.02 1591 -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_358_add__le__cancel__left) # label(axiom). [clausify(784)]. 7.80/8.02 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(1591,a,1580,a)]. 7.80/8.02 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(1591,a,1583,a)]. 7.80/8.02 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),plus_plus(nat,A,B)),plus_plus(nat,A,C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)). [resolve(1591,a,1590,a)]. 7.80/8.02 1592 -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_358_add__le__cancel__left) # label(axiom). [clausify(784)]. 7.80/8.02 1593 -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(872)]. 7.80/8.02 1594 -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(986)]. 7.80/8.02 1595 linordered_ring(int) # label(arity_Int_Oint___Rings_Olinordered__ring) # label(axiom). [assumption]. 8.23/8.43 1596 -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(238)]. 8.23/8.43 1597 -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(247)]. 8.23/8.43 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(1595,a,1596,a)]. 8.23/8.43 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),times_times(int,A,A))). [resolve(1595,a,1597,a)]. 8.23/8.43 1598 -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(615)]. 8.23/8.43 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),times_times(int,A,A)),zero_zero(int))). [resolve(1598,a,1595,a)]. 8.23/8.43 1599 -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(639)]. 8.23/8.43 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(1599,a,1595,a)]. 8.23/8.43 1600 linordered_ring(real) # label(arity_RealDef_Oreal___Rings_Olinordered__ring) # label(axiom). [assumption]. 8.23/8.43 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(1600,a,1596,a)]. 8.23/8.43 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),times_times(real,A,A))). [resolve(1600,a,1597,a)]. 8.23/8.43 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),times_times(real,A,A)),zero_zero(real))). [resolve(1600,a,1598,a)]. 8.23/8.43 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(1600,a,1599,a)]. 8.23/8.43 1601 -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(936)]. 8.23/8.43 1602 ring_char_0(real) # label(arity_RealDef_Oreal___Int_Oring__char__0) # label(axiom). [assumption]. 8.23/8.43 1603 ring_char_0(int) # label(arity_Int_Oint___Int_Oring__char__0) # label(axiom). [assumption]. 8.23/8.43 Derived: -number_ring(real) | number_number_of(real,A) != number_number_of(real,B) | ti(int,A) = ti(int,B). [resolve(1601,a,1602,a)]. 8.23/8.43 Derived: -number_ring(int) | number_number_of(int,A) != number_number_of(int,B) | ti(int,A) = ti(int,B). [resolve(1601,a,1603,a)]. 8.23/8.43 1604 -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(936)]. 8.23/8.43 Derived: -number_ring(real) | number_number_of(real,A) = number_number_of(real,B) | ti(int,A) != ti(int,B). [resolve(1604,a,1602,a)]. 8.23/8.43 Derived: -number_ring(int) | number_number_of(int,A) = number_number_of(int,B) | ti(int,A) != ti(int,B). [resolve(1604,a,1603,a)]. 8.23/8.43 1605 -one(A) | one_one(A) = ti(A,one_one(A)) # label(tsy_c_Groups_Oone__class_Oone_res) # label(axiom). [clausify(641)]. 8.23/8.43 1606 one(real) # label(arity_RealDef_Oreal___Groups_Oone) # label(axiom). [assumption]. 8.23/8.43 1607 one(int) # label(arity_Int_Oint___Groups_Oone) # label(axiom). [assumption]. 8.23/8.43 Derived: one_one(real) = ti(real,one_one(real)). [resolve(1605,a,1606,a)]. 8.23/8.43 Derived: one_one(int) = ti(int,one_one(int)). [resolve(1605,a,1607,a)]. 8.23/8.43 1608 one(nat) # label(arity_Nat_Onat___Groups_Oone) # label(axiom). [assumption]. 8.23/8.43 Derived: one_one(nat) = ti(nat,one_one(nat)). [resolve(1608,a,1605,a)]. 8.23/8.43 1609 ring_11004092258visors(int) # label(arity_Int_Oint___Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 8.23/8.43 1610 -ring_11004092258visors(A) | zero_zero(A) = ti(A,B) | zero_zero(A) != hAPP(nat,A,power_power(A,B),C) # label(fact_165_field__power__not__zero) # label(axiom). [clausify(298)]. 8.59/8.80 1611 -ring_11004092258visors(A) | zero_zero(A) != ti(A,B) | zero_zero(A) = hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))) # label(fact_5_zero__eq__power2) # label(axiom). [clausify(343)]. 8.59/8.80 1612 -ring_11004092258visors(A) | zero_zero(A) = ti(A,B) | zero_zero(A) != hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))) # label(fact_5_zero__eq__power2) # label(axiom). [clausify(343)]. 8.59/8.80 Derived: zero_zero(int) = ti(int,A) | zero_zero(int) != hAPP(nat,int,power_power(int,A),B). [resolve(1609,a,1610,a)]. 8.59/8.80 Derived: zero_zero(int) != ti(int,A) | zero_zero(int) = hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))). [resolve(1609,a,1611,a)]. 8.59/8.80 1613 ring_11004092258visors(real) # label(arity_RealDef_Oreal___Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 8.59/8.80 Derived: zero_zero(real) = ti(real,A) | zero_zero(real) != hAPP(nat,real,power_power(real,A),B). [resolve(1613,a,1610,a)]. 8.59/8.80 Derived: zero_zero(real) != ti(real,A) | zero_zero(real) = hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))). [resolve(1613,a,1611,a)]. 8.59/8.80 1614 -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(418)]. 8.59/8.80 1615 linord893533164strict(nat) # label(arity_Nat_Onat___Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 8.59/8.80 1616 linord893533164strict(int) # label(arity_Int_Oint___Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 8.59/8.80 1617 linord893533164strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 8.59/8.80 1618 -linord1278240602ring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),F)) | one_one(A) != plus_plus(A,E,F) | 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(656)]. 8.59/8.80 1619 linord1278240602ring_1(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring__1) # label(axiom). [assumption]. 8.59/8.80 1620 linord1278240602ring_1(int) # label(arity_Int_Oint___Rings_Olinordered__semiring__1) # label(axiom). [assumption]. 8.59/8.80 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),D)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),E)) | one_one(real) != plus_plus(real,D,E) | 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(1618,a,1619,a)]. 8.59/8.80 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),D)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),E)) | one_one(int) != plus_plus(int,D,E) | 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(1618,a,1620,a)]. 8.59/8.80 1621 -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(539)]. 8.59/8.80 1622 ordere1490568538miring(int) # label(arity_Int_Oint___Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 8.72/8.95 1623 ordere1490568538miring(real) # label(arity_RealDef_Oreal___Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 8.72/8.95 1624 ordere1490568538miring(nat) # label(arity_Nat_Onat___Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 8.72/8.95 1625 -idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),times_times(A,B,C)),times_times(A,D,C))) | zero_zero(A) = ti(A,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(682)]. 8.72/8.95 1626 idom(int) # label(arity_Int_Oint___Rings_Oidom) # label(axiom). [assumption]. 8.72/8.95 1627 idom(real) # label(arity_RealDef_Oreal___Rings_Oidom) # label(axiom). [assumption]. 8.72/8.95 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),times_times(int,A,B)),times_times(int,C,B))) | zero_zero(int) = ti(int,B) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),C)). [resolve(1625,a,1626,a)]. 8.72/8.95 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),times_times(real,A,B)),times_times(real,C,B))) | zero_zero(real) = ti(real,B) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),C)). [resolve(1625,a,1627,a)]. 8.72/8.95 1628 -idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),times_times(A,B,C)),times_times(A,D,C))) | zero_zero(A) != ti(A,C) # label(fact_741_dvd__mult__cancel__right) # label(axiom). [clausify(682)]. 8.72/8.95 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),times_times(int,A,B)),times_times(int,C,B))) | zero_zero(int) != ti(int,B). [resolve(1628,a,1626,a)]. 8.72/8.95 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),times_times(real,A,B)),times_times(real,C,B))) | zero_zero(real) != ti(real,B). [resolve(1628,a,1627,a)]. 8.72/8.95 1629 -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(682)]. 8.72/8.95 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(1629,a,1626,a)]. 8.72/8.95 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(1629,a,1627,a)]. 8.72/8.95 1630 -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)) | zero_zero(A) = ti(A,B) # label(fact_740_dvd__mult__cancel__left) # label(axiom). [clausify(886)]. 8.72/8.95 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)) | zero_zero(int) = ti(int,A). [resolve(1630,a,1626,a)]. 8.72/8.95 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)) | zero_zero(real) = ti(real,A). [resolve(1630,a,1627,a)]. 8.72/8.95 1631 -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(886)]. 8.72/8.95 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(1631,a,1626,a)]. 8.72/8.95 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(1631,a,1627,a)]. 8.72/8.95 1632 -idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),times_times(A,B,C)),times_times(A,B,D))) | zero_zero(A) != ti(A,B) # label(fact_740_dvd__mult__cancel__left) # label(axiom). [clausify(886)]. 8.72/8.95 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),times_times(int,A,B)),times_times(int,A,C))) | zero_zero(int) != ti(int,A). [resolve(1632,a,1626,a)]. 11.07/11.30 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),times_times(real,A,B)),times_times(real,A,C))) | zero_zero(real) != ti(real,A). [resolve(1632,a,1627,a)]. 11.07/11.30 11.07/11.30 ============================== end predicate elimination ============= 11.07/11.30 11.07/11.30 Auto_denials: (non-Horn, no changes). 11.07/11.30 11.07/11.30 Term ordering decisions: 11.07/11.30 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. ti=1. number_number_of=1. power_power=1. abs_abs=1. zcong=1. legendre=1. f7=1. f10=1. f13=1. f16=1. f18=1. f19=1. f20=1. f25=1. f28=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. f4=1. f5=1. f6=1. f14=1. f15=1. f21=1. f22=1. f24=1. f30=1. f31=1. times_times=1. plus_plus=1. minus_minus=1. f1=1. f2=1. f3=1. f8=1. f9=1. f11=1. f12=1. f17=1. f23=1. f26=1. f27=1. f29=1. hAPP=1. if=1. 11.07/11.30 11.07/11.30 ============================== end of process initial clauses ======== 11.07/11.30 11.07/11.30 ============================== CLAUSES FOR SEARCH ==================== 11.07/11.30 11.07/11.30 ============================== end of clauses for search ============= 11.07/11.30 11.07/11.30 ============================== SEARCH ================================ 11.07/11.30 11.07/11.30 % Starting search at 7.45 seconds. 11.07/11.30 11.07/11.30 Low Water (keep): wt=76.000, iters=3418 11.07/11.30 11.07/11.30 Low Water (keep): wt=72.000, iters=3349 11.07/11.30 11.07/11.30 Low Water (keep): wt=70.000, iters=3457 11.07/11.30 11.07/11.30 Low Water (keep): wt=68.000, iters=3417 11.07/11.30 11.07/11.30 Low Water (keep): wt=67.000, iters=3393 11.07/11.30 11.07/11.30 Low Water (keep): wt=66.000, iters=3387 11.07/11.30 11.07/11.30 Low Water (keep): wt=61.000, iters=3363 11.07/11.30 11.07/11.30 Low Water (keep): wt=60.000, iters=3347 11.07/11.30 11.07/11.30 Low Water (keep): wt=59.000, iters=3390 11.07/11.30 11.07/11.30 Low Water (keep): wt=57.000, iters=3339 11.07/11.30 11.07/11.30 Low Water (keep): wt=55.000, iters=3339 11.07/11.30 11.07/11.30 Low Water (keep): wt=54.000, iters=3336 11.07/11.30 11.07/11.30 Low Water (keep): wt=53.000, iters=3399 11.07/11.30 11.07/11.30 Low Water (keep): wt=52.000, iters=3344 11.07/11.30 11.07/11.30 Low Water (keep): wt=51.000, iters=3339 11.07/11.30 11.07/11.30 Low Water (keep): wt=50.000, iters=3340 11.07/11.30 11.07/11.30 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 41 (0.00 of 8.08 sec). 11.07/11.30 11.07/11.30 Low Water (keep): wt=45.000, iters=3351 11.07/11.30 11.07/11.30 Low Water (keep): wt=44.000, iters=3452 11.07/11.30 11.07/11.30 Low Water (keep): wt=43.000, iters=3333 11.07/11.30 11.07/11.30 Low Water (keep): wt=42.000, iters=3376 11.07/11.30 11.07/11.30 Low Water (keep): wt=41.000, iters=3371 11.07/11.30 11.07/11.30 Low Water (keep): wt=40.000, iters=3402 11.07/11.30 11.07/11.30 Low Water (keep): wt=39.000, iters=3342 11.07/11.30 11.07/11.30 Low Water (keep): wt=38.000, iters=3337 11.07/11.30 11.07/11.30 Low Water (keep): wt=37.000, iters=3367 11.07/11.30 11.07/11.30 Low Water (keep): wt=36.000, iters=3356 11.07/11.30 11.07/11.30 Low Water (keep): wt=35.000, iters=3402 11.07/11.30 11.07/11.30 Low Water (keep): wt=34.000, iters=3357 11.07/11.30 11.07/11.30 Low Water (keep): wt=33.000, iters=3342 11.07/11.30 11.07/11.30 Low Water (keep): wt=32.000, iters=3344 11.07/11.30 11.07/11.30 Low Water (keep): wt=31.000, iters=3334 11.07/11.30 11.07/11.30 Low Water (keep): wt=30.000, iters=3345 11.07/11.30 11.07/11.30 Low Water (keep): wt=29.000, iters=3394 11.07/11.30 11.07/11.30 Low Water (keep): wt=28.000, iters=3335 11.07/11.30 11.07/11.30 Low Water (keep): wt=27.000, iters=3339 11.07/11.30 11.07/11.30 ============================== PROOF ================================= 11.07/11.30 % SZS status Theorem 11.07/11.30 % SZS output start Refutation 11.07/11.30 11.07/11.30 % Proof 1 at 9.34 (+ 0.04) seconds. 11.07/11.30 % Length of proof is 88. 11.07/11.30 % Level of proof is 9. 11.07/11.30 % Maximum clause weight is 28.000. 11.07/11.30 % Given clauses 1415. 11.07/11.30 11.07/11.30 65 (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]. 11.07/11.30 113 (all B_1_1 all B_2_1 all X_a (order(X_a) -> hAPP(X_a,bool,B_1_1,ti(X_a,B_2_1)) = hAPP(X_a,bool,B_1_1,B_2_1))) # label(tsy_c_hAPP_1_arg2) # label(axiom) # label(non_clause). [assumption]. 11.07/11.30 162 (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]. 11.07/11.30 257 (all K_1 bit0(K_1) = plus_plus(int,K_1,K_1)) # label(fact_78_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 11.07/11.30 380 (all K_1 ti(int,K_1) = number_number_of(int,K_1)) # label(fact_120_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 11.07/11.30 414 (all K_1 plus_plus(int,plus_plus(int,one_one(int),K_1),K_1) = bit1(K_1)) # label(fact_92_Bit1__def) # label(axiom) # label(non_clause). [assumption]. 11.07/11.30 462 (all B_1_1 all B_2_1 all X_a (order(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_0_arg2) # label(axiom) # label(non_clause). [assumption]. 11.07/11.30 484 (all X_a (mult_zero(X_a) & no_zero_divisors(X_a) & zero_neq_one(X_a) & power(X_a) -> (all A_2 all Na (zero_zero(X_a) = hAPP(nat,X_a,power_power(X_a,A_2),Na) <-> zero_zero(X_a) = ti(X_a,A_2) & Na != zero_zero(nat))))) # label(fact_185_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 11.07/11.30 498 (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]. 11.07/11.30 523 (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]. 11.07/11.30 532 (all K_1 succ(K_1) = minus_minus(int,K_1,min)) # label(fact_777_diff__bin__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 11.07/11.30 558 (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]. 11.07/11.30 577 (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_eq(X_a),number_number_of(X_a,Va)),number_number_of(X_a,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))))))) # label(fact_448_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 11.07/11.30 579 (all T all A ti(T,A) = ti(T,ti(T,A))) # label(help_ti_idem) # label(axiom) # label(non_clause). [assumption]. 11.07/11.30 663 (all B_1_1 nat_1(ti(int,B_1_1)) = nat_1(B_1_1)) # label(tsy_c_Int_Onat_arg1) # label(axiom) # label(non_clause). [assumption]. 11.07/11.30 790 (all K_1 plus_plus(int,pls,K_1) = ti(int,K_1)) # label(fact_76_add__Pls) # label(axiom) # label(non_clause). [assumption]. 11.07/11.30 825 (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]. 11.07/11.30 827 (all Z plus_plus(int,zero_zero(int),Z) = ti(int,Z)) # label(fact_80_zadd__0) # label(axiom) # label(non_clause). [assumption]. 11.07/11.30 955 (all X_a (group_add(X_a) -> (all A_1 zero_zero(X_a) = minus_minus(X_a,A_1,A_1)))) # label(fact_322_diff__self) # label(axiom) # label(non_clause). [assumption]. 11.07/11.30 1011 -(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) # label(non_clause). [assumption]. 11.07/11.30 1012 -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(162)]. 11.07/11.30 1015 -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(498)]. 11.07/11.30 1019 monoid_add(int) # label(arity_Int_Oint___Groups_Omonoid__add) # label(axiom). [assumption]. 11.07/11.30 1243 no_zero_divisors(int) # label(arity_Int_Oint___Rings_Ono__zero__divisors) # label(axiom). [assumption]. 11.07/11.30 1246 -mult_zero(A) | -no_zero_divisors(A) | -zero_neq_one(A) | -power(A) | zero_zero(A) != hAPP(nat,A,power_power(A,B),C) | zero_zero(A) = ti(A,B) # label(fact_185_power__eq__0__iff) # label(axiom). [clausify(484)]. 11.07/11.30 1272 order(int) # label(arity_Int_Oint___Orderings_Oorder) # label(axiom). [assumption]. 11.07/11.30 1275 -order(A) | hAPP(A,bool,B,ti(A,C)) = hAPP(A,bool,B,C) # label(tsy_c_hAPP_1_arg2) # label(axiom). [clausify(113)]. 11.07/11.30 1278 -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(462)]. 11.07/11.30 1494 group_add(int) # label(arity_Int_Oint___Groups_Ogroup__add) # label(axiom). [assumption]. 11.07/11.30 1498 -group_add(A) | zero_zero(A) = minus_minus(A,B,B) # label(fact_322_diff__self) # label(axiom). [clausify(955)]. 11.07/11.30 1519 -linorder(A) | -number(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(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,C)),number_number_of(A,B))) # label(fact_448_le__number__of__eq__not__less) # label(axiom). [clausify(577)]. 11.07/11.30 1520 linorder(int) # label(arity_Int_Oint___Orderings_Olinorder) # label(axiom). [assumption]. 11.07/11.30 1714 plus_plus(int,A,B) = plus_plus(int,B,A) # label(fact_47_zadd__commute) # label(axiom). [clausify(65)]. 11.07/11.30 1790 zero_zero(int) != one_one(int) # label(fact_74_int__0__neq__1) # label(axiom). [assumption]. 11.07/11.30 1791 one_one(int) != zero_zero(int). [copy(1790),flip(a)]. 11.07/11.30 1952 bit0(A) = plus_plus(int,A,A) # label(fact_78_Bit0__def) # label(axiom). [clausify(257)]. 11.07/11.30 2115 number_number_of(int,A) = ti(int,A) # label(fact_120_number__of__is__id) # label(axiom). [clausify(380)]. 11.07/11.30 2116 succ(min) = pls # label(fact_776_succ__Min) # label(axiom). [assumption]. 11.07/11.30 2117 pls = succ(min). [copy(2116),flip(a)]. 11.07/11.30 2160 bit1(A) = plus_plus(int,plus_plus(int,one_one(int),A),A) # label(fact_92_Bit1__def) # label(axiom). [clausify(414)]. 11.07/11.30 2161 bit1(A) = plus_plus(int,A,plus_plus(int,A,one_one(int))). [copy(2160),rewrite([1714(6),1714(7)])]. 11.07/11.30 2181 zero_neq_one(int) # label(arity_Int_Oint___Rings_Ozero__neq__one) # label(axiom). [assumption]. 11.07/11.30 2311 number_number_of(nat,A) = nat_1(number_number_of(int,A)) # label(fact_427_nat__number__of__def) # label(axiom). [clausify(523)]. 11.07/11.30 2312 nat_1(ti(int,A)) = number_number_of(nat,A). [copy(2311),rewrite([2115(4)]),flip(a)]. 11.07/11.30 2319 succ(A) = minus_minus(int,A,min) # label(fact_777_diff__bin__simps_I2_J) # label(axiom). [clausify(532)]. 11.07/11.30 2349 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),A)) # label(fact_251_zle__refl) # label(axiom). [clausify(558)]. 11.07/11.30 2369 ti(A,ti(A,B)) = ti(A,B) # label(help_ti_idem) # label(axiom). [clausify(579)]. 11.07/11.30 2480 nat_1(ti(int,A)) = nat_1(A) # label(tsy_c_Int_Onat_arg1) # label(axiom). [clausify(663)]. 11.07/11.30 2481 number_number_of(nat,A) = nat_1(A). [copy(2480),rewrite([2312(3)])]. 11.07/11.30 2494 one_one(int) = number_number_of(int,bit1(pls)) # label(fact_37_one__is__num__one) # label(axiom). [assumption]. 11.07/11.30 2495 ti(int,plus_plus(int,minus_minus(int,min,min),plus_plus(int,one_one(int),minus_minus(int,min,min)))) = one_one(int). [copy(2494),rewrite([2117(4),2319(5),2161(8),1714(16),2115(18)]),flip(a)]. 11.07/11.30 2516 power(int) # label(arity_Int_Oint___Power_Opower) # label(axiom). [assumption]. 11.07/11.30 2539 number(int) # label(arity_Int_Oint___Int_Onumber) # label(axiom). [assumption]. 11.07/11.30 2547 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]. 11.07/11.30 2647 ti(int,A) = plus_plus(int,pls,A) # label(fact_76_add__Pls) # label(axiom). [clausify(790)]. 11.07/11.30 2648 plus_plus(int,A,minus_minus(int,min,min)) = ti(int,A). [copy(2647),rewrite([2117(4),2319(5),1714(8)]),flip(a)]. 11.07/11.30 2691 succ(A) = plus_plus(int,A,one_one(int)) # label(fact_519_succ__def) # label(axiom). [clausify(825)]. 11.07/11.30 2692 plus_plus(int,A,one_one(int)) = minus_minus(int,A,min). [copy(2691),rewrite([2319(1)]),flip(a)]. 11.07/11.30 2697 ti(int,A) = plus_plus(int,zero_zero(int),A) # label(fact_80_zadd__0) # label(axiom). [clausify(827)]. 11.07/11.30 2698 plus_plus(int,A,zero_zero(int)) = ti(int,A). [copy(2697),rewrite([1714(6)]),flip(a)]. 11.07/11.30 2721 mult_zero(int) # label(arity_Int_Oint___Rings_Omult__zero) # label(axiom). [assumption]. 11.07/11.30 2725 bit1(pls) = succ(pls) # label(fact_513_succ__Pls) # label(axiom). [assumption]. 11.07/11.30 2726 minus_minus(int,minus_minus(int,min,min),min) = ti(int,one_one(int)). [copy(2725),rewrite([2117(1),2319(2),2161(5),1714(13),2648(13),1714(10),2648(10),2369(6),2117(5),2319(6),2319(9)]),flip(a)]. 11.07/11.30 2763 zero_zero(int) = pls # label(fact_73_Pls__def) # label(axiom). [assumption]. 11.07/11.30 2764 minus_minus(int,min,min) = zero_zero(int). [copy(2763),rewrite([2117(3),2319(4)]),flip(a)]. 11.07/11.30 2773 plus_plus(nat,one_one(nat),one_one(nat)) = number_number_of(nat,bit0(bit1(pls))) # label(fact_29_nat__1__add__1) # label(axiom). [assumption]. 11.07/11.30 2774 nat_1(plus_plus(int,plus_plus(int,zero_zero(int),minus_minus(int,zero_zero(int),min)),plus_plus(int,zero_zero(int),minus_minus(int,zero_zero(int),min)))) = plus_plus(nat,one_one(nat),one_one(nat)). [copy(2773),rewrite([2117(8),2319(9),2764(11),2161(10),2692(16),1952(17),2481(28)]),flip(a)]. 11.07/11.30 2934 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). [clausify(1011)]. 11.07/11.30 2935 hAPP(nat,int,power_power(int,plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n))),plus_plus(nat,one_one(nat),one_one(nat))) = zero_zero(int). [copy(2934),rewrite([2117(18),2319(19),2764(21),2161(20),2692(26),1952(27),2481(38),2774(37)]),flip(a)]. 11.07/11.30 2949 plus_plus(int,A,ti(int,B)) = plus_plus(int,A,B). [resolve(1019,a,1012,a)]. 11.07/11.30 2951 ti(int,plus_plus(int,A,B)) = plus_plus(int,A,B). [resolve(1019,a,1015,a)]. 11.07/11.30 3476 -mult_zero(int) | -zero_neq_one(int) | -power(int) | zero_zero(int) != hAPP(nat,int,power_power(int,A),B) | zero_zero(int) = ti(int,A). [resolve(1246,b,1243,a)]. 11.07/11.30 3477 hAPP(nat,int,power_power(int,A),B) != zero_zero(int) | ti(int,A) = zero_zero(int). [copy(3476),flip(d),flip(e),unit_del(a,2721),unit_del(b,2181),unit_del(c,2516)]. 11.07/11.30 3535 hAPP(int,bool,A,ti(int,B)) = hAPP(int,bool,A,B). [resolve(1272,a,1275,a)]. 11.07/11.30 3538 hAPP(int,fun(int,bool),A,ti(int,B)) = hAPP(int,fun(int,bool),A,B). [resolve(1272,a,1278,a)]. 11.07/11.30 4024 zero_zero(int) = minus_minus(int,A,A). [resolve(1498,a,1494,a)]. 11.07/11.30 4025 minus_minus(int,A,A) = zero_zero(int). [copy(4024),flip(a)]. 11.07/11.30 4083 -number(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,A)),number_number_of(int,B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,B)),number_number_of(int,A))). [resolve(1519,a,1520,a)]. 11.07/11.30 4084 -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),B),A)). [copy(4083),rewrite([2115(12),3538(13),2115(13),3535(14),2115(23),3538(24),2115(24),3535(25)]),unit_del(a,2539)]. 11.07/11.30 4233 one_one(int) = c_0. [new_symbol(1791)]. 11.07/11.30 4400 ti(int,c_0) = c_0. [back_rewrite(2495),rewrite([4025(6),4233(7),4025(10),2698(9),2949(8),1714(6),2698(6),2369(5),4233(5)])]. 11.07/11.30 4422 minus_minus(int,zero_zero(int),min) = c_0. [back_rewrite(2726),rewrite([4025(5),4233(8),4400(8)])]. 11.07/11.30 4546 plus_plus(nat,one_one(nat),one_one(nat)) = nat_1(plus_plus(int,c_0,c_0)). [back_rewrite(2774),rewrite([4422(9),1714(6),2698(6),4400(4),4422(10),1714(7),2698(7),4400(5)]),flip(a)]. 11.07/11.30 4701 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(2935),rewrite([4233(6),4546(19)])]. 11.07/11.30 4717 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(2547),rewrite([4233(14)])]. 11.07/11.30 10786 -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),A)). [resolve(4084,a,2349,a)]. 11.07/11.30 12392 plus_plus(int,c_0,hAPP(nat,int,semiring_1_of_nat(int),n)) = zero_zero(int). [resolve(4701,a,3477,a),rewrite([2951(11)])]. 11.07/11.30 12399 $F. [back_rewrite(4717),rewrite([12392(20)]),unit_del(a,10786)]. 11.07/11.30 11.07/11.30 % SZS output end Refutation 11.07/11.30 ============================== end of proof ========================== 11.07/11.30 11.07/11.30 ============================== STATISTICS ============================ 11.07/11.30 11.07/11.30 Given=1415. Generated=46163. Kept=9822. proofs=1. 11.07/11.30 Usable=1398. Sos=7089. Demods=871. Limbo=7, Disabled=3935. Hints=0. 11.07/11.30 Megabytes=38.19. 11.07/11.30 User_CPU=9.34, System_CPU=0.04, Wall_clock=9. 11.07/11.30 11.07/11.30 ============================== end of statistics ===================== 11.07/11.30 11.07/11.30 ============================== end of search ========================= 11.07/11.30 11.07/11.30 THEOREM PROVED 11.07/11.30 % SZS status Theorem 11.07/11.30 11.07/11.30 Exiting with 1 proof. 11.07/11.30 11.07/11.30 Process 4051 exit (max_proofs) Thu Jul 2 07:36:02 2020 11.07/11.30 Prover9 interrupted 11.07/11.30 EOF