0.03/0.11 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.03/0.12 % Command : tptp2X_and_run_prover9 %d %s 0.12/0.33 % Computer : n022.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 : 1440 0.12/0.33 % WCLimit : 180 0.12/0.33 % DateTime : Mon Jul 3 05:29:59 EDT 2023 0.12/0.33 % CPUTime : 1.49/1.85 ============================== Prover9 =============================== 1.49/1.85 Prover9 (32) version 2009-11A, November 2009. 1.49/1.85 Process 2138 was started by sandbox2 on n022.cluster.edu, 1.49/1.85 Mon Jul 3 05:30:00 2023 1.49/1.85 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 1440 -f /tmp/Prover9_1977_n022.cluster.edu". 1.49/1.85 ============================== end of head =========================== 1.49/1.85 1.49/1.85 ============================== INPUT ================================= 1.49/1.85 1.49/1.85 % Reading from file /tmp/Prover9_1977_n022.cluster.edu 1.49/1.85 1.49/1.85 set(prolog_style_variables). 1.49/1.85 set(auto2). 1.49/1.85 % set(auto2) -> set(auto). 1.49/1.85 % set(auto) -> set(auto_inference). 1.49/1.85 % set(auto) -> set(auto_setup). 1.49/1.85 % set(auto_setup) -> set(predicate_elim). 1.49/1.85 % set(auto_setup) -> assign(eq_defs, unfold). 1.49/1.85 % set(auto) -> set(auto_limits). 1.49/1.85 % set(auto_limits) -> assign(max_weight, "100.000"). 1.49/1.85 % set(auto_limits) -> assign(sos_limit, 20000). 1.49/1.85 % set(auto) -> set(auto_denials). 1.49/1.85 % set(auto) -> set(auto_process). 1.49/1.85 % set(auto2) -> assign(new_constants, 1). 1.49/1.85 % set(auto2) -> assign(fold_denial_max, 3). 1.49/1.85 % set(auto2) -> assign(max_weight, "200.000"). 1.49/1.85 % set(auto2) -> assign(max_hours, 1). 1.49/1.85 % assign(max_hours, 1) -> assign(max_seconds, 3600). 1.49/1.85 % set(auto2) -> assign(max_seconds, 0). 1.49/1.85 % set(auto2) -> assign(max_minutes, 5). 1.49/1.85 % assign(max_minutes, 5) -> assign(max_seconds, 300). 1.49/1.85 % set(auto2) -> set(sort_initial_sos). 1.49/1.85 % set(auto2) -> assign(sos_limit, -1). 1.49/1.85 % set(auto2) -> assign(lrs_ticks, 3000). 1.49/1.85 % set(auto2) -> assign(max_megs, 400). 1.49/1.85 % set(auto2) -> assign(stats, some). 1.49/1.85 % set(auto2) -> clear(echo_input). 1.49/1.85 % set(auto2) -> set(quiet). 1.49/1.85 % set(auto2) -> clear(print_initial_clauses). 1.49/1.85 % set(auto2) -> clear(print_given). 1.49/1.85 assign(lrs_ticks,-1). 1.49/1.85 assign(sos_limit,10000). 1.49/1.85 assign(order,kbo). 1.49/1.85 set(lex_order_vars). 1.49/1.85 clear(print_given). 1.49/1.85 1.49/1.85 % formulas(sos). % not echoed (1254 formulas) 1.49/1.85 1.49/1.85 ============================== end of input ========================== 1.49/1.85 1.49/1.85 % From the command line: assign(max_seconds, 1440). 1.49/1.85 1.49/1.85 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 1.49/1.85 1.49/1.85 % Formulas that are not ordinary clauses: 1.49/1.85 1 (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]. 1.49/1.85 2 (all B_1_1 all B_2_1 all X_a (dvd(X_a) -> hAPP(X_a,fun(X_a,bool),B_1_1,B_2_1) = ti(fun(X_a,bool),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]. 1.49/1.85 3 (all K all Ma all Na (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(nat),zero_zero(nat)),K)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na))))) # label(fact_586_mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 4 (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]. 1.49/1.85 5 (all X_a (semiri456707255roduct(X_a) -> (all C_1 all D all A_2 all B (plus_plus(X_a,times_times(X_a,A_2,D),times_times(X_a,B,C_1)) != plus_plus(X_a,times_times(X_a,A_2,C_1),times_times(X_a,B,D)) <-> ti(X_a,C_1) != ti(X_a,D) & ti(X_a,A_2) != ti(X_a,B))))) # label(fact_344_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 6 (all B_1_1 (hBOOL(ti(bool,B_1_1)) <-> hBOOL(B_1_1))) # label(tsy_c_hBOOL_arg1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 7 (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]. 1.49/1.85 8 (all Na (zero_zero(int) = hAPP(nat,int,semiring_1_of_nat(int),Na) <-> Na = zero_zero(nat))) # label(fact_27_int__eq__0__conv) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 9 (all D_1 all C all A_1 all B_1 all M (hBOOL(hAPP(int,bool,zcong(A_1,B_1),M)) -> (ti(int,C) = ti(int,B_1) -> (hBOOL(hAPP(int,bool,zcong(C,D_1),M)) -> hBOOL(hAPP(int,bool,zcong(A_1,D_1),M)))))) # label(fact_818_zcong__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 10 (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]. 1.49/1.85 11 (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]. 1.49/1.85 12 (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]. 1.49/1.85 13 (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]. 1.49/1.85 14 (all Na (zero_zero(nat) != Na <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)))) # label(fact_145_neq0__conv) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 15 (all X_a (number_semiring(X_a) -> (all N hAPP(nat,X_a,semiring_1_of_nat(X_a),N) = number_number_of(X_a,hAPP(nat,int,semiring_1_of_nat(int),N))))) # label(fact_94_number__of__int) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 16 (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]. 1.49/1.85 17 (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))) -> (zero_zero(int) != ti(int,K_1) -> 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]. 1.49/1.85 18 (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]. 1.49/1.85 19 (all X_a all B_1_1 all B_2_1 all B_3 if(X_a,ti(bool,B_1_1),B_2_1,B_3) = if(X_a,B_1_1,B_2_1,B_3)) # label(tsy_c_If_arg1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 20 (all Z_1 (ti(int,Z_1) = number_number_of(int,min) | one_one(int) = ti(int,Z_1) <-> abs_abs(int,Z_1) = one_one(int))) # label(fact_957_abs__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 21 (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]. 1.49/1.85 22 (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]. 1.49/1.85 23 (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]. 1.49/1.85 24 (all X_a (linordered_semidom(X_a) -> (all X all Y (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)),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]. 1.49/1.85 25 (all Na all Ma (one_one(nat) = times_times(nat,Na,Ma) <-> one_one(nat) = Ma & one_one(nat) = Na)) # label(fact_625_nat__mult__eq__one) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 26 (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]. 1.49/1.85 27 (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]. 1.49/1.85 28 (all Ya all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Ya)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Xa),Ya)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),nat_1(Xa)),nat_1(Ya))))))) # label(fact_833_transfer__nat__int__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 29 (all M all N (N = M | 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_605_less__or__eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 30 (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]. 1.49/1.85 31 (all Ma all K all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),times_times(nat,Ma,K)),times_times(nat,Na,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),zero_zero(nat)),K)))) # label(fact_650_mult__less__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 32 (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]. 1.49/1.85 33 (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]. 1.49/1.85 34 (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]. 1.49/1.85 35 (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]. 1.49/1.85 36 (all Wa all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Wa),minus_minus(int,Z_1,one_one(int)))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z_1)))) # label(fact_680_zle__diff1__eq) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 37 (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),Ma),plus_plus(nat,times_times(nat,minus_minus(nat,J,I_1),U),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_990_nat__le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 38 (all X_a (comm_semiring_1(X_a) -> (all M times_times(X_a,plus_plus(X_a,one_one(X_a),one_one(X_a)),M) = plus_plus(X_a,M,M)))) # label(fact_445_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 39 (all B_1_1 all B_2_1 all X_a (dvd(X_a) -> hAPP(X_a,bool,B_1_1,B_2_1) = hAPP(X_a,bool,B_1_1,ti(X_a,B_2_1)))) # label(tsy_c_hAPP_3_arg2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 40 (all X all Y all Z hAPP(nat,int,power_power(int,X),times_times(nat,Y,Z)) = hAPP(nat,int,power_power(int,hAPP(nat,int,power_power(int,X),Y)),Z)) # label(fact_641_zpower__zpower) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 41 (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]. 1.49/1.85 42 (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(int),Xa),Ya)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),nat_1(Xa)),nat_1(Ya))))))) # label(fact_387_transfer__nat__int__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 43 (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]. 1.49/1.85 44 (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]. 1.49/1.85 45 (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]. 1.49/1.85 46 (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]. 1.49/1.85 47 (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]. 1.49/1.85 48 (all A_1 hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit1(bit1(pls)))) = times_times(int,A_1,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls)))))) # label(fact_543_cube__square) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 49 (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]. 1.49/1.85 50 (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]. 1.49/1.85 51 (all X_a all B_1_1 all B_2_1 all B_3 if(X_a,B_1_1,ti(X_a,B_2_1),B_3) = if(X_a,B_1_1,B_2_1,B_3)) # label(tsy_c_If_arg2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 52 (all B_1_1 all B_2_1 legendre(B_1_1,B_2_1) = legendre(ti(int,B_1_1),B_2_1)) # label(tsy_c_Residues_OLegendre_arg1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 53 (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]. 1.49/1.85 54 (all X_a (number_ring(X_a) -> (all X all Y hAPP(nat,X_a,power_power(X_a,minus_minus(X_a,X,Y)),number_number_of(nat,bit0(bit1(pls)))) = minus_minus(X_a,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,X_a,power_power(X_a,Y),number_number_of(nat,bit0(bit1(pls))))),times_times(X_a,times_times(X_a,number_number_of(X_a,bit0(bit1(pls))),X),Y))))) # label(fact_520_power2__diff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 55 (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]. 1.49/1.85 56 (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]. 1.49/1.85 57 (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]. 1.49/1.85 58 (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]. 1.49/1.85 59 (all N N = plus_plus(nat,zero_zero(nat),N)) # label(fact_156_plus__nat_Oadd__0) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 60 (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]. 1.49/1.85 61 (all Ma all Na (one_one(int) = times_times(int,Ma,Na) <-> number_number_of(int,min) = ti(int,Ma) & ti(int,Na) = number_number_of(int,min) | ti(int,Na) = one_one(int) & one_one(int) = ti(int,Ma))) # label(fact_785_zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 62 (all V_1 all W number_number_of(int,times_times(int,V_1,W)) = times_times(int,number_number_of(int,V_1),number_number_of(int,W))) # label(fact_384_times__numeral__code_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 63 (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]. 1.49/1.85 64 (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]. 1.49/1.85 65 (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]. 1.49/1.85 66 (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)) -> (minus_minus(nat,Ma,K) = minus_minus(nat,Na,K) <-> Ma = Na)))) # label(fact_635_eq__diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 67 (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]. 1.49/1.85 68 (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]. 1.49/1.85 69 (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]. 1.49/1.85 70 (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]. 1.49/1.85 71 (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]. 1.49/1.85 72 (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]. 1.49/1.85 73 (all Xa all Ya (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),Xa)),hAPP(nat,int,semiring_1_of_nat(int),Ya))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Xa),Ya)))) # label(fact_677_Nat__Transfer_Otransfer__int__nat__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 74 (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]. 1.49/1.85 75 (all X_a (comm_semiring(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_339_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 76 (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]. 1.49/1.85 77 (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]. 1.49/1.85 78 (all X_a (monoid_add(X_a) -> (all A_1 plus_plus(X_a,A_1,zero_zero(X_a)) = ti(X_a,A_1)))) # label(fact_215_add__0__right) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 79 (all A_1 all B_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1)) -> (A_1 != B_1 -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B_1),A_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1))))) # label(fact_863_dvd_Ole__neq__trans) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 80 (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]. 1.49/1.85 81 (all X_a (idom(X_a) -> (all C_1 all A_2 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),times_times(X_a,C_1,A_2)),times_times(X_a,C_1,B))) <-> zero_zero(X_a) = ti(X_a,C_1) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_2),B)))))) # label(fact_740_dvd__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 82 (all X_a (power(X_a) & semiring_0(X_a) -> (all N ((N != zero_zero(nat) -> hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),N) = zero_zero(X_a)) & (N = zero_zero(nat) -> hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),N) = one_one(X_a)))))) # label(fact_109_power__0__left) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 83 (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_216_double__zero__sym) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 84 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K1)),bit1(K2))))) # label(fact_463_less__eq__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 85 (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]. 1.49/1.85 86 (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]. 1.49/1.85 87 (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]. 1.49/1.85 88 (all X_a (linordered_idom(X_a) -> abs_abs(X_a,one_one(X_a)) = one_one(X_a))) # label(fact_933_abs__one) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 89 (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]. 1.49/1.85 90 (all Z_3 all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_3)) -> (ti(int,Z_1) = ti(int,Z_3) <-> nat_1(Z_3) = nat_1(Z_1))))) # label(fact_261_eq__nat__nat__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 91 (all X_a (group_add(X_a) -> (all A_2 all B (ti(X_a,A_2) = ti(X_a,B) <-> zero_zero(X_a) = minus_minus(X_a,A_2,B))))) # label(fact_320_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 92 (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]. 1.49/1.85 93 (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]. 1.49/1.85 94 (all X_a (comm_semiring_1(X_a) -> (all M all A_1 times_times(X_a,plus_plus(X_a,A_1,one_one(X_a)),M) = plus_plus(X_a,M,times_times(X_a,A_1,M))))) # label(fact_446_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 95 (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]. 1.49/1.85 96 (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),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]. 1.49/1.85 97 (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]. 1.49/1.85 98 (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]. 1.49/1.85 99 (all Ma all K all Na (Ma = Na <-> plus_plus(nat,Ma,K) = plus_plus(nat,Na,K))) # label(fact_126_nat__add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 100 (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]. 1.49/1.85 101 (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]. 1.49/1.85 102 (all Va (number_number_of(nat,Va) = zero_zero(nat) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Va),pls)))) # label(fact_502_eq__number__of__0) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 103 (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]. 1.49/1.85 104 (all B_1_1 all B_2_1 all X_a (group_add(X_a) -> minus_minus(X_a,B_1_1,B_2_1) = minus_minus(X_a,B_1_1,ti(X_a,B_2_1)))) # label(tsy_c_Groups_Ominus__class_Ominus_0_arg2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 105 (all Z times_times(real,one_one(real),Z) = Z) # label(fact_632_real__mult__1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 106 (all X_a (zero_neq_one(X_a) & no_zero_divisors(X_a) & mult_zero(X_a) & power(X_a) -> (all A_2 all Wa (ti(X_a,A_2) = zero_zero(X_a) & number_number_of(nat,Wa) != zero_zero(nat) <-> zero_zero(X_a) = hAPP(nat,X_a,power_power(X_a,A_2),number_number_of(nat,Wa)))))) # label(fact_86_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 107 (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]. 1.49/1.85 108 (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]. 1.49/1.85 109 (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,minus_minus(X_a,B,A_2),E_1),D) = ti(X_a,C_1) <-> 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_288_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 110 (all Xa all P_3 (hBOOL(hAPP(int,bool,zcong(Xa,zero_zero(int)),P_3)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_3),Xa)))) # label(fact_801_zcong__eq__zdvd__prop) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 111 (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]. 1.49/1.85 112 (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]. 1.49/1.85 113 (all I_1 (zero_zero(nat) = nat_1(I_1) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_1),zero_zero(int))))) # label(fact_315_nat__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 114 (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]. 1.49/1.85 115 (all X_a (monoid_add(X_a) -> (all A_1 plus_plus(X_a,zero_zero(X_a),A_1) = ti(X_a,A_1)))) # label(fact_219_add__0__left) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 116 (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(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))))) # label(fact_905_conj__le__cong) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 117 (all X_a (semiri456707255roduct(X_a) -> (all C all D_1 all A_1 all B_1 all R_1 (zero_zero(X_a) != ti(X_a,R_1) -> (ti(X_a,A_1) = ti(X_a,B_1) & ti(X_a,C) != ti(X_a,D_1) -> plus_plus(X_a,A_1,times_times(X_a,R_1,C)) != plus_plus(X_a,B_1,times_times(X_a,R_1,D_1))))))) # label(fact_429_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 118 (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]. 1.49/1.85 119 (all N (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,N))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,bit0(N)))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,bit1(N)))))) # label(fact_914_number__of1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 120 (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]. 1.49/1.85 121 (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,W) = ti(int,Z)))) # label(fact_256_zle__antisym) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 122 (all Xa all Ya (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),Xa)),hAPP(nat,int,semiring_1_of_nat(int),Ya))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Xa),Ya)))) # label(fact_148_Nat__Transfer_Otransfer__int__nat__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 123 (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)))) -> 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_839_Legendre__1mod4) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 124 (all X_a (ordere142940540dd_abs(X_a) -> (all A_2 (abs_abs(X_a,A_2) = zero_zero(X_a) <-> ti(X_a,A_2) = zero_zero(X_a))))) # label(fact_931_abs__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 125 (all K_1 all L bit1(K_1) != bit0(L)) # label(fact_68_rel__simps_I50_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 126 (all X all Y (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y),X)) | ti(int,X) = ti(int,Y) | 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]. 1.49/1.85 127 (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]. 1.49/1.85 128 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) -> minus_minus(nat,M,N) = zero_zero(nat))) # label(fact_659_diff__is__0__eq_H) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 129 (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]. 1.49/1.85 130 (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]. 1.49/1.85 131 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),bit0(K))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),K)))) # label(fact_754_rel__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 132 (all X_a (semiri456707255roduct(X_a) -> (all Wa all Ya all Xa all Z_1 (ti(X_a,Ya) = ti(X_a,Z_1) | ti(X_a,Wa) = ti(X_a,Xa) <-> 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]. 1.49/1.85 133 (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_674_diff__add__assoc) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 134 (all X_a (linord219039673up_add(X_a) -> (all A_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,A_2,A_2)),zero_zero(X_a))))))) # label(fact_433_double__add__le__zero__iff__single__add__le__zero) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 135 (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]. 1.49/1.85 136 (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]. 1.49/1.85 137 (all X_a (comm_semiring_1(X_a) -> (all A_1 zero_zero(X_a) = times_times(X_a,A_1,zero_zero(X_a))))) # label(fact_333_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 138 (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]. 1.49/1.85 139 (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]. 1.49/1.85 140 (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]. 1.49/1.85 141 (all X_a (number_semiring(X_a) -> (all V_2 all V_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_2)) -> times_times(X_a,number_number_of(X_a,V_1),number_number_of(X_a,V_2)) = number_number_of(X_a,times_times(int,V_1,V_2))))))) # label(fact_250_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 142 (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]. 1.49/1.85 143 (all V_1 all W number_number_of(int,plus_plus(int,V_1,W)) = plus_plus(int,number_number_of(int,V_1),number_number_of(int,W))) # label(fact_17_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 144 (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]. 1.49/1.85 145 (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]. 1.49/1.85 146 (all L bit1(L) != pls) # label(fact_67_rel__simps_I39_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 147 (all Na (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),Na)),zero_zero(int))) <-> zero_zero(nat) = Na)) # label(fact_512_int__le__0__conv) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 148 (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]. 1.49/1.85 149 (all K_1 bit0(succ(K_1)) = succ(bit1(K_1))) # label(fact_515_succ__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 150 (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]. 1.49/1.85 151 (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]. 1.49/1.85 152 (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]. 1.49/1.85 153 (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]. 1.49/1.85 154 (all Na all K all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),Ma)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),Na)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),minus_minus(nat,Ma,K)),minus_minus(nat,Na,K))))))) # label(fact_662_less__diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 155 (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]. 1.49/1.85 156 (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]. 1.49/1.85 157 (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]. 1.49/1.85 158 (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))) <-> ti(X_a,Xa) = zero_zero(X_a) & ti(X_a,Ya) = zero_zero(X_a))))) # label(fact_553_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 159 (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]. 1.49/1.85 160 (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]. 1.49/1.85 161 (all X_a (number_ring(X_a) -> (all Z times_times(X_a,Z,number_number_of(X_a,bit0(bit1(pls)))) = plus_plus(X_a,Z,Z)))) # label(fact_535_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 162 (all P_1 ((all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) -> hBOOL(hAPP(nat,bool,P_1,nat_1(X_1))))) <-> (all X1 hBOOL(hAPP(nat,bool,P_1,X1))))) # label(fact_292_all__nat) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 163 (all Z all W times_times(real,Z,W) = times_times(real,W,Z)) # label(fact_713_real__mult__commute) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 164 (all M all N (M = plus_plus(nat,M,N) -> N = zero_zero(nat))) # label(fact_153_add__eq__self__zero) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 165 (all N all M ((zero_zero(nat) = M -> times_times(nat,M,N) = zero_zero(nat)) & (M != zero_zero(nat) -> times_times(nat,M,N) = plus_plus(nat,N,times_times(nat,minus_minus(nat,M,one_one(nat)),N))))) # label(fact_687_mult__eq__if) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 166 (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),times_times(X_a,C_1,A_2)),times_times(X_a,C_1,B))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_2))))))) # label(fact_308_mult__le__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 167 (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]. 1.49/1.85 168 (all X_a (linord581940658strict(X_a) -> (all Xa all 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) = ti(X_a,Xa) & ti(X_a,Ya) = zero_zero(X_a))))) # label(fact_428_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 169 (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)) <-> Ya != Xa & 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]. 1.49/1.85 170 (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]. 1.49/1.85 171 (all B_1_1 all B_2_1 all X_a (ab_semigroup_mult(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_5_res) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 172 (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]. 1.49/1.85 173 (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]. 1.49/1.85 174 (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]. 1.49/1.85 175 (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]. 1.49/1.85 176 (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]. 1.49/1.85 177 (all X_a (order(X_a) -> ti(fun(X_a,fun(X_a,bool)),ord_less(X_a)) = ord_less(X_a))) # label(tsy_c_Orderings_Oord__class_Oless_res) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 178 (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]. 1.49/1.85 179 (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]. 1.49/1.85 180 (all X_a (linordered_semidom(X_a) -> (all A_1 all N all B_1 (hAPP(nat,X_a,power_power(X_a,B_1),N) = hAPP(nat,X_a,power_power(X_a,A_1),N) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> ti(X_a,A_1) = ti(X_a,B_1)))))))) # label(fact_526_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 181 (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]. 1.49/1.85 182 (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]. 1.49/1.85 183 (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]. 1.49/1.85 184 (all K (ti(int,K) = min <-> bit1(K) = min)) # label(fact_771_rel__simps_I47_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 185 (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]. 1.49/1.85 186 (all I_2 all J_1 all K_1 minus_minus(nat,minus_minus(nat,I_2,J_1),K_1) = minus_minus(nat,minus_minus(nat,I_2,K_1),J_1)) # label(fact_267_diff__commute) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 187 (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]. 1.49/1.85 188 (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]. 1.49/1.85 189 (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]. 1.49/1.85 190 (all Z ti(int,Z) = times_times(int,Z,one_one(int))) # label(fact_379_zmult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.49/1.85 191 (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]. 1.49/1.86 192 (all X_a (ring_11004092258visors(X_a) -> (all A_2 (zero_zero(X_a) = hAPP(nat,X_a,power_power(X_a,A_2),number_number_of(nat,bit0(bit1(pls)))) <-> ti(X_a,A_2) = zero_zero(X_a))))) # label(fact_5_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 193 (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]. 1.49/1.86 194 (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]. 1.49/1.86 195 (all X_a (monoid_mult(X_a) -> (all A_1 times_times(X_a,times_times(X_a,A_1,A_1),A_1) = hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,bit1(bit1(pls))))))) # label(fact_527_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 196 (all Xa all Ya (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,semiring_1_of_nat(int),Xa)),hAPP(nat,int,semiring_1_of_nat(int),Ya))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)))) # label(fact_811_zdvd__int) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 197 (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]. 1.49/1.86 198 (all K all L_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K)),bit1(L_1))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L_1)))) # label(fact_505_rel__simps_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 199 (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]. 1.49/1.86 200 (all X_a (dvd(X_a) -> ti(fun(X_a,fun(X_a,bool)),dvd_dvd(X_a)) = dvd_dvd(X_a))) # label(tsy_c_Rings_Odvd__class_Odvd_res) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 201 (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]. 1.49/1.86 202 (all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),P_3)) & (all M_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)) -> one_one(int) = ti(int,M_1) | ti(int,M_1) = ti(int,P_3))))) # label(fact_901_zprime__def) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 203 (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]. 1.49/1.86 204 (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]. 1.49/1.86 205 (all Z1 all Z2 all Z3 plus_plus(int,Z1,plus_plus(int,Z2,Z3)) = plus_plus(int,plus_plus(int,Z1,Z2),Z3)) # label(fact_45_zadd__assoc) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 206 (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]. 1.49/1.86 207 (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]. 1.49/1.86 208 (all B_1_1 all X_a (power(X_a) -> power_power(X_a,B_1_1) = ti(fun(nat,X_a),power_power(X_a,B_1_1)))) # label(tsy_c_Power_Opower__class_Opower_res) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 209 (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)) & Na != Ma)) # label(fact_609_nat__less__le) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 210 (all X_a (number_semiring(X_a) -> (all V_2 all V_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_2)) -> plus_plus(X_a,number_number_of(X_a,V_1),number_number_of(X_a,V_2)) = number_number_of(X_a,plus_plus(int,V_1,V_2))))))) # label(fact_528_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 211 (all N N = times_times(nat,one_one(nat),N)) # label(fact_629_nat__mult__1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 212 (all Z all W times_times(int,Z,W) = times_times(int,W,Z)) # label(fact_252_zmult__commute) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 213 (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]. 1.49/1.86 214 (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)) -> ti(int,Z) = hAPP(nat,int,semiring_1_of_nat(int),nat_1(Z))))) # label(fact_317_int__nat__eq) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 215 (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]. 1.49/1.86 216 (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]. 1.49/1.86 217 (all M hAPP(nat,int,semiring_1_of_nat(int),M) = abs_abs(int,hAPP(nat,int,semiring_1_of_nat(int),M))) # label(fact_936_abs__int__eq) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 218 (all K all F all Na ((all I (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),Na)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),abs_abs(int,minus_minus(int,hAPP(nat,int,F,plus_plus(nat,I,one_one(nat))),hAPP(nat,int,F,I)))),one_one(int))))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,F,zero_zero(nat))),K)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),hAPP(nat,int,F,Na))) -> (exists I (ti(int,K) = hAPP(nat,int,F,I) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),Na)))))))) # label(fact_993_nat0__intermed__int__val) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 219 (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]. 1.49/1.86 220 (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]. 1.49/1.86 221 (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]. 1.49/1.86 222 (all P_1 all A_2 all B ((all D_3 (plus_plus(nat,B,D_3) = A_2 -> 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)))) <-> hBOOL(hAPP(nat,bool,P_1,minus_minus(nat,A_2,B))))) # label(fact_500_nat__diff__split) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 223 (all B_1_1 all B_2_1 all X_a (group_add(X_a) -> minus_minus(X_a,B_1_1,B_2_1) = ti(X_a,minus_minus(X_a,B_1_1,B_2_1)))) # label(tsy_c_Groups_Ominus__class_Ominus_0_res) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 224 (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]. 1.49/1.86 225 (all X_a (linord581940658strict(X_a) -> (all Xa all Ya (ti(X_a,Xa) = zero_zero(X_a) & ti(X_a,Ya) = zero_zero(X_a) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_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]. 1.49/1.86 226 (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]. 1.49/1.86 227 (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]. 1.49/1.86 228 (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]. 1.49/1.86 229 (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]. 1.49/1.86 230 (all A_2 all Ma (hBOOL(hAPP(int,bool,zcong(A_2,zero_zero(int)),Ma)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Ma),A_2)))) # label(fact_802_zcong__zero__equiv__div) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 231 (all N N = times_times(nat,N,one_one(nat))) # label(fact_627_nat__mult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 232 (all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),zero_zero(int))) -> zero_zero(nat) = nat_1(Z))) # label(fact_314_nat__le__0) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 233 (all X_a (number_ring(X_a) -> number_number_of(X_a,pls) = zero_zero(X_a))) # label(fact_82_number__of__Pls) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 234 (all M all N (one_one(int) = abs_abs(int,times_times(int,M,N)) -> abs_abs(int,M) = one_one(int))) # label(fact_938_abs__zmult__eq__1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 235 (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]. 1.49/1.86 236 (all N all K_1 all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),M)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N)) -> minus_minus(nat,minus_minus(nat,M,K_1),minus_minus(nat,N,K_1)) = minus_minus(nat,M,N)))) # label(fact_634_Nat_Odiff__diff__eq) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 237 (all Xa all Ya all P_1 ((hBOOL(P_1) -> nat_1(if(int,P_1,Xa,Ya)) = nat_1(Xa)) & (-hBOOL(P_1) -> nat_1(if(int,P_1,Xa,Ya)) = nat_1(Ya)))) # label(fact_268_nat__if__cong) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 238 (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]. 1.49/1.86 239 (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]. 1.49/1.86 240 (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),times_times(nat,K,Ma)),times_times(nat,K,Na))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na))))) # label(fact_982_nat__mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 241 (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) <-> zero_zero(X_a) = ti(X_a,Ya) & ti(X_a,Xa) = zero_zero(X_a))))) # label(fact_2_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 242 (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]. 1.49/1.86 243 (all Ma all K all Na (times_times(nat,Na,K) = times_times(nat,Ma,K) <-> Ma = Na | K = zero_zero(nat))) # label(fact_598_mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 244 (all X all Y (X != Y -> (-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]. 1.49/1.86 245 (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]. 1.49/1.86 246 (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)) -> zero_zero(int) = ti(int,X))))) # label(fact_824_Int2_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 247 (all B_1_1 ti(nat,nat_1(B_1_1)) = nat_1(B_1_1)) # label(tsy_c_Int_Onat_res) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 248 (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]. 1.49/1.86 249 (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]. 1.49/1.86 250 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B_1 all C plus_plus(X_a,plus_plus(X_a,A_1,B_1),C) = plus_plus(X_a,plus_plus(X_a,A_1,C),B_1)))) # label(fact_205_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 251 (all Z all X all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),Z)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z),X)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Z))))) # label(fact_847_dvd_Oless__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 252 (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]. 1.49/1.86 253 (all X_a (ring(X_a) -> (all A_2 all E_1 all C_1 all B all D (ti(X_a,D) = plus_plus(X_a,times_times(X_a,minus_minus(X_a,A_2,B),E_1),C_1) <-> 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_286_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 254 (all B_1_1 nat_1(B_1_1) = nat_1(ti(int,B_1_1))) # label(tsy_c_Int_Onat_arg1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 255 (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,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),plus_plus(nat,times_times(nat,minus_minus(nat,I_1,J),U),Ma)),Na))))) # label(fact_978_nat__less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 256 (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]. 1.49/1.86 257 (all B_1_1 ti(fun(int,bool),quadRes(B_1_1)) = quadRes(B_1_1)) # label(tsy_c_Residues_OQuadRes_res) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 258 (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]. 1.49/1.86 259 (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]. 1.49/1.86 260 (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]. 1.49/1.86 261 (all M hBOOL(hAPP(int,bool,zcong(M,zero_zero(int)),M))) # label(fact_816_zcong__id) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 262 (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]. 1.49/1.86 263 (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]. 1.49/1.86 264 (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]. 1.49/1.86 265 (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]. 1.49/1.86 266 (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]. 1.49/1.86 267 (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]. 1.49/1.86 268 (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]. 1.49/1.86 269 (all Ma all Ta all K (zero_zero(int) != ti(int,K) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Ma),Ta)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),times_times(int,K,Ma)),times_times(int,K,Ta)))))) # label(fact_907_zdvd__mono) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 270 (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]. 1.49/1.86 271 (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,B_1),plus_plus(X_a,C,D_1)) = plus_plus(X_a,plus_plus(X_a,A_1,C),plus_plus(X_a,B_1,D_1))))) # label(fact_202_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 272 (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]. 1.49/1.86 273 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K1)),bit1(K2))))) # label(fact_506_less__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 274 (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]. 1.49/1.86 275 (all X_a ti(X_a,undefined(X_a)) = undefined(X_a)) # label(tsy_c_HOL_Oundefined_res) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 276 (all Xa all Na (Na = zero_zero(nat) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Xa)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,power_power(nat,Xa),Na))))) # label(fact_147_zero__less__power__nat__eq) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 277 (all X_a (number_semiring(X_a) -> number_number_of(X_a,bit1(pls)) = one_one(X_a))) # label(fact_101_semiring__numeral__1__eq__1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 278 (all X_a (linordered_idom(X_a) -> (all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,Xa),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,X_a,power_power(X_a,Ya),number_number_of(nat,bit0(bit1(pls))))))) <-> zero_zero(X_a) != ti(X_a,Xa) | zero_zero(X_a) != ti(X_a,Ya))))) # label(fact_97_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 279 (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]. 1.49/1.86 280 (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),Xa),Ya)) <-> 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)))))) # label(fact_588_real__mult__le__cancel__iff2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 281 (all A_1 all B_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1)) -> zero_zero(nat) = B_1 | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A_1),B_1)))) # label(fact_804_divides__ge) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 282 (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)))) # label(fact_253_zle__linear) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 283 (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]. 1.49/1.86 284 (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]. 1.49/1.86 285 (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]. 1.49/1.86 286 (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]. 1.49/1.86 287 (all X all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) -> -(-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X))))) # label(fact_844_dvd_Oless__asym) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 288 (all Xa all Ya (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),Ya),Xa)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)))) # label(fact_870_dvd_Ole__less) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 289 (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]. 1.49/1.86 290 (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]. 1.49/1.86 291 (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]. 1.49/1.86 292 (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]. 1.49/1.86 293 (all Z plus_plus(int,zero_zero(int),Z) = ti(int,Z)) # label(fact_80_zadd__0) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 294 (all B_1_1 all B_2_1 all X_a (power(X_a) -> ti(X_a,times_times(X_a,B_1_1,B_2_1)) = times_times(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_1_res) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 295 (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]. 1.49/1.86 296 (all Ma all Na (one_one(nat) = times_times(nat,Ma,Na) <-> Ma = one_one(nat) & Na = one_one(nat))) # label(fact_628_nat__1__eq__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 297 (all N all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N),M)) -> M = zero_zero(nat) | 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 = N)) # label(fact_838_divides__cases) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 298 (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]. 1.49/1.86 299 (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]. 1.49/1.86 300 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),bit0(K))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),K)))) # label(fact_55_rel__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 301 (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]. 1.49/1.86 302 (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]. 1.49/1.86 303 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> abs_abs(X_a,number_number_of(X_a,min)) = one_one(X_a))) # label(fact_929_abs__minus__one) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 304 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),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]. 1.49/1.86 305 (all M times_times(nat,M,zero_zero(nat)) = zero_zero(nat)) # label(fact_595_mult__0__right) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 306 (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),B_1),zero_zero(X_a))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),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_269_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 307 (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) = ti(X_a,if(X_a,B_1_1,B_2_1,B_3))) # label(tsy_c_If_res) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 308 (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]. 1.49/1.86 309 (all B_1_1 all B_2_1 all X_a (cancel_semigroup_add(X_a) -> plus_plus(X_a,B_1_1,B_2_1) = plus_plus(X_a,ti(X_a,B_1_1),B_2_1))) # label(tsy_c_Groups_Oplus__class_Oplus_2_arg1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 310 (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]. 1.49/1.86 311 (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]. 1.49/1.86 312 (all X_a (linordered_semidom(X_a) -> (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),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))))))) # label(fact_266_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 313 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all N one_one(X_a) = abs_abs(X_a,hAPP(nat,X_a,power_power(X_a,number_number_of(X_a,min)),N))))) # label(fact_958_abs__power__minus__one) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 314 (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]. 1.49/1.86 315 (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]. 1.49/1.86 316 (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]. 1.49/1.86 317 (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]. 1.49/1.86 318 (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]. 1.49/1.86 319 (all X_a (comm_semiring_1(X_a) -> (all X all Y all Z plus_plus(X_a,times_times(X_a,X,Y),times_times(X_a,X,Z)) = times_times(X_a,X,plus_plus(X_a,Y,Z))))) # label(fact_345_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 320 (all X_a (monoid_mult(X_a) -> (all A_1 times_times(X_a,one_one(X_a),A_1) = ti(X_a,A_1)))) # label(fact_352_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 321 (all K (ti(int,K) = pls <-> pls = bit0(K))) # label(fact_70_rel__simps_I44_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 322 (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]. 1.49/1.86 323 (all M all N minus_minus(nat,plus_plus(nat,M,N),N) = M) # label(fact_365_diff__add__inverse2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 324 (all Y all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y)) -> plus_plus(nat,nat_1(X),nat_1(Y)) = nat_1(plus_plus(int,X,Y))))) # label(fact_396_Nat__Transfer_Otransfer__nat__int__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 325 (all Z plus_plus(int,Z,zero_zero(int)) = ti(int,Z)) # label(fact_79_zadd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 326 (all X_a (one(X_a) -> (all Xa (ti(X_a,Xa) = one_one(X_a) <-> one_one(X_a) = ti(X_a,Xa))))) # label(fact_211_one__reorient) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 327 (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]. 1.49/1.86 328 (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]. 1.49/1.86 329 (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]. 1.49/1.86 330 (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]. 1.49/1.86 331 (all B_1_1 all B_2_1 all X_a (power(X_a) -> hAPP(nat,X_a,B_1_1,B_2_1) = ti(X_a,hAPP(nat,X_a,B_1_1,B_2_1)))) # label(tsy_c_hAPP_4_res) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 332 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) | Ma = Na)) # label(fact_608_le__eq__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 333 (all Xa all Ya all P_1 ((-hBOOL(P_1) -> hAPP(nat,int,semiring_1_of_nat(int),if(nat,P_1,Xa,Ya)) = hAPP(nat,int,semiring_1_of_nat(int),Ya)) & (hBOOL(P_1) -> hAPP(nat,int,semiring_1_of_nat(int),if(nat,P_1,Xa,Ya)) = hAPP(nat,int,semiring_1_of_nat(int),Xa)))) # label(fact_160_int__if__cong) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 334 (all X_a (ab_group_add(X_a) -> (all A_1 all C all B_1 all D_1 minus_minus(X_a,plus_plus(X_a,A_1,C),plus_plus(X_a,B_1,D_1)) = plus_plus(X_a,minus_minus(X_a,A_1,B_1),minus_minus(X_a,C,D_1))))) # label(fact_325_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 335 (all X_a (semiring_1(X_a) -> zero_zero(X_a) = hAPP(nat,X_a,semiring_1_of_nat(X_a),zero_zero(nat)))) # label(fact_188_of__nat__0) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 336 (all X_a (monoid_mult(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(nat,X_a,power_power(X_a,A_1),one_one(nat))))) # label(fact_157_power__one__right) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 337 (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]. 1.49/1.86 338 (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]. 1.49/1.86 339 (all P_1 ((exists X_1 hBOOL(hAPP(int,bool,P_1,hAPP(nat,int,semiring_1_of_nat(int),X_1)))) <-> (exists X_1 (hBOOL(hAPP(int,bool,P_1,X_1)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)))))) # label(fact_473_transfer__int__nat__quantifiers_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 340 (all Ma all Na (hAPP(nat,int,semiring_1_of_nat(int),Ma) = hAPP(nat,int,semiring_1_of_nat(int),Na) <-> Ma = Na)) # label(fact_48_int__int__eq) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 341 (all X_a (linord219039673up_add(X_a) -> (all A_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),plus_plus(X_a,A_2,A_2))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_2)))))) # label(fact_432_zero__le__double__add__iff__zero__le__single__add) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 342 (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),A_1),B_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B_1),A_1))))) # label(fact_867_dvd_Oneq__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 343 (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]. 1.49/1.86 344 (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(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]. 1.49/1.86 345 (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]. 1.49/1.86 346 (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]. 1.49/1.86 347 (all K_1 succ(K_1) = plus_plus(int,K_1,one_one(int))) # label(fact_519_succ__def) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 348 (all Z_1 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Z_1),hAPP(nat,int,semiring_1_of_nat(int),Ma))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),nat_1(abs_abs(int,Z_1))),Ma)))) # label(fact_963_dvd__int__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 349 (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]. 1.49/1.86 350 (all B_1_1 all B_2_1 all X_a (dvd(X_a) -> hAPP(X_a,fun(X_a,bool),B_1_1,B_2_1) = hAPP(X_a,fun(X_a,bool),ti(fun(X_a,fun(X_a,bool)),B_1_1),B_2_1))) # label(tsy_c_hAPP_2_arg1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 351 (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]. 1.49/1.86 352 (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]. 1.49/1.86 353 (all B_1_1 all B_2_1 all X_a (dvd(X_a) -> times_times(X_a,B_1_1,B_2_1) = ti(X_a,times_times(X_a,B_1_1,B_2_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_0_res) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 354 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K)),min)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),min)))) # label(fact_760_rel__simps_I28_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 355 (all X_a (number_ring(X_a) -> (all A_1 times_times(X_a,number_number_of(X_a,bit1(pls)),A_1) = ti(X_a,A_1)))) # label(fact_498_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 356 (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,B_1) = abs_abs(int,A_1)))) # label(fact_922_zdvd__antisym__abs) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 357 (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]. 1.49/1.86 358 (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]. 1.49/1.86 359 (all Ma all K all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),Na)) | K = zero_zero(nat) <-> 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]. 1.49/1.86 360 (all K_1 bit0(K_1) != min) # label(fact_766_rel__simps_I45_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 361 (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]. 1.49/1.86 362 (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]. 1.49/1.86 363 (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]. 1.49/1.86 364 (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]. 1.49/1.86 365 (all X_a (number_ring(X_a) & linordered_idom(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]. 1.49/1.86 366 (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]. 1.49/1.86 367 (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]. 1.49/1.86 368 (all C all A_1 all B_1 (A_1 = B_1 -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B_1),C)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),C),B_1)) -> 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]. 1.49/1.86 369 (all A_1 all P_2 (hBOOL(hAPP(int,bool,zprime,P_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,zcong(times_times(int,A_1,A_1),one_one(int)),P_2)) -> hBOOL(hAPP(int,bool,zcong(A_1,minus_minus(int,P_2,one_one(int))),P_2)) | hBOOL(hAPP(int,bool,zcong(A_1,one_one(int)),P_2)))))) # label(fact_902_zcong__square) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 370 (all X_a (ordere142940540dd_abs(X_a) -> (all A_2 (ti(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),abs_abs(X_a,A_2)),zero_zero(X_a))))))) # label(fact_948_abs__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 371 (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]. 1.49/1.86 372 (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]. 1.49/1.86 373 (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]. 1.49/1.86 374 (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,A_2)),Ma)) <-> hBOOL(hAPP(int,bool,zcong(C_1,times_times(int,D,B)),Ma))))) # label(fact_823_zcong__zmult__prop2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.86 375 (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]. 1.49/1.86 376 (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]. 1.49/1.87 377 (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]. 1.49/1.87 378 (all X_b (number(X_b) & ring(X_b) -> (all A_1 all B_1 all V_1 times_times(X_b,minus_minus(X_b,A_1,B_1),number_number_of(X_b,V_1)) = minus_minus(X_b,times_times(X_b,A_1,number_number_of(X_b,V_1)),times_times(X_b,B_1,number_number_of(X_b,V_1)))))) # label(fact_263_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 379 (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]. 1.49/1.87 380 (all N (one_one(int) = hAPP(nat,int,power_power(int,number_number_of(int,min)),N) | number_number_of(int,min) = hAPP(nat,int,power_power(int,number_number_of(int,min)),N))) # label(fact_876_neg__one__power) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 381 (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]. 1.49/1.87 382 (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]. 1.49/1.87 383 (all X_a (power(X_a) -> (all M all N ((zero_zero(nat) = N -> one_one(X_a) = hAPP(nat,X_a,power_power(X_a,M),N)) & (zero_zero(nat) != N -> times_times(X_a,M,hAPP(nat,X_a,power_power(X_a,M),minus_minus(nat,N,one_one(nat)))) = hAPP(nat,X_a,power_power(X_a,M),N)))))) # label(fact_558_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 384 (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]. 1.49/1.87 385 (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]. 1.49/1.87 386 (all M M = minus_minus(nat,M,zero_zero(nat))) # label(fact_337_minus__nat_Odiff__0) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 387 (all W all Z1 all Z2 plus_plus(int,times_times(int,W,Z1),times_times(int,W,Z2)) = times_times(int,W,plus_plus(int,Z1,Z2))) # label(fact_381_zadd__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 388 (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]. 1.49/1.87 389 (all X hAPP(nat,real,power_power(real,times_times(real,number_number_of(real,bit0(bit1(pls))),X)),number_number_of(nat,bit0(bit1(pls)))) = times_times(real,number_number_of(real,bit0(bit0(bit1(pls)))),hAPP(nat,real,power_power(real,X),number_number_of(nat,bit0(bit1(pls)))))) # label(fact_695_four__x__squared) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 390 (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]. 1.49/1.87 391 (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),bit1(K)),min)))) # label(fact_756_rel__simps_I30_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 392 (all X_a (semiring_1(X_a) -> one_one(X_a) = hAPP(nat,X_a,power_power(X_a,one_one(X_a)),number_number_of(nat,bit0(bit1(pls)))))) # label(fact_3_one__power2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 393 (all K all L_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L_1)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K)),bit1(L_1))))) # label(fact_462_rel__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 394 (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]. 1.49/1.87 395 (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]. 1.49/1.87 396 (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]. 1.49/1.87 397 (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]. 1.49/1.87 398 (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]. 1.49/1.87 399 (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) <-> Ma = Na)))) # label(fact_158_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 400 (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]. 1.49/1.87 401 (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]. 1.49/1.87 402 (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]. 1.49/1.87 403 -(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]. 1.49/1.87 404 (all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),number_number_of(int,bit0(bit1(pls))))) -> ti(int,X) = zero_zero(int) | ti(int,X) = one_one(int)))) # label(fact_554_int__pos__lt__two__imp__zero__or__one) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 405 (all X_b (ring(X_b) & number(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]. 1.49/1.87 406 (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]. 1.49/1.87 407 (all Va (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Va),pls)) <-> zero_zero(nat) = number_number_of(nat,Va))) # label(fact_501_eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 408 (all B_1_1 all B_2_1 all X_a (no_zero_divisors(X_a) -> times_times(X_a,B_1_1,B_2_1) = times_times(X_a,ti(X_a,B_1_1),B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_3_arg1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 409 (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]. 1.49/1.87 410 (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]. 1.49/1.87 411 (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]. 1.49/1.87 412 (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]. 1.49/1.87 413 (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]. 1.49/1.87 414 (all P_1 all I_1 (hBOOL(hAPP(nat,bool,P_1,nat_1(I_1))) <-> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),zero_zero(int))) -> hBOOL(hAPP(nat,bool,P_1,zero_zero(nat)))) & (all N_1 (ti(int,I_1) = hAPP(nat,int,semiring_1_of_nat(int),N_1) -> hBOOL(hAPP(nat,bool,P_1,N_1)))))) # label(fact_523_split__nat) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 415 (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]. 1.49/1.87 416 (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]. 1.49/1.87 417 (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]. 1.49/1.87 418 (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]. 1.49/1.87 419 (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]. 1.49/1.87 420 (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]. 1.49/1.87 421 (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]. 1.49/1.87 422 (all Ma all Na (minus_minus(nat,Ma,Na) = zero_zero(nat) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)))) # label(fact_658_diff__is__0__eq) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 423 (all X_a (comm_semiring_1(X_a) -> (all A_1 all C all D_1 plus_plus(X_a,C,plus_plus(X_a,A_1,D_1)) = plus_plus(X_a,A_1,plus_plus(X_a,C,D_1))))) # label(fact_209_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 424 (all Z1 all Z2 all W minus_minus(int,times_times(int,Z1,W),times_times(int,Z2,W)) = times_times(int,minus_minus(int,Z1,Z2),W)) # label(fact_592_zdiff__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 425 (all X_a (linord219039673up_add(X_a) -> (all A_2 (ti(X_a,A_2) = zero_zero(X_a) <-> zero_zero(X_a) = plus_plus(X_a,A_2,A_2))))) # label(fact_65_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 426 (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)) & (all I (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),K_2)) -> -hBOOL(hAPP(nat,bool,P_1,I)))) & hBOOL(hAPP(nat,bool,P_1,plus_plus(nat,K_2,one_one(nat))))))))) # label(fact_976_ex__least__nat__less) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 427 (all X_a (comm_semiring_1(X_a) -> (all X all Y all Q times_times(X_a,hAPP(nat,X_a,power_power(X_a,X),Q),hAPP(nat,X_a,power_power(X_a,Y),Q)) = hAPP(nat,X_a,power_power(X_a,times_times(X_a,X,Y)),Q)))) # label(fact_364_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 428 (all I_2 all J_1 all K_1 minus_minus(nat,minus_minus(nat,I_2,J_1),K_1) = minus_minus(nat,I_2,plus_plus(nat,J_1,K_1))) # label(fact_367_diff__diff__left) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 429 (all A_1 all B_1 hAPP(nat,int,power_power(int,plus_plus(int,A_1,B_1)),number_number_of(nat,bit1(bit1(pls)))) = plus_plus(int,plus_plus(int,plus_plus(int,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit1(bit1(pls)))),times_times(int,times_times(int,number_number_of(int,bit1(bit1(pls))),hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls))))),B_1)),times_times(int,times_times(int,number_number_of(int,bit1(bit1(pls))),A_1),hAPP(nat,int,power_power(int,B_1),number_number_of(nat,bit0(bit1(pls)))))),hAPP(nat,int,power_power(int,B_1),number_number_of(nat,bit1(bit1(pls)))))) # label(fact_234_zadd__power3) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 430 (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]. 1.49/1.87 431 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> abs_abs(X_a,A_1) = ti(X_a,A_1))))) # label(fact_944_abs__of__pos) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 432 (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]. 1.49/1.87 433 (all K_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K_1)) -> (exists N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) & ti(int,K_1) = hAPP(nat,int,semiring_1_of_nat(int),N_1))))) # label(fact_190_zero__less__imp__eq__int) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 434 (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]. 1.49/1.87 435 (all N (N != zero_zero(nat) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)))) # label(fact_139_gr0I) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 436 (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]. 1.49/1.87 437 (all N all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N),M)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),M),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),N),M)))) # label(fact_738_zdvd__bounds) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 438 (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]. 1.49/1.87 439 (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]. 1.49/1.87 440 (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)) -> one_one(int) = ti(int,A_1) | minus_minus(int,P_2,one_one(int)) = ti(int,A_1)))))) # label(fact_880_zcong__square__zless) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 441 (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)) -> hAPP(nat,X_a,power_power(X_a,X),N) = times_times(X_a,hAPP(nat,X_a,power_power(X_a,X),minus_minus(nat,N,one_one(nat))),X))))) # label(fact_405_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 442 (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]. 1.49/1.87 443 (all Na all Ma (zero_zero(int) != ti(int,Ma) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),times_times(int,Ma,Na)),Ma)) <-> abs_abs(int,Na) = one_one(int)))) # label(fact_961_zdvd__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 444 (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]. 1.49/1.87 445 (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]. 1.49/1.87 446 (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]. 1.49/1.87 447 (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]. 1.49/1.87 448 (all I_1 all J all K (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,I_1,K)),J)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),minus_minus(nat,J,K))))) # label(fact_455_less__diff__conv) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 449 (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]. 1.49/1.87 450 (all B_1_1 all B_2_1 all X_a (dvd(X_a) -> hAPP(X_a,bool,B_1_1,B_2_1) = hAPP(X_a,bool,ti(fun(X_a,bool),B_1_1),B_2_1))) # label(tsy_c_hAPP_3_arg1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 451 (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]. 1.49/1.87 452 (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]. 1.49/1.87 453 (all B_1_1 all B_2_1 all X_a (monoid_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_0_res) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 454 (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]. 1.49/1.87 455 (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),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]. 1.49/1.87 456 (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]. 1.49/1.87 457 (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]. 1.49/1.87 458 (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]. 1.49/1.87 459 (all A_2 all B all Ma (hBOOL(hAPP(int,bool,zcong(A_2,B),Ma)) <-> hBOOL(hAPP(int,bool,zcong(B,A_2),Ma)))) # label(fact_883_zcong__sym) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 460 (all X_a (zero(X_a) -> (all Xa (ti(X_a,Xa) = zero_zero(X_a) <-> ti(X_a,Xa) = zero_zero(X_a))))) # label(fact_198_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 461 (all Xa all 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)))) <-> number_number_of(nat,Wa) = zero_zero(nat) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Xa)))) # label(fact_149_zero__less__power__nat__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 462 (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]. 1.49/1.87 463 (all X_a (number_ring(X_a) -> (all V_1 all W minus_minus(X_a,number_number_of(X_a,V_1),number_number_of(X_a,W)) = number_number_of(X_a,minus_minus(int,V_1,W))))) # label(fact_264_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 464 (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]. 1.49/1.87 465 (all K_1 bit1(K_1) = succ(bit0(K_1))) # label(fact_514_succ__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 466 (all L_1 (ti(int,L_1) = pls <-> pls = bit0(L_1))) # label(fact_71_rel__simps_I38_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 467 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) <-> (exists K_2 Na = plus_plus(nat,Ma,K_2)))) # label(fact_622_le__iff__add) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 468 (all X_a (linordered_idom(X_a) -> (all Ma all K (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Ma),K)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Ma),abs_abs(X_a,K))))))) # label(fact_921_dvd__abs__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 469 (all K_1 minus_minus(int,K_1,min) = succ(K_1)) # label(fact_777_diff__bin__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 470 (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]. 1.49/1.87 471 (all X_a (comm_semiring_1(X_a) -> (all A_1 all C plus_plus(X_a,A_1,C) = plus_plus(X_a,C,A_1)))) # label(fact_210_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 472 (all Xa all Ya (Ya = Xa | 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)))) # label(fact_611_less__eq__real__def) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 473 (all M all N (one_one(int) = times_times(int,M,N) -> one_one(int) = ti(int,M) | number_number_of(int,min) = ti(int,M))) # label(fact_786_pos__zmult__eq__1__iff__lemma) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 474 (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]. 1.49/1.87 475 (all B_1_1 all B_2_1 all X_a (semiring(X_a) -> times_times(X_a,B_1_1,B_2_1) = ti(X_a,times_times(X_a,B_1_1,B_2_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_2_res) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 476 (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]. 1.49/1.87 477 (all X_a (comm_monoid_mult(X_a) -> (all A_1 times_times(X_a,one_one(X_a),A_1) = ti(X_a,A_1)))) # label(fact_350_mult__1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 478 (all Ma all Va (number_number_of(int,Va) = hAPP(nat,int,semiring_1_of_nat(int),Ma) <-> Ma = nat_1(number_number_of(int,Va)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,Va))))) # label(fact_398_int__eq__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 479 (all M zero_zero(nat) = minus_minus(nat,M,M)) # label(fact_336_diff__self__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 480 (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]. 1.49/1.87 481 (all X_a (power(X_a) & mult_zero(X_a) & zero_neq_one(X_a) & no_zero_divisors(X_a) -> (all A_2 all Na (hAPP(nat,X_a,power_power(X_a,A_2),Na) = zero_zero(X_a) <-> Na != zero_zero(nat) & zero_zero(X_a) = ti(X_a,A_2))))) # label(fact_185_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 482 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),min)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K)),min)))) # label(fact_751_rel__simps_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 483 (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]. 1.49/1.87 484 (all N times_times(nat,zero_zero(nat),N) = zero_zero(nat)) # label(fact_594_mult__0) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 485 (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]. 1.49/1.87 486 (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]. 1.49/1.87 487 (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]. 1.49/1.87 488 (all K all L_1 (bit1(K) = bit1(L_1) <-> ti(int,K) = ti(int,L_1))) # label(fact_42_rel__simps_I51_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 489 (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]. 1.49/1.87 490 (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]. 1.49/1.87 491 (all B_1_1 all B_2_1 all X_a (dvd(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_0_arg2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 492 (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]. 1.49/1.87 493 (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]. 1.49/1.87 494 (all L_1 (ti(int,L_1) = min <-> min = bit1(L_1))) # label(fact_770_rel__simps_I43_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 495 (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]. 1.49/1.87 496 (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]. 1.49/1.87 497 (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]. 1.49/1.87 498 (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]. 1.49/1.87 499 (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]. 1.49/1.87 500 (all X_a (one(X_a) -> one_one(X_a) = ti(X_a,one_one(X_a)))) # label(tsy_c_Groups_Oone__class_Oone_res) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 501 (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]. 1.49/1.87 502 (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]. 1.49/1.87 503 (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]. 1.49/1.87 504 (all V_2 all K_1 all V_1 ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls)) -> times_times(nat,number_number_of(nat,times_times(int,V_1,V_2)),K_1) = times_times(nat,number_number_of(nat,V_1),times_times(nat,number_number_of(nat,V_2),K_1))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls)) -> 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]. 1.49/1.87 505 (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]. 1.49/1.87 506 (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]. 1.49/1.87 507 (all L minus_minus(int,pls,bit1(L)) = bit1(minus_minus(int,min,L))) # label(fact_782_diff__bin__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 508 (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]. 1.49/1.87 509 (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]. 1.49/1.87 510 (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]. 1.49/1.87 511 (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]. 1.49/1.87 512 (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),abs_abs(X_a,Ma)),K)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Ma),K)))))) # label(fact_920_abs__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 513 (all X_a (no_zero_divisors(X_a) -> (all B_1 all A_1 (ti(X_a,A_1) != zero_zero(X_a) -> (zero_zero(X_a) != ti(X_a,B_1) -> zero_zero(X_a) != times_times(X_a,A_1,B_1)))))) # label(fact_331_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 514 (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]. 1.49/1.87 515 (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]. 1.49/1.87 516 (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(nat),Xa),Ya)) <-> 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)))))))) # label(fact_182_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 517 (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]. 1.49/1.87 518 (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]. 1.49/1.87 519 (all Z plus_plus(nat,Z,Z) = times_times(nat,Z,number_number_of(nat,bit0(bit1(pls))))) # label(fact_692_nat__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 520 (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]. 1.49/1.87 521 (all X_a (ordered_ab_group_add(X_a) -> (all A_2 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),minus_minus(X_a,A_2,B)),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B)))))) # label(fact_238_le__iff__diff__le__0) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 522 (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]. 1.49/1.87 523 (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]. 1.49/1.87 524 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),bit1(K))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),K)))) # label(fact_755_rel__simps_I26_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 525 (all X_a (ordered_ring_abs(X_a) -> (all B_1 all A_1 ((hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))) & (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),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))) -> 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]. 1.49/1.87 526 (all K all Ma all Na (Ma = Na | zero_zero(nat) = K <-> times_times(nat,K,Ma) = times_times(nat,K,Na))) # label(fact_597_mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 527 (all Wa all Z_1 (ti(int,Z_1) = ti(int,Wa) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z_1)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),plus_plus(int,Z_1,one_one(int)))))) # label(fact_60_zless__add1__eq) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 528 (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]. 1.49/1.87 529 (all X all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) -> 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_864_dvd_Ole__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 530 (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]. 1.49/1.87 531 (all M plus_plus(nat,M,zero_zero(nat)) = M) # label(fact_155_Nat_Oadd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 532 (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]. 1.49/1.87 533 (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]. 1.49/1.87 534 (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]. 1.49/1.87 535 (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]. 1.49/1.87 536 (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]. 1.49/1.87 537 (all A_1 all P_2 ((-hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_2)) -> (hBOOL(hAPP(int,bool,quadRes(P_2),A_1)) -> one_one(int) = legendre(A_1,P_2)) & (-hBOOL(hAPP(int,bool,quadRes(P_2),A_1)) -> number_number_of(int,min) = legendre(A_1,P_2))) & (hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_2)) -> zero_zero(int) = legendre(A_1,P_2)))) # label(fact_877_Legendre__def) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 538 (all Va (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),number_number_of(nat,Va))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),Va)))) # label(fact_161_less__0__number__of) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 539 (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,ti(X_a,B_1_1),B_2_1))) # label(tsy_c_Groups_Oplus__class_Oplus_1_arg1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 540 (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]. 1.49/1.87 541 (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]. 1.49/1.87 542 (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]. 1.49/1.87 543 (all X_a (semiri456707255roduct(X_a) -> (all B all A_2 (zero_zero(X_a) = ti(X_a,A_2) <-> ti(X_a,B) = plus_plus(X_a,B,A_2))))) # label(fact_212_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 544 (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]. 1.49/1.87 545 (all K_1 ti(int,K_1) = plus_plus(int,pls,K_1)) # label(fact_76_add__Pls) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 546 (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)) -> times_times(nat,nat_1(X),nat_1(Y)) = nat_1(times_times(int,X,Y))))) # label(fact_686_Nat__Transfer_Otransfer__nat__int__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 547 (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]. 1.49/1.87 548 (all X_a (cancel146912293up_add(X_a) -> (all A_1 all B_1 all C (plus_plus(X_a,A_1,B_1) = plus_plus(X_a,A_1,C) -> ti(X_a,C) = ti(X_a,B_1))))) # label(fact_200_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 549 (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]. 1.49/1.87 550 (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]. 1.49/1.87 551 (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),B_1),zero_zero(X_a))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a))) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),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_270_split__mult__pos__le) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 552 (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]. 1.49/1.87 553 (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,K_1),J_1) = minus_minus(nat,I_2,minus_minus(nat,J_1,K_1)))) # label(fact_666_diff__diff__right) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 554 (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]. 1.49/1.87 555 (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]. 1.49/1.87 556 (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]. 1.49/1.87 557 (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]. 1.49/1.87 558 (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]. 1.49/1.87 559 (all A_1 all B_1 minus_minus(int,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,int,power_power(int,B_1),number_number_of(nat,bit0(bit1(pls))))) = times_times(int,plus_plus(int,A_1,B_1),minus_minus(int,A_1,B_1))) # label(fact_698_zspecial__product) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 560 (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]. 1.49/1.87 561 (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]. 1.49/1.87 562 (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),bit0(K)),bit0(L_1))))) # label(fact_34_rel__simps_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 563 (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]. 1.49/1.87 564 (all N N = nat_1(hAPP(nat,int,semiring_1_of_nat(int),N))) # label(fact_346_nat__int) # label(axiom) # label(non_clause). [assumption]. 1.49/1.87 565 (all M all K_1 all N minus_minus(nat,M,N) = minus_minus(nat,plus_plus(nat,M,K_1),plus_plus(nat,N,K_1))) # label(fact_369_diff__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 566 (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]. 1.49/1.88 567 (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]. 1.49/1.88 568 (all Xa all Ya (plus_plus(real,times_times(real,Xa,Xa),times_times(real,Ya,Ya)) = zero_zero(real) <-> Xa = zero_zero(real) & Ya = zero_zero(real))) # label(fact_660_real__two__squares__add__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 569 (all Z_1 (zero_zero(int) = ti(int,Z_1) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),abs_abs(int,Z_1)),one_one(int))))) # label(fact_956_zabs__less__one__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 570 (all X_a (number_ring(X_a) -> (all W number_number_of(X_a,bit1(W)) = plus_plus(X_a,plus_plus(X_a,one_one(X_a),number_number_of(X_a,W)),number_number_of(X_a,W))))) # label(fact_100_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 571 (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]. 1.49/1.88 572 (all B_1_1 all X_a (ordere142940540dd_abs(X_a) -> abs_abs(X_a,B_1_1) = abs_abs(X_a,ti(X_a,B_1_1)))) # label(tsy_c_Groups_Oabs__class_Oabs_arg1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 573 (all Wa all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,Wa,one_one(int))),Z_1)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z_1)))) # label(fact_517_add1__zle__eq) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 574 (all L minus_minus(int,min,bit1(L)) = bit0(minus_minus(int,min,L))) # label(fact_783_diff__bin__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 575 (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]. 1.49/1.88 576 (all B_1_1 all B_2_1 all X_a (monoid_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_0_arg2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 577 (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]. 1.49/1.88 578 (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]. 1.49/1.88 579 (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]. 1.49/1.88 580 (all A_2 all B all C_1 (zero_zero(real) != C_1 -> (times_times(real,A_2,C_1) = times_times(real,B,C_1) <-> A_2 = B))) # label(fact_604_real__mult__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 581 (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]. 1.49/1.88 582 (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,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),L_1),plus_plus(X_a,X_1,zero_zero(X_a)))) & hBOOL(hAPP(X_a,bool,P_1,X_1)))))))) # label(fact_908_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 583 (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]. 1.49/1.88 584 (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]. 1.49/1.88 585 (all X_a (monoid_mult(X_a) -> (all A_1 hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,bit0(bit1(pls)))) = times_times(X_a,A_1,A_1)))) # label(fact_540_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 586 (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]. 1.49/1.88 587 (all K_1 all L bit1(L) != bit0(K_1)) # label(fact_69_rel__simps_I49_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 588 (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]. 1.49/1.88 589 (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]. 1.49/1.88 590 (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]. 1.49/1.88 591 (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]. 1.49/1.88 592 (all K_1 pls != bit1(K_1)) # label(fact_66_rel__simps_I46_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 593 (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]. 1.49/1.88 594 (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]. 1.49/1.88 595 (all Xa all Ya (hAPP(nat,int,semiring_1_of_nat(int),Xa) = hAPP(nat,int,semiring_1_of_nat(int),Ya) <-> Xa = Ya)) # label(fact_159_Nat__Transfer_Otransfer__int__nat__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 596 (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]. 1.49/1.88 597 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),Ma)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) <-> Ma != Na)) # label(fact_121_nat__neq__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 598 (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]. 1.49/1.88 599 (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]. 1.49/1.88 600 (all X_a (group_add(X_a) -> (all A_1 minus_minus(X_a,A_1,A_1) = zero_zero(X_a)))) # label(fact_322_diff__self) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 601 (all Xa (-hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),times_times(real,Xa,Xa))) <-> zero_zero(real) = Xa)) # label(fact_653_not__real__square__gt__zero) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 602 (all X_a (comm_monoid_add(X_a) -> (all A_1 plus_plus(X_a,A_1,zero_zero(X_a)) = ti(X_a,A_1)))) # label(fact_213_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 603 (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]. 1.49/1.88 604 (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]. 1.49/1.88 605 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 all B_1 abs_abs(X_a,minus_minus(X_a,A_1,B_1)) = abs_abs(X_a,minus_minus(X_a,B_1,A_1))))) # label(fact_950_abs__minus__commute) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 606 (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),B),A_2)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a))) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),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]. 1.49/1.88 607 (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]. 1.49/1.88 608 (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]. 1.49/1.88 609 (all A_1 all B_1 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))))) = hAPP(nat,int,power_power(int,minus_minus(int,A_1,B_1)),number_number_of(nat,bit0(bit1(pls))))) # label(fact_703_zdiff__power2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 610 (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]. 1.49/1.88 611 (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]. 1.49/1.88 612 (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]. 1.49/1.88 613 (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]. 1.49/1.88 614 (all A_1 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1)) -> (exists X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1)) & (all Y_1 (hAPP(nat,real,power_power(real,Y_1),N) = A_1 & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y_1)) -> X_1 = Y_1)) & hAPP(nat,real,power_power(real,X_1),N) = A_1))))) # label(fact_229_realpow__pos__nth__unique) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 615 (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]. 1.49/1.88 616 (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]. 1.49/1.88 617 (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),times_times(X_a,C_1,A_2)),times_times(X_a,C_1,B))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B))))))) # label(fact_309_mult__le__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 618 (all K_1 all L minus_minus(int,bit1(K_1),bit0(L)) = bit1(minus_minus(int,K_1,L))) # label(fact_644_diff__bin__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 619 (all X all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) -> Y != X)) # label(fact_850_dvd_Oless__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 620 (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]. 1.49/1.88 621 (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]. 1.49/1.88 622 (all M all N hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,power_power(nat,M),N)) = hAPP(nat,int,power_power(int,hAPP(nat,int,semiring_1_of_nat(int),M)),N)) # label(fact_20_zpower__int) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 623 (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]. 1.49/1.88 624 (all K_1 bit1(K_1) = plus_plus(int,plus_plus(int,one_one(int),K_1),K_1)) # label(fact_92_Bit1__def) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 625 (all X_a (number_ring(X_a) -> (all V_1 all W plus_plus(X_a,number_number_of(X_a,V_1),number_number_of(X_a,W)) = number_number_of(X_a,plus_plus(int,V_1,W))))) # label(fact_89_number__of__add) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 626 (all B_1_1 all B_2_1 all X_a (cancel_semigroup_add(X_a) -> plus_plus(X_a,B_1_1,B_2_1) = plus_plus(X_a,B_1_1,ti(X_a,B_2_1)))) # label(tsy_c_Groups_Oplus__class_Oplus_2_arg2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 627 (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,minus_minus(nat,J,I_1),U),Na) = Ma <-> plus_plus(nat,times_times(nat,I_1,U),Ma) = plus_plus(nat,times_times(nat,J,U),Na)))) # label(fact_992_nat__eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 628 (all X_a (comm_semiring_1(X_a) -> (all X hAPP(nat,X_a,power_power(X_a,X),zero_zero(nat)) = one_one(X_a)))) # label(fact_193_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 629 (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]. 1.49/1.88 630 (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]. 1.49/1.88 631 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),K)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),bit0(K))))) # label(fact_780_rel__simps_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 632 (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)) & nat_1(Z_1) = Ma)) # label(fact_318_int__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 633 (all X_a (semiring_1(X_a) -> (all M all 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)) = hAPP(nat,X_a,semiring_1_of_nat(X_a),times_times(nat,M,N))))) # label(fact_265_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 634 (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(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))))) # label(fact_484_zless__nat__conj) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 635 (all X_a (linordered_idom(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)),hAPP(nat,X_a,power_power(X_a,A_2),number_number_of(nat,bit0(bit1(pls)))))))))) # label(fact_95_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 636 (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]. 1.49/1.88 637 -(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]. 1.49/1.88 638 (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]. 1.49/1.88 639 (all K all Ma 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,K,Ma)),times_times(nat,K,Na))))) # label(fact_984_nat__mult__dvd__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 640 (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]. 1.49/1.88 641 (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]. 1.49/1.88 642 (all X_a (linord219039673up_add(X_a) -> (all A_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,A_2,A_2)),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),zero_zero(X_a))))))) # label(fact_195_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 643 (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]. 1.49/1.88 644 t = one_one(int) -> (exists X_1 exists Y_1 plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)) = 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_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]. 1.49/1.88 645 (all V_1 ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls)) -> plus_plus(nat,one_one(nat),number_number_of(nat,V_1)) = one_one(nat)) & (-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]. 1.49/1.88 646 (all K all Ma all Na (zero_zero(nat) = K | Ma = Na <-> times_times(nat,K,Na) = times_times(nat,K,Ma))) # label(fact_980_nat__mult__eq__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 647 (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) <-> one_one(int) = ti(int,Na) & one_one(int) = ti(int,Ma)))) # label(fact_510_pos__zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 648 (all K all L_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K)),bit1(L_1))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L_1)))) # label(fact_31_rel__simps_I17_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 649 (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]. 1.49/1.88 650 (all X_a (monoid_mult(X_a) -> (all A_1 ti(X_a,A_1) = times_times(X_a,A_1,one_one(X_a))))) # label(fact_349_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 651 (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]. 1.49/1.88 652 (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),number_number_of(int,K)),number_number_of(int,L_1))))) # label(fact_16_less__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 653 (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]. 1.49/1.88 654 (all B_1_1 bit0(B_1_1) = bit0(ti(int,B_1_1))) # label(tsy_c_Int_OBit0_arg1) # label(hypothesis) # label(non_clause). [assumption]. 1.49/1.88 655 (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,ti(X_a,B_1_1),B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_0_arg1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 656 (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]. 1.49/1.88 657 (all X_b (number(X_b) & semiring(X_b) -> (all A_1 all B_1 all V_1 plus_plus(X_b,times_times(X_b,A_1,number_number_of(X_b,V_1)),times_times(X_b,B_1,number_number_of(X_b,V_1))) = times_times(X_b,plus_plus(X_b,A_1,B_1),number_number_of(X_b,V_1))))) # label(fact_444_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 658 (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)) -> one_one(nat) = plus_plus(nat,number_number_of(nat,V_1),one_one(nat))))) # label(fact_232_nat__number__of__add__1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 659 (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,B_1) = ti(X_a,C))))) # label(fact_199_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 660 (all N all P_2 all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M)) -> (hBOOL(hAPP(int,bool,zprime,P_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),times_times(int,M,N))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),N)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),M)))))) # label(fact_900_zprime__zdvd__zmult) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 661 (all A_1 hAPP(nat,int,power_power(int,abs_abs(int,A_1)),number_number_of(nat,bit0(bit1(pls)))) = abs_abs(int,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls)))))) # label(fact_966_abs__power2__distrib) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 662 (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]. 1.49/1.88 663 (all B_1_1 all B_2_1 all X_a (mult_zero(X_a) -> times_times(X_a,ti(X_a,B_1_1),B_2_1) = times_times(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_4_arg1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 664 (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]. 1.49/1.88 665 (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) <-> plus_plus(nat,times_times(nat,minus_minus(nat,I_1,J),U),Ma) = Na))) # label(fact_989_nat__eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 666 (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]. 1.49/1.88 667 (all Ma all Na (Ma = zero_zero(nat) & Na = zero_zero(nat) <-> zero_zero(nat) = plus_plus(nat,Ma,Na))) # label(fact_154_add__is__0) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 668 (all P_1 all Xa all Ya (hBOOL(hAPP(int,bool,P_1,hAPP(nat,int,semiring_1_of_nat(int),minus_minus(nat,Xa,Ya)))) <-> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Xa),Ya)) -> hBOOL(hAPP(int,bool,P_1,zero_zero(int)))) & (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ya),Xa)) -> hBOOL(hAPP(int,bool,P_1,minus_minus(int,hAPP(nat,int,semiring_1_of_nat(int),Xa),hAPP(nat,int,semiring_1_of_nat(int),Ya))))))) # label(fact_903_zdiff__int__split) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 669 (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]. 1.49/1.88 670 (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]. 1.49/1.88 671 (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]. 1.49/1.88 672 (all X_a (cancel_semigroup_add(X_a) -> (all A_1 all B_1 all C (plus_plus(X_a,A_1,C) = plus_plus(X_a,A_1,B_1) -> ti(X_a,C) = ti(X_a,B_1))))) # label(fact_201_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 673 (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]. 1.49/1.88 674 (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]. 1.49/1.88 675 (all I_2 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),N)) -> minus_minus(nat,N,minus_minus(nat,N,I_2)) = I_2)) # label(fact_636_diff__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 676 (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]. 1.49/1.88 677 (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]. 1.49/1.88 678 (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]. 1.49/1.88 679 (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)) -> M = N))) # label(fact_860_dvd__antisym) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 680 (all X_a (comm_semiring_1(X_a) -> (all X times_times(X_a,X,X) = hAPP(nat,X_a,power_power(X_a,X),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_541_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 681 (all X_a (linord581940658strict(X_a) -> (all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),plus_plus(X_a,times_times(X_a,Xa,Xa),times_times(X_a,Ya,Ya)))) <-> ti(X_a,Xa) != zero_zero(X_a) | ti(X_a,Ya) != zero_zero(X_a))))) # label(fact_486_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 682 (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),A_2),B)) | 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),times_times(X_a,A_2,C_1)),times_times(X_a,B,C_1))))))) # label(fact_741_dvd__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 683 (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]. 1.49/1.88 684 (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]. 1.49/1.88 685 (all K all L_1 (ti(int,K) = ti(int,L_1) <-> bit0(L_1) = bit0(K))) # label(fact_43_rel__simps_I48_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 686 (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]. 1.49/1.88 687 (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]. 1.49/1.88 688 -(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]. 1.49/1.88 689 (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]. 1.49/1.88 690 (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]. 1.49/1.88 691 (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) & ti(X_a,Ya) = zero_zero(X_a))))))) # label(fact_435_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 692 (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]. 1.49/1.88 693 (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]. 1.49/1.88 694 (all X_a (ordered_ab_group_add(X_a) -> (all A_2 all B all C_1 all D (minus_minus(X_a,A_2,B) = minus_minus(X_a,C_1,D) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),D))))))) # label(fact_249_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 695 (all Ma all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,semiring_1_of_nat(int),Ma)),Z_1)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),nat_1(abs_abs(int,Z_1)))))) # label(fact_962_int__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 696 (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]. 1.49/1.88 697 (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]. 1.49/1.88 698 (all K 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(nat),zero_zero(nat)),K)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),times_times(nat,K,Ma)),times_times(nat,K,Na))))) # label(fact_651_mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 699 (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]. 1.49/1.88 700 (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),Ma),plus_plus(nat,times_times(nat,minus_minus(nat,J,I_1),U),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_979_nat__less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 701 (all X_a (comm_semiring_1(X_a) -> (all X all N (ti(X_a,X) = one_one(X_a) | 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]. 1.49/1.88 702 (all Xa all Ya (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_873_divides__antisym) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 703 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K1)),bit0(K2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)))) # label(fact_376_less__eq__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 704 (all Wa all Ma (nat_1(Wa) = Ma <-> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa)) -> ti(int,Wa) = hAPP(nat,int,semiring_1_of_nat(int),Ma)) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa)) -> zero_zero(nat) = Ma))) # label(fact_395_nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 705 (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]. 1.49/1.88 706 (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]. 1.49/1.88 707 (all B_1_1 all B_2_1 all X_a (power(X_a) -> hAPP(nat,X_a,B_1_1,ti(nat,B_2_1)) = hAPP(nat,X_a,B_1_1,B_2_1))) # label(tsy_c_hAPP_4_arg2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 708 (all X_a (comm_semiring_1(X_a) -> (all X ti(X_a,X) = hAPP(nat,X_a,power_power(X_a,X),one_one(nat))))) # label(fact_227_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 709 (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]. 1.49/1.88 710 (all A_2 all P_3 (hBOOL(hAPP(int,bool,zcong(A_2,minus_minus(int,P_3,one_one(int))),P_3)) <-> hBOOL(hAPP(int,bool,zcong(times_times(int,A_2,minus_minus(int,P_3,one_one(int))),one_one(int)),P_3)))) # label(fact_912_inv__not__p__minus__1__aux) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 711 (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]. 1.49/1.88 712 (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]. 1.49/1.88 713 (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]. 1.49/1.88 714 (all C all A_1 all B_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B_1),A_1)) -> (C = B_1 -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),C),A_1))))) # label(fact_848_dvd_Oord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 715 (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]. 1.49/1.88 716 (all Wa all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Wa),Z_1)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),plus_plus(int,Z_1,one_one(int)))))) # label(fact_516_zle__add1__eq__le) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 717 (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]. 1.49/1.88 718 (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]. 1.49/1.88 719 (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]. 1.49/1.88 720 (all X_a (ab_group_add(X_a) -> (all A_2 all B (minus_minus(X_a,A_2,B) = zero_zero(X_a) <-> ti(X_a,B) = ti(X_a,A_2))))) # label(fact_321_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 721 (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]. 1.49/1.88 722 (all Ma all Na all K (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K)) -> (Ma = Na <-> times_times(nat,K,Ma) = times_times(nat,K,Na)))) # label(fact_983_nat__mult__eq__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 723 (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]. 1.49/1.88 724 (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,N) = ti(int,M)))))) # label(fact_747_zdvd__antisym__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 725 (all X_a (number_semiring(X_a) -> (all Z times_times(X_a,number_number_of(X_a,bit0(bit1(pls))),Z) = plus_plus(X_a,Z,Z)))) # label(fact_538_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 726 (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]. 1.49/1.88 727 (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]. 1.49/1.88 728 (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]. 1.49/1.88 729 (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]. 1.49/1.88 730 (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]. 1.49/1.88 731 (all A_2 all B all Ma ((exists K_2 plus_plus(int,A_2,times_times(int,Ma,K_2)) = ti(int,B)) <-> hBOOL(hAPP(int,bool,zcong(A_2,B),Ma)))) # label(fact_895_zcong__iff__lin) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 732 (all K all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),Na)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),plus_plus(nat,Na,K))))) # label(fact_814_dvd__reduce) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 733 (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]. 1.49/1.88 734 (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]. 1.49/1.88 735 (all B_1_1 all B_2_1 all X_a (ab_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_1_arg2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 736 -(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]. 1.49/1.88 737 (all Xa all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya)) & Ya != Xa <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya)))) # label(fact_610_real__less__def) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 738 (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]. 1.49/1.88 739 (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]. 1.49/1.88 740 (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]. 1.49/1.88 741 (all W times_times(int,pls,W) = pls) # label(fact_370_mult__Pls) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 742 (all K all L_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L_1)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K)),bit0(L_1))))) # label(fact_375_rel__simps_I31_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 743 (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]. 1.49/1.88 744 (all A_1 (one_one(int) = hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls)))) -> one_one(int) = abs_abs(int,A_1))) # label(fact_968_power2__eq1__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 745 (all X_a (ordere142940540dd_abs(X_a) -> (all A_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),abs_abs(X_a,A_2))) <-> zero_zero(X_a) != ti(X_a,A_2))))) # label(fact_945_zero__less__abs__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 746 (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]. 1.49/1.88 747 (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]. 1.49/1.88 748 (all X_b (number(X_b) & semiring(X_b) -> (all V_1 all B_1 all C times_times(X_b,number_number_of(X_b,V_1),plus_plus(X_b,B_1,C)) = plus_plus(X_b,times_times(X_b,number_number_of(X_b,V_1),B_1),times_times(X_b,number_number_of(X_b,V_1),C))))) # label(fact_443_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 749 (all X_a (mult_zero(X_a) -> (all A_1 zero_zero(X_a) = times_times(X_a,A_1,zero_zero(X_a))))) # label(fact_329_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 750 (all X_a (ordere236663937imp_le(X_a) -> (all C_1 all A_2 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,C_1,A_2)),plus_plus(X_a,C_1,B))))))) # label(fact_358_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 751 (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)),plus_plus(nat,Ma,Na))))) # label(fact_144_add__gr__0) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 752 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),pls)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K)),pls)))) # label(fact_465_rel__simps_I27_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 753 (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]. 1.49/1.88 754 (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]. 1.49/1.88 755 (all N zero_zero(nat) = minus_minus(nat,zero_zero(nat),N)) # label(fact_338_diff__0__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 756 (all A_1 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1)) -> (exists R (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R)) & A_1 = hAPP(nat,real,power_power(real,R),N)))))) # label(fact_230_realpow__pos__nth) # label(axiom) # label(non_clause). [assumption]. 1.49/1.88 757 (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]. 1.49/1.88 758 (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]. 1.49/1.89 759 (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]. 1.49/1.89 760 (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]. 1.49/1.89 761 (all K all Ma all Na (Ma = Na <-> plus_plus(nat,K,Na) = plus_plus(nat,K,Ma))) # label(fact_125_nat__add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 762 (all V_1 number_number_of(nat,V_1) = nat_1(number_number_of(int,V_1))) # label(fact_427_nat__number__of__def) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 763 (all X_a (ordered_ring(X_a) -> (all A_2 all E_1 all C_1 all B all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,times_times(X_a,A_2,E_1),C_1)),plus_plus(X_a,times_times(X_a,B,E_1),D))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),plus_plus(X_a,times_times(X_a,minus_minus(X_a,A_2,B),E_1),C_1)),D)))))) # label(fact_295_less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 764 (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]. 1.49/1.89 765 (all X_a (comm_semiring_1(X_a) -> (all A_1 zero_zero(X_a) = times_times(X_a,zero_zero(X_a),A_1)))) # label(fact_334_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 766 (all I_2 all U_1 all J_1 all K_1 plus_plus(nat,times_times(nat,I_2,U_1),plus_plus(nat,times_times(nat,J_1,U_1),K_1)) = plus_plus(nat,times_times(nat,plus_plus(nat,I_2,J_1),U_1),K_1)) # label(fact_981_left__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 767 (all X_a (power(X_a) -> (all A_1 one_one(X_a) = hAPP(nat,X_a,power_power(X_a,A_1),zero_zero(nat))))) # label(fact_187_power__0) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 768 (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]. 1.49/1.89 769 (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]. 1.49/1.89 770 (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)),A_2)) & 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)),B)) & 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),times_times(X_a,A_2,B)),zero_zero(X_a))))))) # label(fact_290_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 771 (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]. 1.49/1.89 772 (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]. 1.49/1.89 773 (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]. 1.49/1.89 774 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all Xa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),number_number_of(X_a,Xa)),zero_zero(X_a))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Xa),pls)))))) # label(fact_49_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 775 (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]. 1.49/1.89 776 (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]. 1.49/1.89 777 (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]. 1.49/1.89 778 (all A_2 all B (ti(int,A_2) = ti(int,B) <-> hBOOL(hAPP(int,bool,zcong(A_2,B),zero_zero(int))))) # label(fact_885_IntPrimes_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 779 (all Z all W plus_plus(int,Z,W) = plus_plus(int,W,Z)) # label(fact_47_zadd__commute) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 780 (all X_a (linordered_idom(X_a) -> (all X all Y (ti(X_a,X) != ti(X_a,Y) -> (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X))))))) # label(fact_152_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 781 (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]. 1.49/1.89 782 (all Na (Na = zero_zero(nat) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),zero_zero(nat))))) # label(fact_602_le__0__eq) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 783 (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) -> zero_zero(X_a) = ti(X_a,B_1) | ti(X_a,A_1) = zero_zero(X_a))))) # label(fact_332_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 784 (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]. 1.49/1.89 785 (all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) -> ti(int,Z) = hAPP(nat,int,semiring_1_of_nat(int),nat_1(Z)))) # label(fact_319_nat__0__le) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 786 (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]. 1.49/1.89 787 (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)) -> abs_abs(X_a,times_times(X_a,Y,X)) = times_times(X_a,abs_abs(X_a,Y),X))))) # label(fact_924_abs__mult__pos) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 788 (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]. 1.49/1.89 789 (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),Va),V_3)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),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]. 1.49/1.89 790 (all M all 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),ord_less_eq(nat),M),N)))) # label(fact_576_nat__le__linear) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 791 -(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]. 1.49/1.89 792 (all Ma all Wa ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa)) -> zero_zero(nat) = Ma) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa)) -> hAPP(nat,int,semiring_1_of_nat(int),Ma) = ti(int,Wa)) <-> nat_1(Wa) = Ma)) # label(fact_394_nat__eq__iff2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 793 (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]. 1.49/1.89 794 (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]. 1.49/1.89 795 (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]. 1.49/1.89 796 (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]. 1.49/1.89 797 (all B_1_1 all B_2_1 all X_a (order(X_a) -> ti(fun(X_a,bool),hAPP(X_a,fun(X_a,bool),B_1_1,B_2_1)) = hAPP(X_a,fun(X_a,bool),B_1_1,B_2_1))) # label(tsy_c_hAPP_0_res) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 798 (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]. 1.49/1.89 799 (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]. 1.49/1.89 800 (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]. 1.49/1.89 801 (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]. 1.49/1.89 802 (all V_2 all V_1 ((-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)) -> plus_plus(nat,number_number_of(nat,V_1),number_number_of(nat,V_2)) = number_number_of(nat,plus_plus(int,V_1,V_2))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_2),pls)) -> number_number_of(nat,V_1) = plus_plus(nat,number_number_of(nat,V_1),number_number_of(nat,V_2)))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls)) -> plus_plus(nat,number_number_of(nat,V_1),number_number_of(nat,V_2)) = number_number_of(nat,V_2)))) # label(fact_36_add__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 803 (all M all N (M = N -> 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]. 1.49/1.89 804 (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]. 1.49/1.89 805 (all Xa all Ya (zero_zero(real) = Ya & zero_zero(real) = Xa <-> zero_zero(real) = plus_plus(real,hAPP(nat,real,power_power(real,Xa),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,real,power_power(real,Ya),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_106_realpow__two__sum__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 806 (all B_1_1 all B_2_1 zcong(ti(int,B_1_1),B_2_1) = zcong(B_1_1,B_2_1)) # label(tsy_c_IntPrimes_Ozcong_arg1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 807 (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]. 1.49/1.89 808 (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)) & hAPP(nat,int,F,I) = ti(int,K))))))) # label(fact_994_int__val__lemma) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 809 (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]. 1.49/1.89 810 (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]. 1.49/1.89 811 (all M all N (M = times_times(nat,M,N) -> N = one_one(nat) | zero_zero(nat) = M)) # label(fact_657_mult__eq__self__implies__10) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 812 (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]. 1.49/1.89 813 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),pls)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(K)),pls)))) # label(fact_504_rel__simps_I29_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 814 (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_21_int__power) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 815 (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]. 1.49/1.89 816 (all K 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(nat),plus_plus(nat,K,Ma)),plus_plus(nat,K,Na))))) # label(fact_127_nat__add__left__cancel__less) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 817 (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]. 1.49/1.89 818 (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]. 1.49/1.89 819 (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]. 1.49/1.89 820 (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,B) = ti(X_a,A_2) <-> ti(X_a,D) = ti(X_a,C_1)))))) # label(fact_248_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 821 -(all S all W_1 -(hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),times_times(int,number_number_of(int,bit0(bit1(pls))),abs_abs(int,W_1))),plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n)))) & ti(int,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)))))) # 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]. 1.49/1.89 822 (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]. 1.49/1.89 823 (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]. 1.49/1.89 824 (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]. 1.49/1.89 825 (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]. 1.49/1.89 826 (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]. 1.49/1.89 827 (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]. 1.49/1.89 828 (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]. 1.49/1.89 829 (all Wa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),zero_zero(int))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(Wa)),zero_zero(int))))) # label(fact_58_bin__less__0__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 830 (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 (J_1 = plus_plus(nat,I_2,K_2) & 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]. 1.49/1.89 831 (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]. 1.49/1.89 832 (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]. 1.49/1.89 833 (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]. 1.49/1.89 834 (all M all N times_times(int,hAPP(nat,int,semiring_1_of_nat(int),M),hAPP(nat,int,semiring_1_of_nat(int),N)) = hAPP(nat,int,semiring_1_of_nat(int),times_times(nat,M,N))) # label(fact_664_zmult__int) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 835 (all X all 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))))) = times_times(nat,plus_plus(nat,X,Y),minus_minus(nat,X,Y))) # label(fact_701_diff__square) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 836 -(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]. 1.49/1.89 837 (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]. 1.49/1.89 838 (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]. 1.49/1.89 839 (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]. 1.49/1.89 840 (all J_1 all K_1 all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),M)) -> (hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,number_number_of(int,min)),J_1),hAPP(nat,int,power_power(int,number_number_of(int,min)),K_1)),M)) -> hAPP(nat,int,power_power(int,number_number_of(int,min)),K_1) = hAPP(nat,int,power_power(int,number_number_of(int,min)),J_1)))) # label(fact_874_neg__one__power__eq__mod__m) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 841 (all X_a (ab_semigroup_mult(X_a) -> (all A_1 all B_1 all C times_times(X_a,A_1,times_times(X_a,B_1,C)) = times_times(X_a,times_times(X_a,A_1,B_1),C)))) # label(fact_242_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 842 (all X all Y (Y = X -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)))) # label(fact_866_dvd_Oeq__refl) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 843 (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]. 1.49/1.89 844 (all K_1 bit0(K_1) = plus_plus(int,K_1,K_1)) # label(fact_78_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 845 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),bit1(K))))) # label(fact_503_rel__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 846 (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]. 1.49/1.89 847 (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),B),A_2)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a))))))) # label(fact_410_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 848 (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]. 1.49/1.89 849 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K1)),bit0(K2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)))) # label(fact_33_less__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 850 (all P_2 all M ((zero_zero(nat) != M -> times_times(nat,P_2,hAPP(nat,nat,power_power(nat,P_2),minus_minus(nat,M,one_one(nat)))) = hAPP(nat,nat,power_power(nat,P_2),M)) & (M = zero_zero(nat) -> hAPP(nat,nat,power_power(nat,P_2),M) = one_one(nat)))) # label(fact_689_power__eq__if) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 851 (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),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_986_nat__mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 852 (all Xa all Na (Xa = one_one(nat) | zero_zero(nat) = Na <-> hAPP(nat,nat,power_power(nat,Xa),Na) = one_one(nat))) # label(fact_192_exp__eq__1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 853 (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)) -> 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)) = hAPP(nat,X_a,semiring_1_of_nat(X_a),minus_minus(nat,M,N)))))) # label(fact_682_of__nat__diff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 854 (all P_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_2)) -> nat_1(minus_minus(int,P_2,number_number_of(int,bit0(bit1(pls))))) = minus_minus(nat,nat_1(P_2),number_number_of(nat,bit0(bit1(pls)))))) # label(fact_709_Int2_Oaux__1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 855 (all B_1_1 succ(B_1_1) = ti(int,succ(B_1_1))) # label(tsy_c_Int_Osucc_res) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 856 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B_1 all C times_times(X_a,plus_plus(X_a,A_1,B_1),C) = plus_plus(X_a,times_times(X_a,A_1,C),times_times(X_a,B_1,C))))) # label(fact_343_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 857 (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]. 1.49/1.89 858 (all X_a (ordered_ab_group_add(X_a) -> (all A_2 all B all C_1 all D (minus_minus(X_a,A_2,B) = minus_minus(X_a,C_1,D) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B))))))) # label(fact_324_diff__eq__diff__less) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 859 (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]. 1.49/1.89 860 (all X all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) -> X = Y))) # label(fact_859_dvd_Oantisym) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 861 (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]. 1.49/1.89 862 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 abs_abs(X_a,abs_abs(X_a,A_1)) = abs_abs(X_a,A_1)))) # label(fact_937_abs__idempotent) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 863 (all B_1_1 all X_a (number(X_a) -> ti(X_a,number_number_of(X_a,B_1_1)) = number_number_of(X_a,B_1_1))) # label(tsy_c_Int_Onumber__class_Onumber__of_res) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 864 (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]. 1.49/1.89 865 (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]. 1.49/1.89 866 (all N all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) -> nat_1(hAPP(nat,int,power_power(int,Z),N)) = hAPP(nat,nat,power_power(nat,nat_1(Z)),N))) # label(fact_400_nat__power__eq) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 867 (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]. 1.49/1.89 868 (all L min != bit0(L)) # label(fact_765_rel__simps_I42_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 869 (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),minus_minus(nat,Ma,K)),minus_minus(nat,Na,K))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)))))) # label(fact_633_le__diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 870 (all Ma all Na (zero_zero(nat) = Ma & Na != zero_zero(nat) <-> 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]. 1.49/1.89 871 (all X_a (number(X_a) & linorder(X_a) -> (all Va all Wa (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),number_number_of(X_a,Wa)),number_number_of(X_a,Va))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,Va)),number_number_of(X_a,Wa))))))) # label(fact_448_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 872 (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),A_2),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)),B)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_2)))))) # label(fact_291_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 873 (all Ya all 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)) <-> Ya = Xa))) # label(fact_865_dvd_Oantisym__conv) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 874 (all X_a (number_ring(X_a) & ring_char_0(X_a) -> (all Xa all Ya (ti(int,Ya) = ti(int,Xa) <-> number_number_of(X_a,Xa) = number_number_of(X_a,Ya))))) # label(fact_40_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 875 (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]. 1.49/1.89 876 (all B_1_1 bit0(B_1_1) = ti(int,bit0(B_1_1))) # label(tsy_c_Int_OBit0_res) # label(hypothesis) # label(non_clause). [assumption]. 1.49/1.89 877 (all X all Y plus_plus(real,plus_plus(real,hAPP(nat,real,power_power(real,X),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,real,power_power(real,Y),number_number_of(nat,bit0(bit1(pls))))),times_times(real,times_times(real,number_number_of(real,bit0(bit1(pls))),X),Y)) = hAPP(nat,real,power_power(real,plus_plus(real,X,Y)),number_number_of(nat,bit0(bit1(pls))))) # label(fact_721_real__sum__squared__expand) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 878 (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]. 1.49/1.89 879 (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]. 1.49/1.89 880 (all B_1_1 all B_2_1 legendre(B_1_1,ti(int,B_2_1)) = legendre(B_1_1,B_2_1)) # label(tsy_c_Residues_OLegendre_arg2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 881 (all B_1_1 all B_2_1 all X_a (power(X_a) -> times_times(X_a,B_1_1,B_2_1) = times_times(X_a,ti(X_a,B_1_1),B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_1_arg1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 882 (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]. 1.49/1.89 883 (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]. 1.49/1.89 884 (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]. 1.49/1.89 885 (all P (fFalse = ti(bool,P) | fTrue = ti(bool,P))) # label(help_If_3_1_T) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 886 (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]. 1.49/1.89 887 (all X all N hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,power_power(nat,X),N)) = hAPP(nat,int,power_power(int,hAPP(nat,int,semiring_1_of_nat(int),X)),N)) # label(fact_175_Nat__Transfer_Otransfer__int__nat__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 888 (all Ma all Xa (hBOOL(hAPP(int,bool,quadRes(Ma),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))))) # label(fact_878_QuadRes__def) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 889 (all V_2 all V_1 ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls)) -> times_times(nat,number_number_of(nat,V_1),number_number_of(nat,V_2)) = number_number_of(nat,times_times(int,V_1,V_2))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls)) -> zero_zero(nat) = times_times(nat,number_number_of(nat,V_1),number_number_of(nat,V_2))))) # label(fact_694_mult__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 890 (all X_a (linordered_semidom(X_a) -> (all Ma all Na (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Ma)),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)))))) # label(fact_167_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 891 (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]. 1.49/1.89 892 (all K_1 all L bit0(plus_plus(int,K_1,succ(L))) = plus_plus(int,bit1(K_1),bit1(L))) # label(fact_533_add__Bit1__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 893 (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]. 1.49/1.89 894 (all X_a (number_semiring(X_a) -> (all Z times_times(X_a,Z,number_number_of(X_a,bit0(bit1(pls)))) = plus_plus(X_a,Z,Z)))) # label(fact_536_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 895 (all Z times_times(int,one_one(int),Z) = ti(int,Z)) # label(fact_378_zmult__1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 896 (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]. 1.49/1.89 897 -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))) -> 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_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]. 1.49/1.89 898 (all M all N all K_1 times_times(nat,plus_plus(nat,M,N),K_1) = plus_plus(nat,times_times(nat,M,K_1),times_times(nat,N,K_1))) # label(fact_612_add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 899 (all X_a (linordered_semidom(X_a) -> (all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na))))))) # label(fact_107_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 900 (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]. 1.49/1.89 901 (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]. 1.49/1.89 902 (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]. 1.49/1.89 903 (all P_4 all P_1 all Xa ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> (hBOOL(P_4) <-> hBOOL(P_1))) -> ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> hBOOL(P_4)) <-> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> hBOOL(P_1))))) # label(fact_906_imp__le__cong) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 904 (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]. 1.49/1.89 905 (all Z_1 all Wa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_1),Wa)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),Wa)) & ti(int,Z_1) != ti(int,Wa))) # label(fact_380_zless__le) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 906 (all Xa all 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))) <-> zero_zero(int) != ti(int,Xa) | zero_zero(nat) = Na)) # label(fact_960_zero__less__zpower__abs__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 907 (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]. 1.49/1.89 908 (all X all Q all N all R_1 (X = plus_plus(nat,times_times(nat,Q,N),R_1) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),R_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),R_1),N)) -> -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]. 1.49/1.89 909 (all P_1 ((all X_1 hBOOL(hAPP(int,bool,P_1,hAPP(nat,int,semiring_1_of_nat(int),X_1)))) <-> (all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) -> hBOOL(hAPP(int,bool,P_1,X_1)))))) # label(fact_474_transfer__int__nat__quantifiers_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 910 (all X_a (number_ring(X_a) -> (all V_1 all W all Z times_times(X_a,number_number_of(X_a,times_times(int,V_1,W)),Z) = times_times(X_a,number_number_of(X_a,V_1),times_times(X_a,number_number_of(X_a,W),Z))))) # label(fact_260_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 911 (all X_a (linord1278240602ring_1(X_a) -> (all V_1 all U_1 all Y all X all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),U_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),V_1)) -> (one_one(X_a) = plus_plus(X_a,U_1,V_1) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,times_times(X_a,U_1,X),times_times(X_a,V_1,Y))),A_1)))))))))) # label(fact_386_convex__bound__le) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 912 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),bit1(K))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K)))) # label(fact_461_rel__simps_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 913 (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]. 1.49/1.89 914 (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]. 1.49/1.89 915 (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]. 1.49/1.89 916 (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]. 1.49/1.89 917 (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]. 1.49/1.89 918 (all Ma all Wa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),hAPP(nat,int,semiring_1_of_nat(int),Ma))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),nat_1(Wa)),Ma))))) # label(fact_402_nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 919 (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]. 1.49/1.89 920 (all X_a (number_ring(X_a) -> (all V_1 all W times_times(X_a,number_number_of(X_a,V_1),number_number_of(X_a,W)) = number_number_of(X_a,times_times(int,V_1,W))))) # label(fact_258_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 921 (all D_1 all I_2 (zero_zero(int) != ti(int,I_2) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),I_2)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),abs_abs(int,D_1)),abs_abs(int,I_2)))))) # label(fact_917_dvd__imp__le__int) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 922 (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]. 1.49/1.89 923 (all X_a (semiring_0(X_a) & power(X_a) -> (all W ((zero_zero(nat) != number_number_of(nat,W) -> zero_zero(X_a) = hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),number_number_of(nat,W))) & (number_number_of(nat,W) = zero_zero(nat) -> 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]. 1.49/1.89 924 (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]. 1.49/1.89 925 (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]. 1.49/1.89 926 (all X_a (ordered_ring(X_a) -> (all A_2 all E_1 all C_1 all B all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),plus_plus(X_a,times_times(X_a,minus_minus(X_a,B,A_2),E_1),D))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),plus_plus(X_a,times_times(X_a,A_2,E_1),C_1)),plus_plus(X_a,times_times(X_a,B,E_1),D))))))) # label(fact_287_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 927 (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]. 1.49/1.89 928 (all X_a (comm_semiring_1(X_a) -> (all A_1 ti(X_a,A_1) = times_times(X_a,A_1,one_one(X_a))))) # label(fact_348_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 929 (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]. 1.49/1.89 930 (all Z_1 all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),nat_1(Z_1)),Ma)) <-> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1)) -> zero_zero(nat) = 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)))))) # label(fact_835_nat__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 931 (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]. 1.49/1.89 932 (all Na all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Ma)) -> (Na = one_one(nat) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),times_times(nat,Ma,Na)),Ma))))) # label(fact_831_dvd__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 933 (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]. 1.49/1.89 934 (all X_a (ring_n68954251visors(X_a) -> (all A_2 all B (times_times(X_a,A_2,B) = zero_zero(X_a) <-> ti(X_a,B) = zero_zero(X_a) | zero_zero(X_a) = ti(X_a,A_2))))) # label(fact_330_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 935 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(K1)),bit0(K2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)))) # label(fact_508_less__eq__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 936 (all A_1 all B_1 all C (ti(int,C) = minus_minus(int,A_1,B_1) -> plus_plus(int,C,B_1) = ti(int,A_1))) # label(fact_719_Int2_Oaux1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.89 937 (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]. 1.49/1.90 938 (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]. 1.49/1.90 939 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all Xa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,Xa)),zero_zero(X_a))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),pls)))))) # label(fact_390_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 940 (all X_a (cancel_semigroup_add(X_a) -> (all B all A_2 all C_1 (plus_plus(X_a,C_1,A_2) = plus_plus(X_a,B,A_2) <-> ti(X_a,C_1) = ti(X_a,B))))) # label(fact_203_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 941 (all X_a (linordered_idom(X_a) -> (all A_1 all B_1 times_times(X_a,abs_abs(X_a,A_1),abs_abs(X_a,B_1)) = abs_abs(X_a,times_times(X_a,A_1,B_1))))) # label(fact_951_abs__mult) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 942 (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]. 1.49/1.90 943 (all X_a (group_add(X_a) -> (all A_1 all B_1 plus_plus(X_a,minus_minus(X_a,A_1,B_1),B_1) = ti(X_a,A_1)))) # label(fact_327_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 944 (all X_a (comm_semiring_1(X_a) -> (all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),zero_zero(X_a)),A_1)) -> ti(X_a,A_1) = zero_zero(X_a))))) # label(fact_775_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 945 (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]. 1.49/1.90 946 (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]. 1.49/1.90 947 (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,minus_minus(nat,I_2,J_1),U_1),M),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_988_nat__diff__add__eq1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 948 (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]. 1.49/1.90 949 (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]. 1.49/1.90 950 (all Xa all Ya (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)) & -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]. 1.49/1.90 951 (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]. 1.49/1.90 952 (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]. 1.49/1.90 953 (all K_1 all L bit1(plus_plus(int,K_1,L)) = plus_plus(int,bit0(K_1),bit1(L))) # label(fact_91_add__Bit0__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 954 (all X_a (comm_semiring_1(X_a) -> (all A_1 plus_plus(X_a,A_1,zero_zero(X_a)) = ti(X_a,A_1)))) # label(fact_214_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 955 (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]. 1.49/1.90 956 (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]. 1.49/1.90 957 (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]. 1.49/1.90 958 (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]. 1.49/1.90 959 (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]. 1.49/1.90 960 (all X_a (number_ring(X_a) -> (all A_1 plus_plus(X_a,A_1,number_number_of(X_a,pls)) = ti(X_a,A_1)))) # label(fact_85_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 961 (all X_a all B_1_1 all B_2_1 all B_3 if(X_a,B_1_1,B_2_1,ti(X_a,B_3)) = if(X_a,B_1_1,B_2_1,B_3)) # label(tsy_c_If_arg3) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 962 (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,Rx),times_times(X_a,Ly,Ry)) = times_times(X_a,times_times(X_a,Lx,Ly),times_times(X_a,Rx,Ry))))) # label(fact_247_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 963 (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_851_dvd_Oless__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 964 (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]. 1.49/1.90 965 (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]. 1.49/1.90 966 (all B_1_1 bit1(B_1_1) = bit1(ti(int,B_1_1))) # label(tsy_c_Int_OBit1_arg1) # label(hypothesis) # label(non_clause). [assumption]. 1.49/1.90 967 (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]. 1.49/1.90 968 (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]. 1.49/1.90 969 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K1)),bit0(K2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)))) # label(fact_52_less__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 970 (all X_a (comm_semiring_1(X_a) -> (all A_1 ti(X_a,A_1) = times_times(X_a,one_one(X_a),A_1)))) # label(fact_351_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 971 (all B_1 all Q all R_1 all B_2 all Q_1 all R_2 (plus_plus(int,times_times(int,B_2,Q_1),R_2) = plus_plus(int,times_times(int,B_1,Q),R_1) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,times_times(int,B_2,Q_1),R_2)),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_1),B_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),B_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_1),Q))))))))) # label(fact_704_zdiv__mono2__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 972 (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]. 1.49/1.90 973 (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]. 1.49/1.90 974 (all N all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) -> hAPP(nat,nat,power_power(nat,nat_1(X)),N) = nat_1(hAPP(nat,int,power_power(int,X),N)))) # label(fact_399_Nat__Transfer_Otransfer__nat__int__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 975 (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,A_2) = abs_abs(int,B))) # label(fact_967_power2__eq__iff__abs__eq) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 976 (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]. 1.49/1.90 977 (all B_1_1 ti(int,bit1(B_1_1)) = bit1(B_1_1)) # label(tsy_c_Int_OBit1_res) # label(hypothesis) # label(non_clause). [assumption]. 1.49/1.90 978 (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]. 1.49/1.90 979 (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]. 1.49/1.90 980 (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,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_eq(X_a),plus_plus(X_a,times_times(X_a,A_2,E_1),C_1)),plus_plus(X_a,times_times(X_a,B,E_1),D))))))) # label(fact_285_le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 981 (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]. 1.49/1.90 982 (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]. 1.49/1.90 983 (exists X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))) & hBOOL(hAPP(int,bool,zcong(s1,X_1),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))) & (all Y_1 (hBOOL(hAPP(int,bool,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)))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_1)) -> ti(int,Y_1) = ti(int,X_1))))) # label(fact_841__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s1) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 984 (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]. 1.49/1.90 985 (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]. 1.49/1.90 986 (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]. 1.49/1.90 987 (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]. 1.49/1.90 988 (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]. 1.49/1.90 989 (all T all A ti(T,A) = ti(T,ti(T,A))) # label(help_ti_idem) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 990 (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]. 1.49/1.90 991 (all A_2 all B all C_1 (zero_zero(real) != C_1 -> (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]. 1.49/1.90 992 (all K_1 number_number_of(int,K_1) = ti(int,K_1)) # label(fact_120_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 993 (all M all N hAPP(nat,int,semiring_1_of_nat(int),times_times(nat,M,N)) = times_times(int,hAPP(nat,int,semiring_1_of_nat(int),M),hAPP(nat,int,semiring_1_of_nat(int),N))) # label(fact_665_int__mult) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 994 (all P_1 all A_2 all B (-(hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_2),B)) & -hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) | (exists D_3 (-hBOOL(hAPP(nat,bool,P_1,D_3)) & A_2 = plus_plus(nat,B,D_3)))) <-> 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]. 1.49/1.90 995 (all W nat_1(number_number_of(int,W)) = number_number_of(nat,W)) # label(fact_426_nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 996 (all P_2 (hBOOL(hAPP(int,bool,zprime,P_2)) -> (ti(int,P_2) != number_number_of(int,bit0(bit1(pls))) -> (number_number_of(int,bit1(bit1(pls))) != ti(int,P_2) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,bit1(bit0(bit1(pls))))),P_2)))))) # label(fact_904_prime__g__5) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 997 (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]. 1.49/1.90 998 (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)) -> (Na = Ma <-> hAPP(nat,X_a,power_power(X_a,A_2),Ma) = hAPP(nat,X_a,power_power(X_a,A_2),Na)))))) # label(fact_181_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 1.49/1.90 999 (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]. 1.49/1.90 1000 (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]. 1.49/1.90 1001 (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]. 1.49/1.93 1002 (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]. 1.49/1.93 1003 (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]. 1.49/1.93 1004 (all X all Y all Z times_times(int,hAPP(nat,int,power_power(int,X),Y),hAPP(nat,int,power_power(int,X),Z)) = hAPP(nat,int,power_power(int,X),plus_plus(nat,Y,Z))) # label(fact_478_zpower__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 1.49/1.93 1005 (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]. 1.49/1.93 1006 (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]. 1.49/1.93 1007 (all X_a (semiring_1(X_a) -> (all M all N hAPP(nat,X_a,semiring_1_of_nat(X_a),plus_plus(nat,M,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))))) # label(fact_170_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 1.49/1.93 1008 (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]. 1.49/1.93 1009 (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]. 1.49/1.93 1010 (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]. 1.49/1.93 1011 -(zero_zero(int) != hAPP(nat,int,power_power(int,plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n))),number_number_of(nat,bit0(bit1(pls))))) # label(conj_0) # label(negated_conjecture) # label(non_clause). [assumption]. 1.49/1.93 1.49/1.93 ============================== end of process non-clausal formulas === 1.49/1.93 1.49/1.93 ============================== PROCESS INITIAL CLAUSES =============== 1.49/1.93 1.49/1.93 ============================== PREDICATE ELIMINATION ================= 1.49/1.93 1012 dvd(int) # label(arity_Int_Oint___Rings_Odvd) # label(axiom). [assumption]. 1.49/1.93 1013 -dvd(A) | ti(fun(A,bool),hAPP(A,fun(A,bool),B,C)) = hAPP(A,fun(A,bool),B,C) # label(tsy_c_hAPP_2_res) # label(axiom). [clausify(2)]. 1.49/1.93 1014 -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(35)]. 1.49/1.93 1015 -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(35)]. 1.49/1.94 1016 -dvd(A) | hAPP(A,bool,B,ti(A,C)) = hAPP(A,bool,B,C) # label(tsy_c_hAPP_3_arg2) # label(axiom). [clausify(39)]. 1.49/1.94 1017 -dvd(A) | dvd_dvd(A) = ti(fun(A,fun(A,bool)),dvd_dvd(A)) # label(tsy_c_Rings_Odvd__class_Odvd_res) # label(axiom). [clausify(200)]. 1.49/1.94 1018 -dvd(A) | hAPP(A,fun(A,bool),B,ti(A,C)) = hAPP(A,fun(A,bool),B,C) # label(tsy_c_hAPP_2_arg2) # label(axiom). [clausify(343)]. 1.49/1.94 1019 -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(349)]. 1.49/1.94 1020 -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(349)]. 1.49/1.94 1021 -dvd(A) | hAPP(A,fun(A,bool),ti(fun(A,fun(A,bool)),B),C) = hAPP(A,fun(A,bool),B,C) # label(tsy_c_hAPP_2_arg1) # label(axiom). [clausify(350)]. 1.49/1.94 1022 -dvd(A) | ti(A,times_times(A,B,C)) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_0_res) # label(axiom). [clausify(353)]. 1.49/1.94 1023 -dvd(A) | hAPP(A,bool,ti(fun(A,bool),B),C) = hAPP(A,bool,B,C) # label(tsy_c_hAPP_3_arg1) # label(axiom). [clausify(450)]. 1.49/1.94 Derived: ti(fun(int,bool),hAPP(int,fun(int,bool),A,B)) = hAPP(int,fun(int,bool),A,B). [resolve(1012,a,1013,a)]. 1.49/1.94 Derived: -comm_ring(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),plus_plus(int,C,D))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),plus_plus(int,minus_minus(int,C,times_times(int,E,B)),D))). [resolve(1012,a,1014,a)]. 1.49/1.94 Derived: -comm_ring(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),plus_plus(int,C,D))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),plus_plus(int,minus_minus(int,C,times_times(int,E,B)),D))). [resolve(1012,a,1015,a)]. 1.49/1.94 Derived: hAPP(int,bool,A,ti(int,B)) = hAPP(int,bool,A,B). [resolve(1012,a,1016,a)]. 1.49/1.94 Derived: dvd_dvd(int) = ti(fun(int,fun(int,bool)),dvd_dvd(int)). [resolve(1012,a,1017,a)]. 1.49/1.94 Derived: hAPP(int,fun(int,bool),A,ti(int,B)) = hAPP(int,fun(int,bool),A,B). [resolve(1012,a,1018,a)]. 1.49/1.94 Derived: hAPP(int,fun(int,bool),ti(fun(int,fun(int,bool)),A),B) = hAPP(int,fun(int,bool),A,B). [resolve(1012,a,1021,a)]. 1.49/1.94 Derived: ti(int,times_times(int,A,B)) = times_times(int,A,B). [resolve(1012,a,1022,a)]. 1.49/1.94 Derived: hAPP(int,bool,ti(fun(int,bool),A),B) = hAPP(int,bool,A,B). [resolve(1012,a,1023,a)]. 1.49/1.94 1024 -dvd(A) | times_times(A,B,ti(A,C)) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_0_arg2) # label(axiom). [clausify(491)]. 1.49/1.94 Derived: times_times(int,A,ti(int,B)) = times_times(int,A,B). [resolve(1024,a,1012,a)]. 1.49/1.94 1025 -dvd(A) | times_times(A,B,C) != D | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) # label(fact_730_dvdI) # label(axiom). [clausify(541)]. 1.49/1.94 Derived: times_times(int,A,B) != C | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),C)). [resolve(1025,a,1012,a)]. 1.49/1.94 1026 -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,f19(A,B,C),zero_zero(A)))) # label(fact_908_unity__coeff__ex) # label(axiom). [clausify(582)]. 1.49/1.94 Derived: -semiring_0(int) | -hBOOL(hAPP(int,bool,A,times_times(int,B,C))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),plus_plus(int,f19(int,A,B),zero_zero(int)))). [resolve(1026,a,1012,a)]. 1.49/1.94 1027 -dvd(A) | -semiring_0(A) | -hBOOL(hAPP(A,bool,B,times_times(A,C,D))) | hBOOL(hAPP(A,bool,B,f19(A,B,C))) # label(fact_908_unity__coeff__ex) # label(axiom). [clausify(582)]. 1.49/1.94 Derived: -semiring_0(int) | -hBOOL(hAPP(int,bool,A,times_times(int,B,C))) | hBOOL(hAPP(int,bool,A,f19(int,A,B))). [resolve(1027,a,1012,a)]. 1.49/2.01 1028 -dvd(A) | -semiring_0(A) | hBOOL(hAPP(A,bool,B,times_times(A,C,f20(A,B,C)))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),plus_plus(A,D,zero_zero(A)))) | -hBOOL(hAPP(A,bool,B,D)) # label(fact_908_unity__coeff__ex) # label(axiom). [clausify(582)]. 1.49/2.01 Derived: -semiring_0(int) | hBOOL(hAPP(int,bool,A,times_times(int,B,f20(int,A,B)))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),plus_plus(int,C,zero_zero(int)))) | -hBOOL(hAPP(int,bool,A,C)). [resolve(1028,a,1012,a)]. 1.49/2.01 1029 -dvd(A) | times_times(A,ti(A,B),C) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_0_arg1) # label(axiom). [clausify(655)]. 1.49/2.01 Derived: times_times(int,ti(int,A),B) = times_times(int,A,B). [resolve(1029,a,1012,a)]. 1.49/2.01 1030 dvd(nat) # label(arity_Nat_Onat___Rings_Odvd) # label(axiom). [assumption]. 1.49/2.01 Derived: ti(fun(nat,bool),hAPP(nat,fun(nat,bool),A,B)) = hAPP(nat,fun(nat,bool),A,B). [resolve(1030,a,1013,a)]. 1.49/2.01 Derived: -comm_ring(nat) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),plus_plus(nat,C,D))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),plus_plus(nat,minus_minus(nat,C,times_times(nat,E,B)),D))). [resolve(1030,a,1014,a)]. 1.49/2.01 Derived: -comm_ring(nat) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),plus_plus(nat,C,D))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),plus_plus(nat,minus_minus(nat,C,times_times(nat,E,B)),D))). [resolve(1030,a,1015,a)]. 1.49/2.01 Derived: hAPP(nat,bool,A,ti(nat,B)) = hAPP(nat,bool,A,B). [resolve(1030,a,1016,a)]. 1.49/2.01 Derived: dvd_dvd(nat) = ti(fun(nat,fun(nat,bool)),dvd_dvd(nat)). [resolve(1030,a,1017,a)]. 1.49/2.01 Derived: hAPP(nat,fun(nat,bool),A,ti(nat,B)) = hAPP(nat,fun(nat,bool),A,B). [resolve(1030,a,1018,a)]. 1.49/2.01 Derived: hAPP(nat,fun(nat,bool),ti(fun(nat,fun(nat,bool)),A),B) = hAPP(nat,fun(nat,bool),A,B). [resolve(1030,a,1021,a)]. 1.49/2.01 Derived: ti(nat,times_times(nat,A,B)) = times_times(nat,A,B). [resolve(1030,a,1022,a)]. 1.49/2.01 Derived: hAPP(nat,bool,ti(fun(nat,bool),A),B) = hAPP(nat,bool,A,B). [resolve(1030,a,1023,a)]. 1.49/2.01 Derived: times_times(nat,A,ti(nat,B)) = times_times(nat,A,B). [resolve(1030,a,1024,a)]. 1.49/2.01 Derived: times_times(nat,A,B) != C | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),C)). [resolve(1030,a,1025,a)]. 1.49/2.01 Derived: -semiring_0(nat) | -hBOOL(hAPP(nat,bool,A,times_times(nat,B,C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),plus_plus(nat,f19(nat,A,B),zero_zero(nat)))). [resolve(1030,a,1026,a)]. 1.49/2.01 Derived: -semiring_0(nat) | -hBOOL(hAPP(nat,bool,A,times_times(nat,B,C))) | hBOOL(hAPP(nat,bool,A,f19(nat,A,B))). [resolve(1030,a,1027,a)]. 1.49/2.01 Derived: -semiring_0(nat) | hBOOL(hAPP(nat,bool,A,times_times(nat,B,f20(nat,A,B)))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),plus_plus(nat,C,zero_zero(nat)))) | -hBOOL(hAPP(nat,bool,A,C)). [resolve(1030,a,1028,a)]. 1.49/2.01 Derived: times_times(nat,ti(nat,A),B) = times_times(nat,A,B). [resolve(1030,a,1029,a)]. 1.49/2.01 1031 -dvd(A) | ti(bool,hAPP(A,bool,B,C)) = hAPP(A,bool,B,C) # label(tsy_c_hAPP_3_res) # label(axiom). [clausify(942)]. 1.49/2.01 Derived: ti(bool,hAPP(int,bool,A,B)) = hAPP(int,bool,A,B). [resolve(1031,a,1012,a)]. 1.49/2.01 Derived: ti(bool,hAPP(nat,bool,A,B)) = hAPP(nat,bool,A,B). [resolve(1031,a,1030,a)]. 1.49/2.01 1032 dvd(real) # label(arity_RealDef_Oreal___Rings_Odvd) # label(axiom). [assumption]. 1.49/2.01 Derived: ti(fun(real,bool),hAPP(real,fun(real,bool),A,B)) = hAPP(real,fun(real,bool),A,B). [resolve(1032,a,1013,a)]. 1.49/2.01 Derived: -comm_ring(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),plus_plus(real,C,D))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),plus_plus(real,minus_minus(real,C,times_times(real,E,B)),D))). [resolve(1032,a,1014,a)]. 1.49/2.01 Derived: -comm_ring(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),plus_plus(real,C,D))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),plus_plus(real,minus_minus(real,C,times_times(real,E,B)),D))). [resolve(1032,a,1015,a)]. 1.49/2.09 Derived: hAPP(real,bool,A,ti(real,B)) = hAPP(real,bool,A,B). [resolve(1032,a,1016,a)]. 1.49/2.09 Derived: dvd_dvd(real) = ti(fun(real,fun(real,bool)),dvd_dvd(real)). [resolve(1032,a,1017,a)]. 1.49/2.09 Derived: hAPP(real,fun(real,bool),A,ti(real,B)) = hAPP(real,fun(real,bool),A,B). [resolve(1032,a,1018,a)]. 1.49/2.09 Derived: hAPP(real,fun(real,bool),ti(fun(real,fun(real,bool)),A),B) = hAPP(real,fun(real,bool),A,B). [resolve(1032,a,1021,a)]. 1.49/2.09 Derived: ti(real,times_times(real,A,B)) = times_times(real,A,B). [resolve(1032,a,1022,a)]. 1.49/2.09 Derived: hAPP(real,bool,ti(fun(real,bool),A),B) = hAPP(real,bool,A,B). [resolve(1032,a,1023,a)]. 1.49/2.09 Derived: times_times(real,A,ti(real,B)) = times_times(real,A,B). [resolve(1032,a,1024,a)]. 1.49/2.09 Derived: times_times(real,A,B) != C | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),C)). [resolve(1032,a,1025,a)]. 1.49/2.09 Derived: -semiring_0(real) | -hBOOL(hAPP(real,bool,A,times_times(real,B,C))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),B),plus_plus(real,f19(real,A,B),zero_zero(real)))). [resolve(1032,a,1026,a)]. 1.49/2.09 Derived: -semiring_0(real) | -hBOOL(hAPP(real,bool,A,times_times(real,B,C))) | hBOOL(hAPP(real,bool,A,f19(real,A,B))). [resolve(1032,a,1027,a)]. 1.49/2.09 Derived: -semiring_0(real) | hBOOL(hAPP(real,bool,A,times_times(real,B,f20(real,A,B)))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),B),plus_plus(real,C,zero_zero(real)))) | -hBOOL(hAPP(real,bool,A,C)). [resolve(1032,a,1028,a)]. 1.49/2.09 Derived: times_times(real,ti(real,A),B) = times_times(real,A,B). [resolve(1032,a,1029,a)]. 1.49/2.09 Derived: ti(bool,hAPP(real,bool,A,B)) = hAPP(real,bool,A,B). [resolve(1032,a,1031,a)]. 1.49/2.09 1033 ordere236663937imp_le(nat) # label(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 1.49/2.09 1034 -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(4)]. 1.49/2.09 1035 -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(23)]. 1.49/2.09 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(1033,a,1034,a)]. 1.49/2.09 1036 -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(229)]. 1.49/2.09 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(1036,a,1033,a)]. 1.49/2.09 1037 -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(410)]. 1.49/2.09 1038 -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(557)]. 1.49/2.09 1039 -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(557)]. 1.49/2.09 1040 ordere236663937imp_le(int) # label(arity_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 1.49/2.12 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(1040,a,1034,a)]. 1.49/2.12 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(1040,a,1035,a)]. 1.49/2.12 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(1040,a,1036,a)]. 1.49/2.12 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,A,B)),plus_plus(int,A,C))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)). [resolve(1040,a,1037,a)]. 1.49/2.12 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(1040,a,1038,a)]. 1.49/2.12 1041 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,D,B)),plus_plus(A,D,C))) # label(fact_358_add__le__cancel__left) # label(axiom). [clausify(750)]. 1.49/2.12 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_eq(nat),plus_plus(nat,C,A)),plus_plus(nat,C,B))). [resolve(1041,a,1033,a)]. 1.49/2.12 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,C,A)),plus_plus(int,C,B))). [resolve(1041,a,1040,a)]. 1.49/2.12 1042 -ordere236663937imp_le(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,D,B)),plus_plus(A,D,C))) # label(fact_358_add__le__cancel__left) # label(axiom). [clausify(750)]. 1.49/2.12 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_eq(nat),plus_plus(nat,C,A)),plus_plus(nat,C,B))). [resolve(1042,a,1033,a)]. 1.49/2.12 1043 ordere236663937imp_le(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 1.49/2.12 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(1043,a,1034,a)]. 1.49/2.12 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(1043,a,1035,a)]. 1.49/2.12 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(1043,a,1036,a)]. 1.49/2.12 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),plus_plus(real,A,B)),plus_plus(real,A,C))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),C)). [resolve(1043,a,1037,a)]. 1.49/2.12 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(1043,a,1038,a)]. 1.49/2.12 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),plus_plus(real,C,A)),plus_plus(real,C,B))). [resolve(1043,a,1041,a)]. 1.49/2.12 1044 -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(865)]. 1.49/2.12 1045 -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(865)]. 1.94/2.18 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(1045,a,1033,a)]. 1.94/2.18 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(1045,a,1040,a)]. 1.94/2.18 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(1045,a,1043,a)]. 1.94/2.18 1046 -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(1010)]. 1.94/2.18 1047 -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(1010)]. 1.94/2.18 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(1047,a,1033,a)]. 1.94/2.18 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(1047,a,1040,a)]. 1.94/2.18 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(1047,a,1043,a)]. 1.94/2.18 1048 semiri456707255roduct(nat) # label(arity_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc) # label(axiom). [assumption]. 1.94/2.18 1049 -semiri456707255roduct(A) | plus_plus(A,times_times(A,B,C),times_times(A,D,E)) = plus_plus(A,times_times(A,B,E),times_times(A,D,C)) | ti(A,C) != ti(A,E) # label(fact_344_crossproduct__noteq) # label(axiom). [clausify(5)]. 1.94/2.18 1050 -semiri456707255roduct(A) | plus_plus(A,times_times(A,B,C),times_times(A,D,E)) = plus_plus(A,times_times(A,B,E),times_times(A,D,C)) | ti(A,D) != ti(A,B) # label(fact_344_crossproduct__noteq) # label(axiom). [clausify(5)]. 1.94/2.18 1051 -semiri456707255roduct(A) | plus_plus(A,times_times(A,B,C),times_times(A,D,E)) != plus_plus(A,times_times(A,B,E),times_times(A,D,C)) | ti(A,C) = ti(A,E) | ti(A,D) = ti(A,B) # label(fact_344_crossproduct__noteq) # label(axiom). [clausify(5)]. 1.94/2.18 Derived: plus_plus(nat,times_times(nat,A,B),times_times(nat,C,D)) = plus_plus(nat,times_times(nat,A,D),times_times(nat,C,B)) | ti(nat,B) != ti(nat,D). [resolve(1048,a,1049,a)]. 1.94/2.18 Derived: plus_plus(nat,times_times(nat,A,B),times_times(nat,C,D)) = plus_plus(nat,times_times(nat,A,D),times_times(nat,C,B)) | ti(nat,C) != ti(nat,A). [resolve(1048,a,1050,a)]. 1.94/2.18 Derived: plus_plus(nat,times_times(nat,A,B),times_times(nat,C,D)) != plus_plus(nat,times_times(nat,A,D),times_times(nat,C,B)) | ti(nat,B) = ti(nat,D) | ti(nat,C) = ti(nat,A). [resolve(1048,a,1051,a)]. 1.94/2.18 1052 -semiri456707255roduct(A) | ti(A,B) = zero_zero(A) | ti(A,C) != ti(A,D) | ti(A,E) = ti(A,F) | plus_plus(A,C,times_times(A,B,E)) != plus_plus(A,D,times_times(A,B,F)) # label(fact_429_add__scale__eq__noteq) # label(axiom). [clausify(117)]. 1.94/2.18 Derived: ti(nat,A) = zero_zero(nat) | ti(nat,B) != ti(nat,C) | ti(nat,D) = ti(nat,E) | plus_plus(nat,B,times_times(nat,A,D)) != plus_plus(nat,C,times_times(nat,A,E)). [resolve(1052,a,1048,a)]. 1.94/2.18 1053 -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(132)]. 1.94/2.18 1054 -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(132)]. 1.94/2.18 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(1054,a,1048,a)]. 2.02/2.24 1055 -semiri456707255roduct(A) | ti(A,B) = ti(A,C) | ti(A,D) = ti(A,E) | 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(132)]. 2.02/2.24 Derived: ti(nat,A) = ti(nat,B) | ti(nat,C) = ti(nat,D) | 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(1055,a,1048,a)]. 2.02/2.24 1056 semiri456707255roduct(int) # label(arity_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc) # label(axiom). [assumption]. 2.02/2.24 Derived: plus_plus(int,times_times(int,A,B),times_times(int,C,D)) = plus_plus(int,times_times(int,A,D),times_times(int,C,B)) | ti(int,B) != ti(int,D). [resolve(1056,a,1049,a)]. 2.02/2.24 Derived: plus_plus(int,times_times(int,A,B),times_times(int,C,D)) = plus_plus(int,times_times(int,A,D),times_times(int,C,B)) | ti(int,C) != ti(int,A). [resolve(1056,a,1050,a)]. 2.02/2.24 Derived: plus_plus(int,times_times(int,A,B),times_times(int,C,D)) != plus_plus(int,times_times(int,A,D),times_times(int,C,B)) | ti(int,B) = ti(int,D) | ti(int,C) = ti(int,A). [resolve(1056,a,1051,a)]. 2.02/2.24 Derived: ti(int,A) = zero_zero(int) | ti(int,B) != ti(int,C) | ti(int,D) = ti(int,E) | plus_plus(int,B,times_times(int,A,D)) != plus_plus(int,C,times_times(int,A,E)). [resolve(1056,a,1052,a)]. 2.02/2.24 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(1056,a,1054,a)]. 2.02/2.24 Derived: ti(int,A) = ti(int,B) | ti(int,C) = ti(int,D) | 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(1056,a,1055,a)]. 2.02/2.24 1057 -semiri456707255roduct(A) | ti(A,B) != zero_zero(A) | plus_plus(A,C,B) = ti(A,C) # label(fact_212_add__0__iff) # label(axiom). [clausify(543)]. 2.02/2.24 Derived: ti(nat,A) != zero_zero(nat) | plus_plus(nat,B,A) = ti(nat,B). [resolve(1057,a,1048,a)]. 2.02/2.24 Derived: ti(int,A) != zero_zero(int) | plus_plus(int,B,A) = ti(int,B). [resolve(1057,a,1056,a)]. 2.02/2.24 1058 -semiri456707255roduct(A) | ti(A,B) = zero_zero(A) | plus_plus(A,C,B) != ti(A,C) # label(fact_212_add__0__iff) # label(axiom). [clausify(543)]. 2.02/2.24 Derived: ti(nat,A) = zero_zero(nat) | plus_plus(nat,B,A) != ti(nat,B). [resolve(1058,a,1048,a)]. 2.02/2.24 Derived: ti(int,A) = zero_zero(int) | plus_plus(int,B,A) != ti(int,B). [resolve(1058,a,1056,a)]. 2.02/2.24 1059 semiri456707255roduct(real) # label(arity_RealDef_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossp) # label(axiom). [assumption]. 2.02/2.24 Derived: plus_plus(real,times_times(real,A,B),times_times(real,C,D)) = plus_plus(real,times_times(real,A,D),times_times(real,C,B)) | ti(real,B) != ti(real,D). [resolve(1059,a,1049,a)]. 2.02/2.24 Derived: plus_plus(real,times_times(real,A,B),times_times(real,C,D)) = plus_plus(real,times_times(real,A,D),times_times(real,C,B)) | ti(real,C) != ti(real,A). [resolve(1059,a,1050,a)]. 2.02/2.24 Derived: plus_plus(real,times_times(real,A,B),times_times(real,C,D)) != plus_plus(real,times_times(real,A,D),times_times(real,C,B)) | ti(real,B) = ti(real,D) | ti(real,C) = ti(real,A). [resolve(1059,a,1051,a)]. 2.02/2.24 Derived: ti(real,A) = zero_zero(real) | ti(real,B) != ti(real,C) | ti(real,D) = ti(real,E) | plus_plus(real,B,times_times(real,A,D)) != plus_plus(real,C,times_times(real,A,E)). [resolve(1059,a,1052,a)]. 2.02/2.24 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(1059,a,1054,a)]. 2.02/2.24 Derived: ti(real,A) = ti(real,B) | ti(real,C) = ti(real,D) | 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(1059,a,1055,a)]. 2.02/2.24 Derived: ti(real,A) != zero_zero(real) | plus_plus(real,B,A) = ti(real,B). [resolve(1059,a,1057,a)]. 2.02/2.24 Derived: ti(real,A) = zero_zero(real) | plus_plus(real,B,A) != ti(real,B). [resolve(1059,a,1058,a)]. 2.14/2.39 1060 -comm_monoid_mult(A) | ti(A,B) = times_times(A,B,one_one(A)) # label(fact_347_mult_Ocomm__neutral) # label(axiom). [clausify(201)]. 2.14/2.39 1061 comm_monoid_mult(int) # label(arity_Int_Oint___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 2.14/2.39 Derived: ti(int,A) = times_times(int,A,one_one(int)). [resolve(1060,a,1061,a)]. 2.14/2.39 1062 comm_monoid_mult(nat) # label(arity_Nat_Onat___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 2.14/2.39 Derived: ti(nat,A) = times_times(nat,A,one_one(nat)). [resolve(1062,a,1060,a)]. 2.14/2.39 1063 -comm_monoid_mult(A) | ti(A,B) = times_times(A,one_one(A),B) # label(fact_350_mult__1) # label(axiom). [clausify(477)]. 2.14/2.39 Derived: ti(nat,A) = times_times(nat,one_one(nat),A). [resolve(1063,a,1062,a)]. 2.14/2.39 1064 comm_monoid_mult(real) # label(arity_RealDef_Oreal___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 2.14/2.39 Derived: ti(real,A) = times_times(real,A,one_one(real)). [resolve(1064,a,1060,a)]. 2.14/2.39 Derived: ti(real,A) = times_times(real,one_one(real),A). [resolve(1064,a,1063,a)]. 2.14/2.39 1065 -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(799)]. 2.14/2.39 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(1065,a,1061,a)]. 2.14/2.39 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(1065,a,1062,a)]. 2.14/2.39 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(1065,a,1064,a)]. 2.14/2.39 1066 comm_semiring_1(nat) # label(arity_Nat_Onat___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 2.14/2.39 1067 -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(12)]. 2.14/2.39 1068 -comm_semiring_1(A) | plus_plus(A,B,B) = times_times(A,plus_plus(A,one_one(A),one_one(A)),B) # label(fact_445_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom). [clausify(38)]. 2.14/2.39 1069 -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(45)]. 2.14/2.39 1070 -comm_semiring_1(A) | plus_plus(A,times_times(A,B,C),C) = times_times(A,plus_plus(A,B,one_one(A)),C) # label(fact_447_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom). [clausify(63)]. 2.14/2.39 1071 -comm_semiring_1(A) | plus_plus(A,B,times_times(A,C,B)) = times_times(A,plus_plus(A,C,one_one(A)),B) # label(fact_446_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom). [clausify(94)]. 2.14/2.39 1072 -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(100)]. 2.14/2.39 1073 -comm_semiring_1(A) | times_times(A,B,zero_zero(A)) = zero_zero(A) # label(fact_333_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom). [clausify(137)]. 2.14/2.39 1074 -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(142)]. 2.14/2.39 1075 -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(197)]. 2.14/2.39 1076 -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(235)]. 2.14/2.39 1077 -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(242)]. 2.14/2.40 1078 -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(250)]. 2.14/2.40 1079 -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(271)]. 2.14/2.40 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(1066,a,1067,a)]. 2.14/2.40 Derived: plus_plus(nat,A,A) = times_times(nat,plus_plus(nat,one_one(nat),one_one(nat)),A). [resolve(1066,a,1068,a)]. 2.14/2.40 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),zero_zero(nat))). [resolve(1066,a,1069,a)]. 2.14/2.40 Derived: plus_plus(nat,times_times(nat,A,B),B) = times_times(nat,plus_plus(nat,A,one_one(nat)),B). [resolve(1066,a,1070,a)]. 2.14/2.40 Derived: plus_plus(nat,A,times_times(nat,B,A)) = times_times(nat,plus_plus(nat,B,one_one(nat)),A). [resolve(1066,a,1071,a)]. 2.14/2.40 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(1066,a,1072,a)]. 2.14/2.40 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(1066,a,1074,a)]. 2.14/2.40 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(1066,a,1075,a)]. 2.14/2.40 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),A)). [resolve(1066,a,1076,a)]. 2.14/2.40 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(1066,a,1077,a)]. 2.14/2.40 Derived: plus_plus(nat,plus_plus(nat,A,B),C) = plus_plus(nat,plus_plus(nat,A,C),B). [resolve(1066,a,1078,a)]. 2.14/2.40 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(1066,a,1079,a)]. 2.14/2.40 1080 -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(291)]. 2.14/2.40 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(1080,a,1066,a)]. 2.14/2.40 1081 -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(308)]. 2.14/2.40 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(1081,a,1066,a)]. 2.14/2.40 1082 -comm_semiring_1(A) | plus_plus(A,times_times(A,B,C),times_times(A,B,D)) = times_times(A,B,plus_plus(A,C,D)) # label(fact_345_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom). [clausify(319)]. 2.14/2.40 Derived: plus_plus(nat,times_times(nat,A,B),times_times(nat,A,C)) = times_times(nat,A,plus_plus(nat,B,C)). [resolve(1082,a,1066,a)]. 2.14/2.40 1083 -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(351)]. 2.14/2.40 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(1083,a,1066,a)]. 2.20/2.42 1084 -comm_semiring_1(A) | plus_plus(A,times_times(A,B,C),times_times(A,D,C)) = times_times(A,plus_plus(A,B,D),C) # label(fact_342_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom). [clausify(379)]. 2.20/2.42 1085 -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(416)]. 2.20/2.42 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(1085,a,1066,a)]. 2.20/2.42 1086 -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(423)]. 2.20/2.42 1087 -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(427)]. 2.20/2.42 1088 -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(471)]. 2.20/2.42 1089 -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(536)]. 2.20/2.42 Derived: times_times(nat,times_times(nat,A,B),C) = times_times(nat,times_times(nat,A,C),B). [resolve(1089,a,1066,a)]. 2.20/2.42 1090 -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(555)]. 2.20/2.42 Derived: times_times(nat,times_times(nat,A,B),C) = times_times(nat,A,times_times(nat,B,C)). [resolve(1090,a,1066,a)]. 2.20/2.42 1091 -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(556)]. 2.20/2.42 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(1091,a,1066,a)]. 2.20/2.42 1092 -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(578)]. 2.20/2.42 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(1092,a,1066,a)]. 2.20/2.42 1093 -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(623)]. 2.20/2.42 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(1093,a,1066,a)]. 2.20/2.42 1094 -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(628)]. 2.20/2.42 Derived: one_one(nat) = hAPP(nat,nat,power_power(nat,A),zero_zero(nat)). [resolve(1094,a,1066,a)]. 2.20/2.42 1095 -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(676)]. 2.20/2.43 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(1095,a,1066,a)]. 2.20/2.43 1096 -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(680)]. 2.20/2.43 Derived: times_times(nat,A,A) = hAPP(nat,nat,power_power(nat,A),number_number_of(nat,bit0(bit1(pls)))). [resolve(1096,a,1066,a)]. 2.20/2.43 1097 -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(699)]. 2.20/2.43 Derived: times_times(nat,A,times_times(nat,B,C)) = times_times(nat,B,times_times(nat,A,C)). [resolve(1097,a,1066,a)]. 2.20/2.43 1098 -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(701)]. 2.20/2.43 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(1098,a,1066,a)]. 2.20/2.43 1099 -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(701)]. 2.20/2.43 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(1099,a,1066,a)]. 2.20/2.43 1100 -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(705)]. 2.20/2.43 Derived: plus_plus(nat,plus_plus(nat,A,B),C) = plus_plus(nat,A,plus_plus(nat,B,C)). [resolve(1100,a,1066,a)]. 2.20/2.43 1101 -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(708)]. 2.20/2.43 Derived: ti(nat,A) = hAPP(nat,nat,power_power(nat,A),one_one(nat)). [resolve(1101,a,1066,a)]. 2.20/2.43 1102 comm_semiring_1(real) # label(arity_RealDef_Oreal___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 2.20/2.43 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(1102,a,1067,a)]. 2.20/2.43 Derived: plus_plus(real,A,A) = times_times(real,plus_plus(real,one_one(real),one_one(real)),A). [resolve(1102,a,1068,a)]. 2.20/2.43 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),zero_zero(real))). [resolve(1102,a,1069,a)]. 2.20/2.43 Derived: plus_plus(real,times_times(real,A,B),B) = times_times(real,plus_plus(real,A,one_one(real)),B). [resolve(1102,a,1070,a)]. 2.20/2.43 Derived: plus_plus(real,A,times_times(real,B,A)) = times_times(real,plus_plus(real,B,one_one(real)),A). [resolve(1102,a,1071,a)]. 2.20/2.43 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(1102,a,1072,a)]. 2.20/2.43 Derived: times_times(real,A,zero_zero(real)) = zero_zero(real). [resolve(1102,a,1073,a)]. 2.20/2.43 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(1102,a,1074,a)]. 2.20/2.43 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(1102,a,1075,a)]. 2.20/2.43 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),A)). [resolve(1102,a,1076,a)]. 2.20/2.43 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(1102,a,1077,a)]. 2.20/2.43 Derived: plus_plus(real,plus_plus(real,A,B),C) = plus_plus(real,plus_plus(real,A,C),B). [resolve(1102,a,1078,a)]. 2.20/2.43 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(1102,a,1079,a)]. 2.20/2.43 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(1102,a,1080,a)]. 2.20/2.43 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(1102,a,1081,a)]. 2.20/2.43 Derived: plus_plus(real,times_times(real,A,B),times_times(real,A,C)) = times_times(real,A,plus_plus(real,B,C)). [resolve(1102,a,1082,a)]. 2.20/2.43 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(1102,a,1083,a)]. 2.20/2.43 Derived: plus_plus(real,times_times(real,A,B),times_times(real,C,B)) = times_times(real,plus_plus(real,A,C),B). [resolve(1102,a,1084,a)]. 2.20/2.43 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(1102,a,1085,a)]. 2.20/2.43 Derived: plus_plus(real,A,plus_plus(real,B,C)) = plus_plus(real,B,plus_plus(real,A,C)). [resolve(1102,a,1086,a)]. 2.20/2.43 Derived: plus_plus(real,A,B) = plus_plus(real,B,A). [resolve(1102,a,1088,a)]. 2.20/2.43 Derived: times_times(real,times_times(real,A,B),C) = times_times(real,times_times(real,A,C),B). [resolve(1102,a,1089,a)]. 2.20/2.43 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(1102,a,1091,a)]. 2.20/2.43 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(1102,a,1092,a)]. 2.20/2.43 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(1102,a,1093,a)]. 2.20/2.43 Derived: one_one(real) = hAPP(nat,real,power_power(real,A),zero_zero(nat)). [resolve(1102,a,1094,a)]. 2.20/2.43 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(1102,a,1095,a)]. 2.20/2.43 Derived: times_times(real,A,A) = hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))). [resolve(1102,a,1096,a)]. 2.20/2.43 Derived: times_times(real,A,times_times(real,B,C)) = times_times(real,B,times_times(real,A,C)). [resolve(1102,a,1097,a)]. 2.20/2.43 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(1102,a,1098,a)]. 2.20/2.43 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(1102,a,1099,a)]. 2.20/2.43 Derived: plus_plus(real,plus_plus(real,A,B),C) = plus_plus(real,A,plus_plus(real,B,C)). [resolve(1102,a,1100,a)]. 2.20/2.43 Derived: ti(real,A) = hAPP(nat,real,power_power(real,A),one_one(nat)). [resolve(1102,a,1101,a)]. 2.20/2.43 1103 -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(712)]. 2.20/2.44 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),times_times(nat,B,A))). [resolve(1103,a,1066,a)]. 2.20/2.44 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),times_times(real,B,A))). [resolve(1103,a,1102,a)]. 2.20/2.44 1104 -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(733)]. 2.20/2.44 1105 -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(734)]. 2.20/2.44 1106 -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(740)]. 2.20/2.44 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(1106,a,1066,a)]. 2.20/2.44 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(1106,a,1102,a)]. 2.20/2.44 1107 -comm_semiring_1(A) | times_times(A,zero_zero(A),B) = zero_zero(A) # label(fact_334_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom). [clausify(765)]. 2.20/2.44 Derived: times_times(nat,zero_zero(nat),A) = zero_zero(nat). [resolve(1107,a,1066,a)]. 2.20/2.44 Derived: times_times(real,zero_zero(real),A) = zero_zero(real). [resolve(1107,a,1102,a)]. 2.20/2.44 1108 -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(773)]. 2.20/2.44 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),one_one(nat)),A)). [resolve(1108,a,1066,a)]. 2.20/2.44 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),one_one(real)),A)). [resolve(1108,a,1102,a)]. 2.20/2.44 1109 comm_semiring_1(int) # label(arity_Int_Oint___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 2.20/2.44 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(1109,a,1067,a)]. 2.20/2.44 Derived: plus_plus(int,A,A) = times_times(int,plus_plus(int,one_one(int),one_one(int)),A). [resolve(1109,a,1068,a)]. 2.20/2.44 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),zero_zero(int))). [resolve(1109,a,1069,a)]. 2.20/2.44 Derived: plus_plus(int,times_times(int,A,B),B) = times_times(int,plus_plus(int,A,one_one(int)),B). [resolve(1109,a,1070,a)]. 2.20/2.44 Derived: plus_plus(int,A,times_times(int,B,A)) = times_times(int,plus_plus(int,B,one_one(int)),A). [resolve(1109,a,1071,a)]. 2.20/2.44 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(1109,a,1072,a)]. 2.20/2.44 Derived: times_times(int,A,zero_zero(int)) = zero_zero(int). [resolve(1109,a,1073,a)]. 2.20/2.44 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(1109,a,1074,a)]. 2.20/2.44 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(1109,a,1075,a)]. 2.20/2.44 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),A)). [resolve(1109,a,1076,a)]. 2.20/2.44 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(1109,a,1077,a)]. 2.24/2.45 Derived: plus_plus(int,plus_plus(int,A,B),C) = plus_plus(int,plus_plus(int,A,C),B). [resolve(1109,a,1078,a)]. 2.24/2.45 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(1109,a,1079,a)]. 2.24/2.45 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(1109,a,1080,a)]. 2.24/2.45 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(1109,a,1081,a)]. 2.24/2.45 Derived: plus_plus(int,times_times(int,A,B),times_times(int,A,C)) = times_times(int,A,plus_plus(int,B,C)). [resolve(1109,a,1082,a)]. 2.24/2.45 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),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(1109,a,1083,a)]. 2.24/2.45 Derived: plus_plus(int,times_times(int,A,B),times_times(int,C,B)) = times_times(int,plus_plus(int,A,C),B). [resolve(1109,a,1084,a)]. 2.24/2.45 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),plus_plus(int,B,C))). [resolve(1109,a,1085,a)]. 2.24/2.45 Derived: plus_plus(int,A,plus_plus(int,B,C)) = plus_plus(int,B,plus_plus(int,A,C)). [resolve(1109,a,1086,a)]. 2.24/2.45 Derived: times_times(int,times_times(int,A,B),C) = times_times(int,times_times(int,A,C),B). [resolve(1109,a,1089,a)]. 2.24/2.45 Derived: times_times(int,times_times(int,A,B),C) = times_times(int,A,times_times(int,B,C)). [resolve(1109,a,1090,a)]. 2.24/2.45 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(1109,a,1091,a)]. 2.24/2.45 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(1109,a,1092,a)]. 2.24/2.45 Derived: one_one(int) = hAPP(nat,int,power_power(int,A),zero_zero(nat)). [resolve(1109,a,1094,a)]. 2.24/2.45 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),C)). [resolve(1109,a,1095,a)]. 2.24/2.45 Derived: times_times(int,A,A) = hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))). [resolve(1109,a,1096,a)]. 2.24/2.45 Derived: times_times(int,A,times_times(int,B,C)) = times_times(int,B,times_times(int,A,C)). [resolve(1109,a,1097,a)]. 2.24/2.45 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(1109,a,1098,a)]. 2.24/2.45 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(1109,a,1099,a)]. 2.24/2.45 Derived: plus_plus(int,plus_plus(int,A,B),C) = plus_plus(int,A,plus_plus(int,B,C)). [resolve(1109,a,1100,a)]. 2.24/2.45 Derived: ti(int,A) = hAPP(nat,int,power_power(int,A),one_one(nat)). [resolve(1109,a,1101,a)]. 2.24/2.45 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),times_times(int,B,A))). [resolve(1109,a,1103,a)]. 2.24/2.45 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(1109,a,1106,a)]. 2.24/2.45 Derived: times_times(int,zero_zero(int),A) = zero_zero(int). [resolve(1109,a,1107,a)]. 2.24/2.45 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),one_one(int)),A)). [resolve(1109,a,1108,a)]. 2.24/2.45 1110 -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(793)]. 2.24/2.48 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),times_times(nat,A,B))). [resolve(1110,a,1066,a)]. 2.24/2.48 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),times_times(real,A,B))). [resolve(1110,a,1102,a)]. 2.24/2.48 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),times_times(int,A,B))). [resolve(1110,a,1109,a)]. 2.24/2.48 1111 -comm_semiring_1(A) | plus_plus(A,zero_zero(A),B) = ti(A,B) # label(fact_218_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom). [clausify(819)]. 2.24/2.48 Derived: plus_plus(nat,zero_zero(nat),A) = ti(nat,A). [resolve(1111,a,1066,a)]. 2.24/2.48 Derived: plus_plus(real,zero_zero(real),A) = ti(real,A). [resolve(1111,a,1102,a)]. 2.24/2.48 Derived: plus_plus(int,zero_zero(int),A) = ti(int,A). [resolve(1111,a,1109,a)]. 2.24/2.48 1112 -comm_semiring_1(A) | plus_plus(A,times_times(A,B,C),times_times(A,D,C)) = times_times(A,plus_plus(A,B,D),C) # label(fact_343_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom). [clausify(856)]. 2.24/2.48 1113 -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(901)]. 2.24/2.48 1114 -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(927)]. 2.24/2.48 Derived: times_times(nat,A,B) = times_times(nat,B,A). [resolve(1114,a,1066,a)]. 2.24/2.48 Derived: times_times(real,A,B) = times_times(real,B,A). [resolve(1114,a,1102,a)]. 2.24/2.48 Derived: times_times(int,A,B) = times_times(int,B,A). [resolve(1114,a,1109,a)]. 2.24/2.48 1115 -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(928)]. 2.24/2.48 1116 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),zero_zero(A)),B)) | ti(A,B) = zero_zero(A) # label(fact_775_dvd__0__left) # label(axiom). [clausify(944)]. 2.24/2.48 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A)) | ti(nat,A) = zero_zero(nat). [resolve(1116,a,1066,a)]. 2.24/2.48 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),zero_zero(real)),A)) | ti(real,A) = zero_zero(real). [resolve(1116,a,1102,a)]. 2.24/2.48 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),zero_zero(int)),A)) | ti(int,A) = zero_zero(int). [resolve(1116,a,1109,a)]. 2.24/2.48 1117 -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(949)]. 2.24/2.48 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,A),C)),hAPP(nat,nat,power_power(nat,B),C))). [resolve(1117,a,1066,a)]. 2.24/2.48 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(1117,a,1102,a)]. 2.24/2.48 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(1117,a,1109,a)]. 2.24/2.48 1118 -comm_semiring_1(A) | plus_plus(A,B,zero_zero(A)) = ti(A,B) # label(fact_214_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom). [clausify(954)]. 2.24/2.48 Derived: plus_plus(nat,A,zero_zero(nat)) = ti(nat,A). [resolve(1118,a,1066,a)]. 2.24/2.48 Derived: plus_plus(real,A,zero_zero(real)) = ti(real,A). [resolve(1118,a,1102,a)]. 2.24/2.48 Derived: plus_plus(int,A,zero_zero(int)) = ti(int,A). [resolve(1118,a,1109,a)]. 2.24/2.48 1119 -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(962)]. 2.38/2.62 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(1119,a,1066,a)]. 2.38/2.62 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(1119,a,1102,a)]. 2.38/2.62 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(1119,a,1109,a)]. 2.38/2.62 1120 -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(970)]. 2.38/2.62 Derived: ti(int,A) = times_times(int,one_one(int),A). [resolve(1120,a,1109,a)]. 2.38/2.62 1121 -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(69)]. 2.38/2.62 1122 semiring(nat) # label(arity_Nat_Onat___Rings_Osemiring) # label(axiom). [assumption]. 2.38/2.62 1123 semiring(real) # label(arity_RealDef_Oreal___Rings_Osemiring) # label(axiom). [assumption]. 2.38/2.62 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(1123,a,1121,a)]. 2.38/2.62 1124 -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(286)]. 2.38/2.62 1125 -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(464)]. 2.38/2.62 1126 -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(475)]. 2.38/2.62 1127 -number(A) | -semiring(A) | plus_plus(A,times_times(A,B,number_number_of(A,C)),times_times(A,D,number_number_of(A,C))) = times_times(A,plus_plus(A,B,D),number_number_of(A,C)) # label(fact_444_left__distrib__number__of) # label(axiom). [clausify(657)]. 2.38/2.62 1128 -number(A) | -semiring(A) | plus_plus(A,times_times(A,number_number_of(A,B),C),times_times(A,number_number_of(A,B),D)) = times_times(A,number_number_of(A,B),plus_plus(A,C,D)) # label(fact_443_right__distrib__number__of) # label(axiom). [clausify(748)]. 2.38/2.62 1129 semiring(int) # label(arity_Int_Oint___Rings_Osemiring) # label(axiom). [assumption]. 2.38/2.62 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(1129,a,1121,a)]. 2.38/2.62 1130 number_semiring(int) # label(arity_Int_Oint___Int_Onumber__semiring) # label(axiom). [assumption]. 2.38/2.62 1131 -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(15)]. 2.38/2.62 Derived: number_number_of(int,hAPP(nat,int,semiring_1_of_nat(int),A)) = hAPP(nat,int,semiring_1_of_nat(int),A). [resolve(1130,a,1131,a)]. 2.38/2.62 1132 number_semiring(real) # label(arity_RealDef_Oreal___Int_Onumber__semiring) # label(axiom). [assumption]. 2.38/2.62 Derived: number_number_of(real,hAPP(nat,int,semiring_1_of_nat(int),A)) = hAPP(nat,real,semiring_1_of_nat(real),A). [resolve(1132,a,1131,a)]. 2.38/2.62 1133 -number_semiring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),C)) | times_times(A,number_number_of(A,B),number_number_of(A,C)) = number_number_of(A,times_times(int,B,C)) # label(fact_250_semiring__mult__number__of) # label(axiom). [clausify(141)]. 2.38/2.62 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) | times_times(int,number_number_of(int,A),number_number_of(int,B)) = number_number_of(int,times_times(int,A,B)). [resolve(1133,a,1130,a)]. 2.38/2.62 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) | times_times(real,number_number_of(real,A),number_number_of(real,B)) = number_number_of(real,times_times(int,A,B)). [resolve(1133,a,1132,a)]. 2.45/2.67 1134 -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)) | plus_plus(A,number_number_of(A,B),number_number_of(A,C)) = number_number_of(A,plus_plus(int,B,C)) # label(fact_528_semiring__add__number__of) # label(axiom). [clausify(210)]. 2.45/2.67 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)) | plus_plus(int,number_number_of(int,A),number_number_of(int,B)) = number_number_of(int,plus_plus(int,A,B)). [resolve(1134,a,1130,a)]. 2.45/2.67 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)) | plus_plus(real,number_number_of(real,A),number_number_of(real,B)) = number_number_of(real,plus_plus(int,A,B)). [resolve(1134,a,1132,a)]. 2.45/2.67 1135 -number_semiring(A) | one_one(A) = number_number_of(A,bit1(pls)) # label(fact_101_semiring__numeral__1__eq__1) # label(axiom). [clausify(277)]. 2.45/2.67 Derived: one_one(real) = number_number_of(real,bit1(pls)). [resolve(1135,a,1132,a)]. 2.45/2.67 1136 -number_semiring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) | plus_plus(A,one_one(A),number_number_of(A,B)) = number_number_of(A,succ(B)) # label(fact_479_semiring__1__add__number__of) # label(axiom). [clausify(456)]. 2.45/2.67 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | plus_plus(int,one_one(int),number_number_of(int,A)) = number_number_of(int,succ(A)). [resolve(1136,a,1130,a)]. 2.45/2.67 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | plus_plus(real,one_one(real),number_number_of(real,A)) = number_number_of(real,succ(A)). [resolve(1136,a,1132,a)]. 2.45/2.67 1137 -number_semiring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) | plus_plus(A,number_number_of(A,B),one_one(A)) = number_number_of(A,succ(B)) # label(fact_480_semiring__number__of__add__1) # label(axiom). [clausify(530)]. 2.45/2.67 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | plus_plus(int,number_number_of(int,A),one_one(int)) = number_number_of(int,succ(A)). [resolve(1137,a,1130,a)]. 2.45/2.67 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | plus_plus(real,number_number_of(real,A),one_one(real)) = number_number_of(real,succ(A)). [resolve(1137,a,1132,a)]. 2.45/2.67 1138 -number_semiring(A) | plus_plus(A,B,B) = times_times(A,number_number_of(A,bit0(bit1(pls))),B) # label(fact_538_semiring__mult__2) # label(axiom). [clausify(725)]. 2.45/2.67 Derived: plus_plus(int,A,A) = times_times(int,number_number_of(int,bit0(bit1(pls))),A). [resolve(1138,a,1130,a)]. 2.45/2.67 Derived: plus_plus(real,A,A) = times_times(real,number_number_of(real,bit0(bit1(pls))),A). [resolve(1138,a,1132,a)]. 2.45/2.67 1139 number_semiring(nat) # label(arity_Nat_Onat___Int_Onumber__semiring) # label(axiom). [assumption]. 2.45/2.67 Derived: number_number_of(nat,hAPP(nat,int,semiring_1_of_nat(int),A)) = hAPP(nat,nat,semiring_1_of_nat(nat),A). [resolve(1139,a,1131,a)]. 2.45/2.67 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) | times_times(nat,number_number_of(nat,A),number_number_of(nat,B)) = number_number_of(nat,times_times(int,A,B)). [resolve(1139,a,1133,a)]. 2.45/2.67 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)) | plus_plus(nat,number_number_of(nat,A),number_number_of(nat,B)) = number_number_of(nat,plus_plus(int,A,B)). [resolve(1139,a,1134,a)]. 2.45/2.67 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | plus_plus(nat,one_one(nat),number_number_of(nat,A)) = number_number_of(nat,succ(A)). [resolve(1139,a,1136,a)]. 2.45/2.67 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | plus_plus(nat,number_number_of(nat,A),one_one(nat)) = number_number_of(nat,succ(A)). [resolve(1139,a,1137,a)]. 2.51/2.75 Derived: plus_plus(nat,A,A) = times_times(nat,number_number_of(nat,bit0(bit1(pls))),A). [resolve(1139,a,1138,a)]. 2.51/2.75 1140 -number_semiring(A) | number_number_of(A,pls) = zero_zero(A) # label(fact_81_semiring__numeral__0__eq__0) # label(axiom). [clausify(784)]. 2.51/2.75 Derived: number_number_of(int,pls) = zero_zero(int). [resolve(1140,a,1130,a)]. 2.51/2.75 Derived: number_number_of(real,pls) = zero_zero(real). [resolve(1140,a,1132,a)]. 2.51/2.75 Derived: number_number_of(nat,pls) = zero_zero(nat). [resolve(1140,a,1139,a)]. 2.51/2.75 1141 -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(823)]. 2.51/2.75 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(1141,a,1130,a)]. 2.51/2.75 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(1141,a,1139,a)]. 2.51/2.75 1142 -number_semiring(A) | plus_plus(A,one_one(A),one_one(A)) = number_number_of(A,bit0(bit1(pls))) # label(fact_9_semiring__one__add__one__is__two) # label(axiom). [clausify(837)]. 2.51/2.75 Derived: plus_plus(int,one_one(int),one_one(int)) = number_number_of(int,bit0(bit1(pls))). [resolve(1142,a,1130,a)]. 2.51/2.75 Derived: plus_plus(real,one_one(real),one_one(real)) = number_number_of(real,bit0(bit1(pls))). [resolve(1142,a,1132,a)]. 2.51/2.75 Derived: plus_plus(nat,one_one(nat),one_one(nat)) = number_number_of(nat,bit0(bit1(pls))). [resolve(1142,a,1139,a)]. 2.51/2.75 1143 -number_semiring(A) | plus_plus(A,B,B) = times_times(A,B,number_number_of(A,bit0(bit1(pls)))) # label(fact_536_semiring__mult__2__right) # label(axiom). [clausify(894)]. 2.51/2.75 Derived: plus_plus(int,A,A) = times_times(int,A,number_number_of(int,bit0(bit1(pls)))). [resolve(1143,a,1130,a)]. 2.51/2.75 Derived: plus_plus(real,A,A) = times_times(real,A,number_number_of(real,bit0(bit1(pls)))). [resolve(1143,a,1132,a)]. 2.51/2.75 Derived: plus_plus(nat,A,A) = times_times(nat,A,number_number_of(nat,bit0(bit1(pls)))). [resolve(1143,a,1139,a)]. 2.51/2.75 1144 ring(real) # label(arity_RealDef_Oreal___Rings_Oring) # label(axiom). [assumption]. 2.51/2.75 1145 -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(16)]. 2.51/2.75 1146 -ring(A) | plus_plus(A,times_times(A,minus_minus(A,B,C),D),E) != ti(A,F) | plus_plus(A,times_times(A,B,D),E) = plus_plus(A,times_times(A,C,D),F) # label(fact_288_eq__add__iff2) # label(axiom). [clausify(109)]. 2.51/2.75 1147 -ring(A) | plus_plus(A,times_times(A,minus_minus(A,B,C),D),E) = ti(A,F) | plus_plus(A,times_times(A,B,D),E) != plus_plus(A,times_times(A,C,D),F) # label(fact_288_eq__add__iff2) # label(axiom). [clausify(109)]. 2.51/2.75 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(1144,a,1145,a)]. 2.51/2.75 Derived: plus_plus(real,times_times(real,minus_minus(real,A,B),C),D) != ti(real,E) | plus_plus(real,times_times(real,A,C),D) = plus_plus(real,times_times(real,B,C),E). [resolve(1144,a,1146,a)]. 2.51/2.75 Derived: plus_plus(real,times_times(real,minus_minus(real,A,B),C),D) = ti(real,E) | plus_plus(real,times_times(real,A,C),D) != plus_plus(real,times_times(real,B,C),E). [resolve(1144,a,1147,a)]. 2.55/2.85 1148 -ring(A) | plus_plus(A,times_times(A,minus_minus(A,B,C),D),E) != ti(A,F) | plus_plus(A,times_times(A,C,D),F) = plus_plus(A,times_times(A,B,D),E) # label(fact_286_eq__add__iff1) # label(axiom). [clausify(253)]. 2.55/2.85 Derived: plus_plus(real,times_times(real,minus_minus(real,A,B),C),D) != ti(real,E) | plus_plus(real,times_times(real,B,C),E) = plus_plus(real,times_times(real,A,C),D). [resolve(1148,a,1144,a)]. 2.55/2.85 1149 -ring(A) | plus_plus(A,times_times(A,minus_minus(A,B,C),D),E) = ti(A,F) | plus_plus(A,times_times(A,C,D),F) != plus_plus(A,times_times(A,B,D),E) # label(fact_286_eq__add__iff1) # label(axiom). [clausify(253)]. 2.55/2.85 Derived: plus_plus(real,times_times(real,minus_minus(real,A,B),C),D) = ti(real,E) | plus_plus(real,times_times(real,B,C),E) != plus_plus(real,times_times(real,A,C),D). [resolve(1149,a,1144,a)]. 2.55/2.85 1150 -number(A) | -ring(A) | minus_minus(A,times_times(A,B,number_number_of(A,C)),times_times(A,D,number_number_of(A,C))) = times_times(A,minus_minus(A,B,D),number_number_of(A,C)) # label(fact_263_left__diff__distrib__number__of) # label(axiom). [clausify(378)]. 2.55/2.85 Derived: -number(real) | minus_minus(real,times_times(real,A,number_number_of(real,B)),times_times(real,C,number_number_of(real,B))) = times_times(real,minus_minus(real,A,C),number_number_of(real,B)). [resolve(1150,b,1144,a)]. 2.55/2.85 1151 -ring(A) | -number(A) | minus_minus(A,times_times(A,number_number_of(A,B),C),times_times(A,number_number_of(A,B),D)) = times_times(A,number_number_of(A,B),minus_minus(A,C,D)) # label(fact_262_right__diff__distrib__number__of) # label(axiom). [clausify(405)]. 2.55/2.85 Derived: -number(real) | minus_minus(real,times_times(real,number_number_of(real,A),B),times_times(real,number_number_of(real,A),C)) = times_times(real,number_number_of(real,A),minus_minus(real,B,C)). [resolve(1151,a,1144,a)]. 2.55/2.85 1152 ring(int) # label(arity_Int_Oint___Rings_Oring) # label(axiom). [assumption]. 2.55/2.85 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(1152,a,1145,a)]. 2.55/2.85 Derived: plus_plus(int,times_times(int,minus_minus(int,A,B),C),D) != ti(int,E) | plus_plus(int,times_times(int,A,C),D) = plus_plus(int,times_times(int,B,C),E). [resolve(1152,a,1146,a)]. 2.55/2.85 Derived: plus_plus(int,times_times(int,minus_minus(int,A,B),C),D) = ti(int,E) | plus_plus(int,times_times(int,A,C),D) != plus_plus(int,times_times(int,B,C),E). [resolve(1152,a,1147,a)]. 2.55/2.85 Derived: plus_plus(int,times_times(int,minus_minus(int,A,B),C),D) != ti(int,E) | plus_plus(int,times_times(int,B,C),E) = plus_plus(int,times_times(int,A,C),D). [resolve(1152,a,1148,a)]. 2.55/2.85 Derived: plus_plus(int,times_times(int,minus_minus(int,A,B),C),D) = ti(int,E) | plus_plus(int,times_times(int,B,C),E) != plus_plus(int,times_times(int,A,C),D). [resolve(1152,a,1149,a)]. 2.55/2.85 Derived: -number(int) | minus_minus(int,times_times(int,A,number_number_of(int,B)),times_times(int,C,number_number_of(int,B))) = times_times(int,minus_minus(int,A,C),number_number_of(int,B)). [resolve(1152,a,1150,b)]. 2.55/2.85 Derived: -number(int) | minus_minus(int,times_times(int,number_number_of(int,A),B),times_times(int,number_number_of(int,A),C)) = times_times(int,number_number_of(int,A),minus_minus(int,B,C)). [resolve(1152,a,1151,a)]. 2.55/2.85 1153 number_ring(int) # label(arity_Int_Oint___Int_Onumber__ring) # label(axiom). [assumption]. 2.55/2.85 1154 -number_ring(A) | minus_minus(A,number_number_of(A,plus_plus(int,B,C)),D) = plus_plus(A,number_number_of(A,B),minus_minus(A,number_number_of(A,C),D)) # label(fact_482_add__number__of__diff1) # label(axiom). [clausify(18)]. 2.55/2.85 1155 -number_ring(A) | minus_minus(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,minus_minus(A,B,C)),number_number_of(nat,bit0(bit1(pls)))) # label(fact_520_power2__diff) # label(axiom). [clausify(54)]. 2.64/2.86 Derived: minus_minus(int,number_number_of(int,plus_plus(int,A,B)),C) = plus_plus(int,number_number_of(int,A),minus_minus(int,number_number_of(int,B),C)). [resolve(1153,a,1154,a)]. 2.64/2.86 Derived: minus_minus(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,minus_minus(int,A,B)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1153,a,1155,a)]. 2.64/2.86 1156 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),number_number_of(A,B))) # label(fact_50_less__special_I1_J) # label(axiom). [clausify(96)]. 2.64/2.86 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),number_number_of(int,A))). [resolve(1156,b,1153,a)]. 2.64/2.86 1157 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),number_number_of(A,B))) # label(fact_50_less__special_I1_J) # label(axiom). [clausify(96)]. 2.64/2.86 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),number_number_of(int,A))). [resolve(1157,b,1153,a)]. 2.64/2.86 1158 -number_ring(A) | plus_plus(A,B,B) = times_times(A,B,number_number_of(A,bit0(bit1(pls)))) # label(fact_535_mult__2__right) # label(axiom). [clausify(161)]. 2.64/2.86 1159 -number_ring(A) | plus_plus(A,one_one(A),number_number_of(A,B)) = number_number_of(A,plus_plus(int,bit1(pls),B)) # label(fact_6_add__special_I2_J) # label(axiom). [clausify(172)]. 2.64/2.86 Derived: plus_plus(int,one_one(int),number_number_of(int,A)) = number_number_of(int,plus_plus(int,bit1(pls),A)). [resolve(1159,a,1153,a)]. 2.64/2.86 1160 -number_ring(A) | number_number_of(A,pls) = zero_zero(A) # label(fact_82_number__of__Pls) # label(axiom). [clausify(233)]. 2.64/2.86 1161 -number_ring(A) | plus_plus(A,number_number_of(A,B),one_one(A)) = number_number_of(A,plus_plus(int,B,bit1(pls))) # label(fact_7_add__special_I3_J) # label(axiom). [clausify(252)]. 2.64/2.86 Derived: plus_plus(int,number_number_of(int,A),one_one(int)) = number_number_of(int,plus_plus(int,A,bit1(pls))). [resolve(1161,a,1153,a)]. 2.64/2.86 1162 -linordered_idom(A) | -number_ring(A) | abs_abs(A,number_number_of(A,min)) = one_one(A) # label(fact_929_abs__minus__one) # label(axiom). [clausify(303)]. 2.64/2.86 Derived: -linordered_idom(int) | abs_abs(int,number_number_of(int,min)) = one_one(int). [resolve(1162,b,1153,a)]. 2.64/2.86 1163 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),number_number_of(A,B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) # label(fact_389_le__special_I1_J) # label(axiom). [clausify(304)]. 2.64/2.86 Derived: -linordered_idom(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(1163,b,1153,a)]. 2.64/2.86 1164 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),number_number_of(A,B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) # label(fact_389_le__special_I1_J) # label(axiom). [clausify(304)]. 2.64/2.86 Derived: -linordered_idom(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(1164,b,1153,a)]. 2.64/2.86 1165 -linordered_idom(A) | -number_ring(A) | abs_abs(A,hAPP(nat,A,power_power(A,number_number_of(A,min)),B)) = one_one(A) # label(fact_958_abs__power__minus__one) # label(axiom). [clausify(313)]. 2.64/2.86 Derived: -linordered_idom(int) | abs_abs(int,hAPP(nat,int,power_power(int,number_number_of(int,min)),A)) = one_one(int). [resolve(1165,b,1153,a)]. 2.64/2.87 1166 -number_ring(A) | ti(A,B) = times_times(A,B,number_number_of(A,bit1(pls))) # label(fact_497_mult__numeral__1__right) # label(axiom). [clausify(337)]. 2.64/2.87 Derived: ti(int,A) = times_times(int,A,number_number_of(int,bit1(pls))). [resolve(1166,a,1153,a)]. 2.64/2.87 1167 -number_ring(A) | ti(A,B) = times_times(A,number_number_of(A,bit1(pls)),B) # label(fact_498_mult__numeral__1) # label(axiom). [clausify(355)]. 2.64/2.87 Derived: ti(int,A) = times_times(int,number_number_of(int,bit1(pls)),A). [resolve(1167,a,1153,a)]. 2.64/2.87 1168 -number_ring(A) | plus_plus(A,number_number_of(A,pls),B) = ti(A,B) # label(fact_84_add__numeral__0) # label(axiom). [clausify(363)]. 2.64/2.87 Derived: plus_plus(int,number_number_of(int,pls),A) = ti(int,A). [resolve(1168,a,1153,a)]. 2.64/2.87 1169 -number_ring(A) | -linordered_idom(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(365)]. 2.64/2.87 Derived: -linordered_idom(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(1169,a,1153,a)]. 2.64/2.87 1170 -number_ring(A) | -linordered_idom(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(365)]. 2.64/2.87 Derived: -linordered_idom(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(1170,a,1153,a)]. 2.64/2.87 1171 -number_ring(A) | one_one(A) = hAPP(nat,A,power_power(A,number_number_of(A,min)),times_times(nat,number_number_of(nat,bit0(bit1(pls))),B)) # label(fact_791_power__m1__even) # label(axiom). [clausify(385)]. 2.64/2.87 Derived: one_one(int) = hAPP(nat,int,power_power(int,number_number_of(int,min)),times_times(nat,number_number_of(nat,bit0(bit1(pls))),A)). [resolve(1171,a,1153,a)]. 2.64/2.87 1172 -number_ring(A) | -linordered_idom(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(455)]. 2.64/2.87 Derived: -linordered_idom(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(1172,a,1153,a)]. 2.64/2.87 1173 -number_ring(A) | -linordered_idom(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(455)]. 2.64/2.87 Derived: -linordered_idom(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(1173,a,1153,a)]. 2.64/2.87 1174 -number_ring(A) | minus_minus(A,number_number_of(A,B),number_number_of(A,C)) = number_number_of(A,minus_minus(int,B,C)) # label(fact_264_number__of__diff) # label(axiom). [clausify(463)]. 2.64/2.87 Derived: minus_minus(int,number_number_of(int,A),number_number_of(int,B)) = number_number_of(int,minus_minus(int,A,B)). [resolve(1174,a,1153,a)]. 2.64/2.87 1175 -number_ring(A) | plus_plus(A,number_number_of(A,B),number_number_of(A,C)) = number_number_of(A,plus_plus(int,B,C)) # label(fact_88_add__number__of__eq) # label(axiom). [clausify(483)]. 2.64/2.87 Derived: plus_plus(int,number_number_of(int,A),number_number_of(int,B)) = number_number_of(int,plus_plus(int,A,B)). [resolve(1175,a,1153,a)]. 2.64/2.87 1176 -number_ring(A) | plus_plus(A,one_one(A),number_number_of(A,B)) = number_number_of(A,succ(B)) # label(fact_532_number__of__succ) # label(axiom). [clausify(520)]. 2.64/2.87 Derived: plus_plus(int,one_one(int),number_number_of(int,A)) = number_number_of(int,succ(A)). [resolve(1176,a,1153,a)]. 2.68/2.89 1177 -number_ring(A) | times_times(A,plus_plus(A,one_one(A),one_one(A)),number_number_of(A,B)) = number_number_of(A,bit0(B)) # label(fact_525_double__number__of__Bit0) # label(axiom). [clausify(532)]. 2.68/2.89 Derived: times_times(int,plus_plus(int,one_one(int),one_one(int)),number_number_of(int,A)) = number_number_of(int,bit0(A)). [resolve(1177,a,1153,a)]. 2.68/2.89 1178 -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(558)]. 2.68/2.89 1179 -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(558)]. 2.68/2.89 1180 -number_ring(A) | plus_plus(A,plus_plus(A,one_one(A),number_number_of(A,B)),number_number_of(A,B)) = number_number_of(A,bit1(B)) # label(fact_100_number__of__Bit1) # label(axiom). [clausify(570)]. 2.68/2.89 Derived: plus_plus(int,plus_plus(int,one_one(int),number_number_of(int,A)),number_number_of(int,A)) = number_number_of(int,bit1(A)). [resolve(1180,a,1153,a)]. 2.68/2.89 1181 -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(588)]. 2.68/2.89 1182 -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(588)]. 2.68/2.89 1183 -number_ring(A) | plus_plus(A,number_number_of(A,B),number_number_of(A,C)) = number_number_of(A,plus_plus(int,B,C)) # label(fact_89_number__of__add) # label(axiom). [clausify(625)]. 2.68/2.89 1184 -number_ring(A) | plus_plus(A,B,B) = times_times(A,number_number_of(A,bit0(bit1(pls))),B) # label(fact_537_mult__2) # label(axiom). [clausify(656)]. 2.68/2.89 1185 -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(684)]. 2.68/2.89 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),number_number_of(int,A))). [resolve(1185,a,1153,a)]. 2.68/2.89 1186 -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(684)]. 2.68/2.89 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),number_number_of(int,A))). [resolve(1186,a,1153,a)]. 2.68/2.89 1187 -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(726)]. 2.68/2.89 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),A)),hAPP(nat,int,power_power(int,number_number_of(int,bit0(bit1(pls)))),A))). [resolve(1187,b,1153,a)]. 2.68/2.89 1188 number_ring(real) # label(arity_RealDef_Oreal___Int_Onumber__ring) # label(axiom). [assumption]. 2.68/2.89 Derived: minus_minus(real,number_number_of(real,plus_plus(int,A,B)),C) = plus_plus(real,number_number_of(real,A),minus_minus(real,number_number_of(real,B),C)). [resolve(1188,a,1154,a)]. 2.68/2.90 Derived: minus_minus(real,plus_plus(real,hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))),hAPP(nat,real,power_power(real,B),number_number_of(nat,bit0(bit1(pls))))),times_times(real,times_times(real,number_number_of(real,bit0(bit1(pls))),A),B)) = hAPP(nat,real,power_power(real,minus_minus(real,A,B)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1188,a,1155,a)]. 2.68/2.90 Derived: -linordered_idom(real) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),number_number_of(real,A))). [resolve(1188,a,1156,b)]. 2.68/2.90 Derived: -linordered_idom(real) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),number_number_of(real,A))). [resolve(1188,a,1157,b)]. 2.68/2.90 Derived: plus_plus(real,one_one(real),number_number_of(real,A)) = number_number_of(real,plus_plus(int,bit1(pls),A)). [resolve(1188,a,1159,a)]. 2.68/2.90 Derived: plus_plus(real,number_number_of(real,A),one_one(real)) = number_number_of(real,plus_plus(int,A,bit1(pls))). [resolve(1188,a,1161,a)]. 2.68/2.90 Derived: -linordered_idom(real) | abs_abs(real,number_number_of(real,min)) = one_one(real). [resolve(1188,a,1162,b)]. 2.68/2.90 Derived: -linordered_idom(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(1188,a,1163,b)]. 2.68/2.90 Derived: -linordered_idom(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(1188,a,1164,b)]. 2.68/2.90 Derived: -linordered_idom(real) | abs_abs(real,hAPP(nat,real,power_power(real,number_number_of(real,min)),A)) = one_one(real). [resolve(1188,a,1165,b)]. 2.68/2.90 Derived: ti(real,A) = times_times(real,A,number_number_of(real,bit1(pls))). [resolve(1188,a,1166,a)]. 2.68/2.90 Derived: ti(real,A) = times_times(real,number_number_of(real,bit1(pls)),A). [resolve(1188,a,1167,a)]. 2.68/2.90 Derived: plus_plus(real,number_number_of(real,pls),A) = ti(real,A). [resolve(1188,a,1168,a)]. 2.68/2.90 Derived: -linordered_idom(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(1188,a,1169,a)]. 2.68/2.90 Derived: -linordered_idom(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(1188,a,1170,a)]. 2.68/2.90 Derived: one_one(real) = hAPP(nat,real,power_power(real,number_number_of(real,min)),times_times(nat,number_number_of(nat,bit0(bit1(pls))),A)). [resolve(1188,a,1171,a)]. 2.68/2.90 Derived: -linordered_idom(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(1188,a,1172,a)]. 2.68/2.90 Derived: -linordered_idom(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(1188,a,1173,a)]. 2.68/2.90 Derived: minus_minus(real,number_number_of(real,A),number_number_of(real,B)) = number_number_of(real,minus_minus(int,A,B)). [resolve(1188,a,1174,a)]. 2.68/2.90 Derived: plus_plus(real,number_number_of(real,A),number_number_of(real,B)) = number_number_of(real,plus_plus(int,A,B)). [resolve(1188,a,1175,a)]. 2.68/2.90 Derived: plus_plus(real,one_one(real),number_number_of(real,A)) = number_number_of(real,succ(A)). [resolve(1188,a,1176,a)]. 2.68/2.90 Derived: times_times(real,plus_plus(real,one_one(real),one_one(real)),number_number_of(real,A)) = number_number_of(real,bit0(A)). [resolve(1188,a,1177,a)]. 2.68/2.90 Derived: -linordered_idom(real) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),number_number_of(real,B))). [resolve(1188,a,1178,a)]. 2.68/2.90 Derived: -linordered_idom(real) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),number_number_of(real,B))). [resolve(1188,a,1179,a)]. 2.68/2.90 Derived: plus_plus(real,plus_plus(real,one_one(real),number_number_of(real,A)),number_number_of(real,A)) = number_number_of(real,bit1(A)). [resolve(1188,a,1180,a)]. 2.68/2.90 Derived: -linordered_idom(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(1188,a,1181,b)]. 2.68/2.90 Derived: -linordered_idom(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(1188,a,1182,b)]. 2.68/2.90 Derived: -linordered_idom(real) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),number_number_of(real,A))). [resolve(1188,a,1185,a)]. 2.68/2.90 Derived: -linordered_idom(real) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),number_number_of(real,A))). [resolve(1188,a,1186,a)]. 2.68/2.90 Derived: -linordered_idom(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(1188,a,1187,b)]. 2.68/2.90 1189 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,B)),zero_zero(A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),pls)) # label(fact_49_less__special_I3_J) # label(axiom). [clausify(774)]. 2.68/2.90 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,A)),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)). [resolve(1189,b,1153,a)]. 2.68/2.90 Derived: -linordered_idom(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,A)),zero_zero(real))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)). [resolve(1189,b,1188,a)]. 2.68/2.90 1190 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,B)),zero_zero(A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),pls)) # label(fact_49_less__special_I3_J) # label(axiom). [clausify(774)]. 2.68/2.90 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,A)),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)). [resolve(1190,b,1153,a)]. 2.68/2.90 Derived: -linordered_idom(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,A)),zero_zero(real))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)). [resolve(1190,b,1188,a)]. 2.68/2.90 1191 -number_ring(A) | number_number_of(A,pls) = zero_zero(A) # label(fact_83_semiring__norm_I112_J) # label(axiom). [clausify(804)]. 2.68/2.90 1192 -number_ring(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,B))) | hAPP(nat,A,semiring_1_of_nat(A),number_number_of(nat,B)) = zero_zero(A) # label(fact_549_of__nat__number__of__lemma) # label(axiom). [clausify(815)]. 2.68/2.90 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,A))) | hAPP(nat,int,semiring_1_of_nat(int),number_number_of(nat,A)) = zero_zero(int). [resolve(1192,a,1153,a)]. 2.68/2.90 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,A))) | hAPP(nat,real,semiring_1_of_nat(real),number_number_of(nat,A)) = zero_zero(real). [resolve(1192,a,1188,a)]. 2.68/2.90 1193 -number_ring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,B))) | number_number_of(A,B) = hAPP(nat,A,semiring_1_of_nat(A),number_number_of(nat,B)) # label(fact_549_of__nat__number__of__lemma) # label(axiom). [clausify(815)]. 2.68/2.92 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,A))) | number_number_of(int,A) = hAPP(nat,int,semiring_1_of_nat(int),number_number_of(nat,A)). [resolve(1193,a,1153,a)]. 2.68/2.92 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,A))) | number_number_of(real,A) = hAPP(nat,real,semiring_1_of_nat(real),number_number_of(nat,A)). [resolve(1193,a,1188,a)]. 2.68/2.92 1194 -number_ring(A) | times_times(A,number_number_of(A,B),number_number_of(A,C)) = number_number_of(A,times_times(int,B,C)) # label(fact_259_arith__simps_I32_J) # label(axiom). [clausify(861)]. 2.68/2.92 Derived: times_times(int,number_number_of(int,A),number_number_of(int,B)) = number_number_of(int,times_times(int,A,B)). [resolve(1194,a,1153,a)]. 2.68/2.92 Derived: times_times(real,number_number_of(real,A),number_number_of(real,B)) = number_number_of(real,times_times(int,A,B)). [resolve(1194,a,1188,a)]. 2.68/2.92 1195 -number_ring(A) | -ring_char_0(A) | ti(int,B) != ti(int,C) | number_number_of(A,B) = number_number_of(A,C) # label(fact_40_eq__number__of) # label(axiom). [clausify(874)]. 2.68/2.92 Derived: -ring_char_0(int) | ti(int,A) != ti(int,B) | number_number_of(int,A) = number_number_of(int,B). [resolve(1195,a,1153,a)]. 2.68/2.92 Derived: -ring_char_0(real) | ti(int,A) != ti(int,B) | number_number_of(real,A) = number_number_of(real,B). [resolve(1195,a,1188,a)]. 2.68/2.92 1196 -number_ring(A) | -ring_char_0(A) | ti(int,B) = ti(int,C) | number_number_of(A,B) != number_number_of(A,C) # label(fact_40_eq__number__of) # label(axiom). [clausify(874)]. 2.68/2.92 Derived: -ring_char_0(int) | ti(int,A) = ti(int,B) | number_number_of(int,A) != number_number_of(int,B). [resolve(1196,a,1153,a)]. 2.68/2.92 Derived: -ring_char_0(real) | ti(int,A) = ti(int,B) | number_number_of(real,A) != number_number_of(real,B). [resolve(1196,a,1188,a)]. 2.68/2.92 1197 -number_ring(A) | plus_plus(A,plus_plus(A,zero_zero(A),number_number_of(A,B)),number_number_of(A,B)) = number_number_of(A,bit0(B)) # label(fact_99_number__of__Bit0) # label(axiom). [clausify(882)]. 2.68/2.92 Derived: plus_plus(int,plus_plus(int,zero_zero(int),number_number_of(int,A)),number_number_of(int,A)) = number_number_of(int,bit0(A)). [resolve(1197,a,1153,a)]. 2.68/2.92 Derived: plus_plus(real,plus_plus(real,zero_zero(real),number_number_of(real,A)),number_number_of(real,A)) = number_number_of(real,bit0(A)). [resolve(1197,a,1188,a)]. 2.68/2.92 1198 -number_ring(A) | times_times(A,number_number_of(A,times_times(int,B,C)),D) = times_times(A,number_number_of(A,B),times_times(A,number_number_of(A,C),D)) # label(fact_260_mult__number__of__left) # label(axiom). [clausify(910)]. 2.68/2.92 Derived: times_times(int,number_number_of(int,times_times(int,A,B)),C) = times_times(int,number_number_of(int,A),times_times(int,number_number_of(int,B),C)). [resolve(1198,a,1153,a)]. 2.68/2.92 Derived: times_times(real,number_number_of(real,times_times(int,A,B)),C) = times_times(real,number_number_of(real,A),times_times(real,number_number_of(real,B),C)). [resolve(1198,a,1188,a)]. 2.68/2.92 1199 -number_ring(A) | times_times(A,number_number_of(A,B),number_number_of(A,C)) = number_number_of(A,times_times(int,B,C)) # label(fact_258_number__of__mult) # label(axiom). [clausify(920)]. 2.68/2.92 1200 -number_ring(A) | plus_plus(A,number_number_of(A,plus_plus(int,B,C)),D) = plus_plus(A,number_number_of(A,B),plus_plus(A,number_number_of(A,C),D)) # label(fact_87_add__number__of__left) # label(axiom). [clausify(925)]. 2.68/2.92 Derived: plus_plus(int,number_number_of(int,plus_plus(int,A,B)),C) = plus_plus(int,number_number_of(int,A),plus_plus(int,number_number_of(int,B),C)). [resolve(1200,a,1153,a)]. 2.68/2.92 Derived: plus_plus(real,number_number_of(real,plus_plus(int,A,B)),C) = plus_plus(real,number_number_of(real,A),plus_plus(real,number_number_of(real,B),C)). [resolve(1200,a,1188,a)]. 2.68/2.92 1201 -number_ring(A) | one_one(A) = number_number_of(A,bit1(pls)) # label(fact_13_numeral__1__eq__1) # label(axiom). [clausify(931)]. 2.73/2.99 Derived: one_one(int) = number_number_of(int,bit1(pls)). [resolve(1201,a,1153,a)]. 2.73/2.99 1202 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,B)),zero_zero(A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),pls)) # label(fact_390_le__special_I3_J) # label(axiom). [clausify(939)]. 2.73/2.99 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,A)),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),pls)). [resolve(1202,b,1153,a)]. 2.73/2.99 Derived: -linordered_idom(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),zero_zero(real))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),pls)). [resolve(1202,b,1188,a)]. 2.73/2.99 1203 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,B)),zero_zero(A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),pls)) # label(fact_390_le__special_I3_J) # label(axiom). [clausify(939)]. 2.73/2.99 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,A)),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),pls)). [resolve(1203,b,1153,a)]. 2.73/2.99 Derived: -linordered_idom(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),zero_zero(real))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),pls)). [resolve(1203,b,1188,a)]. 2.73/2.99 1204 -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(951)]. 2.73/2.99 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,A)),one_one(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),bit1(pls))). [resolve(1204,a,1153,a)]. 2.73/2.99 Derived: -linordered_idom(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),one_one(real))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),bit1(pls))). [resolve(1204,a,1188,a)]. 2.73/2.99 1205 -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(951)]. 2.73/2.99 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,A)),one_one(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),bit1(pls))). [resolve(1205,a,1153,a)]. 2.73/2.99 Derived: -linordered_idom(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),one_one(real))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),bit1(pls))). [resolve(1205,a,1188,a)]. 2.73/2.99 1206 -number_ring(A) | plus_plus(A,B,number_number_of(A,pls)) = ti(A,B) # label(fact_85_add__numeral__0__right) # label(axiom). [clausify(960)]. 2.73/2.99 Derived: plus_plus(int,A,number_number_of(int,pls)) = ti(int,A). [resolve(1206,a,1153,a)]. 2.73/2.99 Derived: plus_plus(real,A,number_number_of(real,pls)) = ti(real,A). [resolve(1206,a,1188,a)]. 2.73/2.99 1207 -number_ring(A) | plus_plus(A,one_one(A),one_one(A)) = number_number_of(A,bit0(bit1(pls))) # label(fact_8_one__add__one__is__two) # label(axiom). [clausify(984)]. 2.73/2.99 1208 -number_ring(A) | one_one(A) = number_number_of(A,bit1(pls)) # label(fact_12_semiring__norm_I110_J) # label(axiom). [clausify(987)]. 2.73/2.99 1209 linordered_semidom(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semidom) # label(axiom). [assumption]. 2.73/2.99 1210 -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(24)]. 2.73/2.99 1211 -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(64)]. 2.73/2.99 1212 -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(93)]. 2.73/2.99 1213 -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(112)]. 2.73/2.99 1214 -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(180)]. 2.73/2.99 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(1209,a,1210,a)]. 2.73/2.99 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(1209,a,1211,a)]. 2.73/2.99 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(1209,a,1212,a)]. 2.73/2.99 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(1209,a,1213,a)]. 2.73/2.99 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(1209,a,1214,a)]. 2.73/2.99 1215 -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(226)]. 2.73/2.99 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(1215,a,1209,a)]. 2.73/2.99 1216 -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(267)]. 2.73/2.99 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(1216,a,1209,a)]. 2.73/3.00 1217 -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),hAPP(nat,A,semiring_1_of_nat(A),B)),hAPP(nat,A,semiring_1_of_nat(A),C))) # label(fact_266_of__nat__le__iff) # label(axiom). [clausify(312)]. 2.73/3.00 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),hAPP(nat,real,semiring_1_of_nat(real),A)),hAPP(nat,real,semiring_1_of_nat(real),B))). [resolve(1217,a,1209,a)]. 2.73/3.00 1218 -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),hAPP(nat,A,semiring_1_of_nat(A),B)),hAPP(nat,A,semiring_1_of_nat(A),C))) # label(fact_266_of__nat__le__iff) # label(axiom). [clausify(312)]. 2.73/3.00 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),hAPP(nat,real,semiring_1_of_nat(real),A)),hAPP(nat,real,semiring_1_of_nat(real),B))). [resolve(1218,a,1209,a)]. 2.73/3.00 1219 -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(327)]. 2.73/3.00 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(1219,a,1209,a)]. 2.73/3.00 1220 linordered_semidom(nat) # label(arity_Nat_Onat___Rings_Olinordered__semidom) # label(axiom). [assumption]. 2.73/3.00 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(1220,a,1210,a)]. 2.73/3.00 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(1220,a,1211,a)]. 2.73/3.00 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(1220,a,1212,a)]. 2.73/3.00 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(1220,a,1213,a)]. 2.73/3.00 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(1220,a,1214,a)]. 2.73/3.00 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(1220,a,1215,a)]. 2.73/3.00 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,semiring_1_of_nat(nat),A)),hAPP(nat,nat,semiring_1_of_nat(nat),B))). [resolve(1220,a,1217,a)]. 2.73/3.00 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,semiring_1_of_nat(nat),A)),hAPP(nat,nat,semiring_1_of_nat(nat),B))). [resolve(1220,a,1218,a)]. 2.73/3.01 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(1220,a,1219,a)]. 2.73/3.01 1221 -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(409)]. 2.73/3.01 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,power_power(real,A),B)),hAPP(nat,real,power_power(real,A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)). [resolve(1221,a,1209,a)]. 2.73/3.01 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,power_power(nat,A),B)),hAPP(nat,nat,power_power(nat,A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)). [resolve(1221,a,1220,a)]. 2.73/3.01 1222 -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(419)]. 2.73/3.01 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(1222,a,1209,a)]. 2.73/3.01 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(1222,a,1220,a)]. 2.73/3.01 1223 -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(420)]. 2.73/3.01 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(1223,a,1209,a)]. 2.73/3.01 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(1223,a,1220,a)]. 2.73/3.01 1224 -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(514)]. 2.73/3.01 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(1224,a,1209,a)]. 2.80/3.01 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(1224,a,1220,a)]. 2.80/3.01 1225 -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(515)]. 2.80/3.01 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(1225,a,1209,a)]. 2.80/3.01 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(1225,a,1220,a)]. 2.80/3.01 1226 -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),C),D)) | 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))) # label(fact_182_power__strict__increasing__iff) # label(axiom). [clausify(516)]. 2.80/3.01 1227 -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),C),D)) | -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))) # label(fact_182_power__strict__increasing__iff) # label(axiom). [clausify(516)]. 2.80/3.01 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),B),C)) | -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))). [resolve(1227,a,1209,a)]. 2.80/3.01 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),B),C)) | -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))). [resolve(1227,a,1220,a)]. 2.80/3.01 1228 -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(533)]. 2.80/3.01 1229 -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(577)]. 2.80/3.01 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(1229,a,1209,a)]. 2.80/3.02 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(1229,a,1220,a)]. 2.80/3.02 1230 -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(586)]. 2.80/3.02 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(1230,a,1209,a)]. 2.80/3.02 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(1230,a,1220,a)]. 2.80/3.02 1231 -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(589)]. 2.80/3.02 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(1231,a,1209,a)]. 2.80/3.02 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(1231,a,1220,a)]. 2.80/3.02 1232 -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(604)]. 2.80/3.02 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(1232,a,1209,a)]. 2.80/3.02 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(1232,a,1220,a)]. 2.80/3.02 1233 -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(610)]. 2.80/3.02 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(1233,a,1209,a)]. 2.80/3.02 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(1233,a,1220,a)]. 2.80/3.02 1234 -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(649)]. 2.80/3.02 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(1234,a,1209,a)]. 2.80/3.02 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(1234,a,1220,a)]. 2.80/3.03 1235 -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(686)]. 2.80/3.03 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(1235,a,1209,a)]. 2.80/3.03 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(1235,a,1220,a)]. 2.80/3.03 1236 -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(697)]. 2.80/3.03 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,semiring_1_of_nat(real),A)),hAPP(nat,real,semiring_1_of_nat(real),B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)). [resolve(1236,a,1209,a)]. 2.80/3.03 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,semiring_1_of_nat(nat),A)),hAPP(nat,nat,semiring_1_of_nat(nat),B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)). [resolve(1236,a,1220,a)]. 2.80/3.03 1237 -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(713)]. 2.80/3.03 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),zero_zero(real))). [resolve(1237,a,1209,a)]. 2.80/3.03 1238 -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(717)]. 2.80/3.03 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(1238,a,1209,a)]. 2.80/3.03 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(1238,a,1220,a)]. 2.80/3.03 1239 -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(746)]. 2.80/3.03 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(1239,a,1209,a)]. 2.80/3.03 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(1239,a,1220,a)]. 2.80/3.05 1240 -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(775)]. 2.80/3.05 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(1240,a,1209,a)]. 2.80/3.05 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(1240,a,1220,a)]. 2.80/3.05 1241 -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(781)]. 2.80/3.05 1242 -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(826)]. 2.80/3.05 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),one_one(real))). [resolve(1242,a,1209,a)]. 2.80/3.05 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),one_one(nat))). [resolve(1242,a,1220,a)]. 2.80/3.05 1243 -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(831)]. 2.80/3.05 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),plus_plus(real,A,one_one(real)))). [resolve(1243,a,1209,a)]. 2.80/3.05 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),plus_plus(nat,A,one_one(nat)))). [resolve(1243,a,1220,a)]. 2.80/3.05 1244 -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(846)]. 2.80/3.05 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(1244,a,1209,a)]. 2.80/3.05 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(1244,a,1220,a)]. 2.80/3.05 1245 -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(846)]. 2.80/3.05 1246 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,semiring_1_of_nat(A),B)),hAPP(nat,A,semiring_1_of_nat(A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)) # label(fact_167_of__nat__less__iff) # label(axiom). [clausify(890)]. 2.80/3.05 1247 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,semiring_1_of_nat(A),B)),hAPP(nat,A,semiring_1_of_nat(A),C))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)) # label(fact_167_of__nat__less__iff) # label(axiom). [clausify(890)]. 2.80/3.05 1248 -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(891)]. 2.80/3.06 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(1248,a,1209,a)]. 2.80/3.06 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(1248,a,1220,a)]. 2.80/3.06 1249 -linordered_semidom(A) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(nat,A,semiring_1_of_nat(A),B))) # label(fact_107_of__nat__0__less__iff) # label(axiom). [clausify(899)]. 2.80/3.06 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,semiring_1_of_nat(real),A))). [resolve(1249,a,1209,a)]. 2.80/3.06 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,semiring_1_of_nat(nat),A))). [resolve(1249,a,1220,a)]. 2.80/3.06 1250 -linordered_semidom(A) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(nat,A,semiring_1_of_nat(A),B))) # label(fact_107_of__nat__0__less__iff) # label(axiom). [clausify(899)]. 2.80/3.06 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,semiring_1_of_nat(real),A))). [resolve(1250,a,1209,a)]. 2.80/3.06 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,semiring_1_of_nat(nat),A))). [resolve(1250,a,1220,a)]. 2.80/3.06 1251 -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(956)]. 2.80/3.06 1252 -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(973)]. 2.80/3.06 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),one_one(real))). [resolve(1252,a,1209,a)]. 2.80/3.06 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),one_one(nat))). [resolve(1252,a,1220,a)]. 2.80/3.06 1253 -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(978)]. 2.80/3.06 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),zero_zero(real))). [resolve(1253,a,1209,a)]. 2.80/3.06 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),zero_zero(nat))). [resolve(1253,a,1220,a)]. 2.80/3.06 1254 linordered_semidom(int) # label(arity_Int_Oint___Rings_Olinordered__semidom) # label(axiom). [assumption]. 2.80/3.06 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(1254,a,1210,a)]. 2.80/3.06 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(1254,a,1211,a)]. 2.80/3.06 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,power_power(int,A),B))). [resolve(1254,a,1212,a)]. 2.80/3.06 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(1254,a,1213,a)]. 2.80/3.06 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(1254,a,1214,a)]. 2.80/3.06 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(1254,a,1215,a)]. 2.80/3.06 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(1254,a,1216,a)]. 2.80/3.06 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),hAPP(nat,int,semiring_1_of_nat(int),A)),hAPP(nat,int,semiring_1_of_nat(int),B))). [resolve(1254,a,1217,a)]. 2.80/3.06 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),hAPP(nat,int,semiring_1_of_nat(int),A)),hAPP(nat,int,semiring_1_of_nat(int),B))). [resolve(1254,a,1218,a)]. 2.80/3.06 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(1254,a,1219,a)]. 2.80/3.06 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,power_power(int,A),B)),hAPP(nat,int,power_power(int,A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)). [resolve(1254,a,1221,a)]. 2.80/3.06 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(1254,a,1222,a)]. 2.80/3.06 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(1254,a,1223,a)]. 2.80/3.06 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(1254,a,1224,a)]. 2.80/3.06 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(1254,a,1225,a)]. 2.85/3.06 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),B),C)) | -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))). [resolve(1254,a,1227,a)]. 2.85/3.06 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(1254,a,1229,a)]. 2.85/3.06 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(1254,a,1231,a)]. 2.85/3.06 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(1254,a,1232,a)]. 2.85/3.06 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(1254,a,1233,a)]. 2.85/3.06 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(1254,a,1234,a)]. 2.85/3.06 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(1254,a,1235,a)]. 2.85/3.06 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),A)),hAPP(nat,int,semiring_1_of_nat(int),B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)). [resolve(1254,a,1236,a)]. 2.85/3.06 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),zero_zero(int))). [resolve(1254,a,1237,a)]. 2.85/3.06 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(1254,a,1238,a)]. 2.85/3.06 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(1254,a,1239,a)]. 2.85/3.06 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(1254,a,1240,a)]. 2.85/3.06 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),one_one(int))). [resolve(1254,a,1242,a)]. 2.85/3.06 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),plus_plus(int,A,one_one(int)))). [resolve(1254,a,1243,a)]. 2.85/3.06 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(1254,a,1244,a)]. 2.85/3.06 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(1254,a,1248,a)]. 2.85/3.13 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),ord_less(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),A))). [resolve(1254,a,1249,a)]. 2.85/3.13 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),ord_less(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),A))). [resolve(1254,a,1250,a)]. 2.85/3.13 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),one_one(int))). [resolve(1254,a,1252,a)]. 2.85/3.13 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),zero_zero(int))). [resolve(1254,a,1253,a)]. 2.85/3.13 1255 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | C != D | hAPP(nat,A,power_power(A,B),C) = hAPP(nat,A,power_power(A,B),D) # label(fact_181_power__inject__exp) # label(axiom). [clausify(998)]. 2.85/3.13 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A)) | B != C | hAPP(nat,real,power_power(real,A),B) = hAPP(nat,real,power_power(real,A),C). [resolve(1255,a,1209,a)]. 2.85/3.13 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),A)) | B != C | hAPP(nat,nat,power_power(nat,A),B) = hAPP(nat,nat,power_power(nat,A),C). [resolve(1255,a,1220,a)]. 2.85/3.13 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A)) | B != C | hAPP(nat,int,power_power(int,A),B) = hAPP(nat,int,power_power(int,A),C). [resolve(1255,a,1254,a)]. 2.85/3.13 1256 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | C = D | hAPP(nat,A,power_power(A,B),C) != hAPP(nat,A,power_power(A,B),D) # label(fact_181_power__inject__exp) # label(axiom). [clausify(998)]. 2.85/3.13 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A)) | B = C | hAPP(nat,real,power_power(real,A),B) != hAPP(nat,real,power_power(real,A),C). [resolve(1256,a,1209,a)]. 2.85/3.13 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),A)) | B = C | hAPP(nat,nat,power_power(nat,A),B) != hAPP(nat,nat,power_power(nat,A),C). [resolve(1256,a,1220,a)]. 2.85/3.13 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A)) | B = C | hAPP(nat,int,power_power(int,A),B) != hAPP(nat,int,power_power(int,A),C). [resolve(1256,a,1254,a)]. 2.85/3.13 1257 ordered_semiring(int) # label(arity_Int_Oint___Rings_Oordered__semiring) # label(axiom). [assumption]. 2.85/3.13 1258 -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(26)]. 2.85/3.13 1259 -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(415)]. 2.85/3.13 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(1257,a,1258,a)]. 2.85/3.13 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(1257,a,1259,a)]. 2.94/3.18 1260 -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(485)]. 2.94/3.18 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),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(1260,a,1257,a)]. 2.94/3.18 1261 ordered_semiring(nat) # label(arity_Nat_Onat___Rings_Oordered__semiring) # label(axiom). [assumption]. 2.94/3.18 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),times_times(nat,A,C)),times_times(nat,B,D))). [resolve(1261,a,1258,a)]. 2.94/3.18 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),times_times(nat,A,C)),times_times(nat,B,D))). [resolve(1261,a,1259,a)]. 2.94/3.18 1262 -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(907)]. 2.94/3.18 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),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(1262,a,1257,a)]. 2.94/3.18 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),times_times(nat,A,C)),times_times(nat,B,C))). [resolve(1262,a,1261,a)]. 2.94/3.18 1263 ordered_semiring(real) # label(arity_RealDef_Oreal___Rings_Oordered__semiring) # label(axiom). [assumption]. 2.94/3.18 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),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(1263,a,1258,a)]. 2.94/3.18 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),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(1263,a,1259,a)]. 2.94/3.18 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),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(1263,a,1260,a)]. 3.11/3.36 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_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(1263,a,1262,a)]. 3.11/3.36 1264 -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(548)]. 3.11/3.36 1265 cancel146912293up_add(nat) # label(arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 3.11/3.36 1266 cancel146912293up_add(real) # label(arity_RealDef_Oreal___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 3.11/3.36 Derived: plus_plus(nat,A,B) != plus_plus(nat,A,C) | ti(nat,B) = ti(nat,C). [resolve(1264,a,1265,a)]. 3.11/3.36 Derived: plus_plus(real,A,B) != plus_plus(real,A,C) | ti(real,B) = ti(real,C). [resolve(1264,a,1266,a)]. 3.11/3.36 1267 cancel146912293up_add(int) # label(arity_Int_Oint___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 3.11/3.36 Derived: plus_plus(int,A,B) != plus_plus(int,A,C) | ti(int,B) = ti(int,C). [resolve(1267,a,1264,a)]. 3.11/3.36 1268 ordere142940540dd_abs(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__group__add__abs) # label(axiom). [assumption]. 3.11/3.36 1269 -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(34)]. 3.11/3.36 1270 -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(68)]. 3.11/3.36 1271 -ordere142940540dd_abs(A) | abs_abs(A,B) != zero_zero(A) | ti(A,B) = zero_zero(A) # label(fact_931_abs__eq__0) # label(axiom). [clausify(124)]. 3.11/3.36 1272 -ordere142940540dd_abs(A) | abs_abs(A,B) = zero_zero(A) | ti(A,B) != zero_zero(A) # label(fact_931_abs__eq__0) # label(axiom). [clausify(124)]. 3.11/3.36 1273 -ordere142940540dd_abs(A) | abs_abs(A,B) = ti(A,abs_abs(A,B)) # label(tsy_c_Groups_Oabs__class_Oabs_res) # label(axiom). [clausify(160)]. 3.11/3.36 1274 -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(194)]. 3.11/3.36 1275 -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(199)]. 3.11/3.36 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(1268,a,1269,a)]. 3.11/3.36 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(1268,a,1270,a)]. 3.11/3.36 Derived: abs_abs(real,A) != zero_zero(real) | ti(real,A) = zero_zero(real). [resolve(1268,a,1271,a)]. 3.11/3.36 Derived: abs_abs(real,A) = zero_zero(real) | ti(real,A) != zero_zero(real). [resolve(1268,a,1272,a)]. 3.11/3.36 Derived: abs_abs(real,A) = ti(real,abs_abs(real,A)). [resolve(1268,a,1273,a)]. 3.11/3.36 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(1268,a,1274,a)]. 3.11/3.36 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(1268,a,1275,a)]. 3.11/3.36 1276 -ordere142940540dd_abs(A) | ti(A,B) != zero_zero(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),abs_abs(A,B)),zero_zero(A))) # label(fact_948_abs__le__zero__iff) # label(axiom). [clausify(370)]. 3.18/3.39 Derived: ti(real,A) != zero_zero(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),abs_abs(real,A)),zero_zero(real))). [resolve(1276,a,1268,a)]. 3.18/3.39 1277 -ordere142940540dd_abs(A) | ti(A,B) = zero_zero(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),abs_abs(A,B)),zero_zero(A))) # label(fact_948_abs__le__zero__iff) # label(axiom). [clausify(370)]. 3.18/3.39 Derived: ti(real,A) = zero_zero(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),abs_abs(real,A)),zero_zero(real))). [resolve(1277,a,1268,a)]. 3.18/3.39 1278 -ordere142940540dd_abs(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | abs_abs(A,B) = ti(A,B) # label(fact_944_abs__of__pos) # label(axiom). [clausify(431)]. 3.18/3.39 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | abs_abs(real,A) = ti(real,A). [resolve(1278,a,1268,a)]. 3.18/3.39 1279 -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(522)]. 3.18/3.39 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),abs_abs(real,A))). [resolve(1279,a,1268,a)]. 3.18/3.39 1280 -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(540)]. 3.18/3.39 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(1280,a,1268,a)]. 3.18/3.39 1281 -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(544)]. 3.18/3.39 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(1281,a,1268,a)]. 3.18/3.39 1282 -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(571)]. 3.18/3.39 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(1282,a,1268,a)]. 3.18/3.39 1283 -ordere142940540dd_abs(A) | abs_abs(A,ti(A,B)) = abs_abs(A,B) # label(tsy_c_Groups_Oabs__class_Oabs_arg1) # label(axiom). [clausify(572)]. 3.18/3.39 Derived: abs_abs(real,ti(real,A)) = abs_abs(real,A). [resolve(1283,a,1268,a)]. 3.18/3.39 1284 -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(605)]. 3.18/3.39 Derived: abs_abs(real,minus_minus(real,A,B)) = abs_abs(real,minus_minus(real,B,A)). [resolve(1284,a,1268,a)]. 3.18/3.39 1285 -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(613)]. 3.18/3.39 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),abs_abs(real,A)),zero_zero(real))). [resolve(1285,a,1268,a)]. 3.18/3.39 1286 -ordere142940540dd_abs(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | abs_abs(A,B) = ti(A,B) # label(fact_947_abs__of__nonneg) # label(axiom). [clausify(690)]. 3.18/3.39 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | abs_abs(real,A) = ti(real,A). [resolve(1286,a,1268,a)]. 3.18/3.39 1287 -ordere142940540dd_abs(A) | abs_abs(A,zero_zero(A)) = zero_zero(A) # label(fact_930_abs__zero) # label(axiom). [clausify(723)]. 3.18/3.39 Derived: abs_abs(real,zero_zero(real)) = zero_zero(real). [resolve(1287,a,1268,a)]. 3.18/3.39 1288 -ordere142940540dd_abs(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),abs_abs(A,B))) | ti(A,B) != zero_zero(A) # label(fact_945_zero__less__abs__iff) # label(axiom). [clausify(745)]. 3.18/3.40 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),abs_abs(real,A))) | ti(real,A) != zero_zero(real). [resolve(1288,a,1268,a)]. 3.18/3.40 1289 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),abs_abs(A,B))) | ti(A,B) = zero_zero(A) # label(fact_945_zero__less__abs__iff) # label(axiom). [clausify(745)]. 3.18/3.40 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),abs_abs(real,A))) | ti(real,A) = zero_zero(real). [resolve(1289,a,1268,a)]. 3.18/3.40 1290 -ordere142940540dd_abs(A) | abs_abs(A,abs_abs(A,B)) = abs_abs(A,B) # label(fact_937_abs__idempotent) # label(axiom). [clausify(862)]. 3.18/3.40 Derived: abs_abs(real,abs_abs(real,A)) = abs_abs(real,A). [resolve(1290,a,1268,a)]. 3.18/3.40 1291 ordere142940540dd_abs(int) # label(arity_Int_Oint___Groups_Oordered__ab__group__add__abs) # label(axiom). [assumption]. 3.18/3.40 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(1291,a,1269,a)]. 3.18/3.40 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(1291,a,1270,a)]. 3.18/3.40 Derived: abs_abs(int,A) != zero_zero(int) | ti(int,A) = zero_zero(int). [resolve(1291,a,1271,a)]. 3.18/3.40 Derived: abs_abs(int,A) = zero_zero(int) | ti(int,A) != zero_zero(int). [resolve(1291,a,1272,a)]. 3.18/3.40 Derived: abs_abs(int,A) = ti(int,abs_abs(int,A)). [resolve(1291,a,1273,a)]. 3.18/3.40 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(1291,a,1274,a)]. 3.18/3.40 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(1291,a,1275,a)]. 3.18/3.40 Derived: ti(int,A) != zero_zero(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),abs_abs(int,A)),zero_zero(int))). [resolve(1291,a,1276,a)]. 3.18/3.40 Derived: ti(int,A) = zero_zero(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),abs_abs(int,A)),zero_zero(int))). [resolve(1291,a,1277,a)]. 3.18/3.40 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | abs_abs(int,A) = ti(int,A). [resolve(1291,a,1278,a)]. 3.18/3.40 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),abs_abs(int,A))). [resolve(1291,a,1279,a)]. 3.18/3.40 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(1291,a,1280,a)]. 3.18/3.40 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(1291,a,1281,a)]. 3.18/3.40 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(1291,a,1282,a)]. 3.18/3.40 Derived: abs_abs(int,ti(int,A)) = abs_abs(int,A). [resolve(1291,a,1283,a)]. 3.18/3.40 Derived: abs_abs(int,minus_minus(int,A,B)) = abs_abs(int,minus_minus(int,B,A)). [resolve(1291,a,1284,a)]. 3.18/3.40 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),abs_abs(int,A)),zero_zero(int))). [resolve(1291,a,1285,a)]. 3.18/3.40 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | abs_abs(int,A) = ti(int,A). [resolve(1291,a,1286,a)]. 3.18/3.40 Derived: abs_abs(int,zero_zero(int)) = zero_zero(int). [resolve(1291,a,1287,a)]. 3.18/3.40 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),abs_abs(int,A))) | ti(int,A) != zero_zero(int). [resolve(1291,a,1288,a)]. 3.18/3.40 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),abs_abs(int,A))) | ti(int,A) = zero_zero(int). [resolve(1291,a,1289,a)]. 3.18/3.40 Derived: abs_abs(int,abs_abs(int,A)) = abs_abs(int,A). [resolve(1291,a,1290,a)]. 3.18/3.40 1292 -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(878)]. 3.34/3.60 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),abs_abs(real,A))). [resolve(1292,a,1268,a)]. 3.34/3.60 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),abs_abs(int,A))). [resolve(1292,a,1291,a)]. 3.34/3.60 1293 -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(1002)]. 3.34/3.60 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(1293,a,1268,a)]. 3.34/3.60 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(1293,a,1291,a)]. 3.34/3.60 1294 -comm_monoid_add(A) | plus_plus(A,zero_zero(A),B) = ti(A,B) # label(fact_217_add__0) # label(axiom). [clausify(268)]. 3.34/3.60 1295 comm_monoid_add(real) # label(arity_RealDef_Oreal___Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 3.34/3.60 1296 -comm_monoid_add(A) | plus_plus(A,B,zero_zero(A)) = ti(A,B) # label(fact_213_add_Ocomm__neutral) # label(axiom). [clausify(602)]. 3.34/3.60 1297 comm_monoid_add(nat) # label(arity_Nat_Onat___Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 3.34/3.60 1298 comm_monoid_add(int) # label(arity_Int_Oint___Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 3.34/3.60 1299 ordere216010020id_add(nat) # label(arity_Nat_Onat___Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 3.34/3.60 1300 -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(46)]. 3.34/3.60 1301 -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(80)]. 3.34/3.60 1302 -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(279)]. 3.34/3.60 1303 -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(318)]. 3.34/3.60 1304 -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(362)]. 3.34/3.60 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(1304,a,1299,a)]. 3.34/3.60 1305 ordere216010020id_add(int) # label(arity_Int_Oint___Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 3.34/3.60 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(1305,a,1300,a)]. 3.34/3.60 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(1305,a,1301,a)]. 3.34/3.60 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(1305,a,1302,a)]. 3.34/3.60 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(1305,a,1303,a)]. 3.34/3.60 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(1305,a,1304,a)]. 3.34/3.60 1306 -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(388)]. 3.34/3.60 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(1306,a,1305,a)]. 3.34/3.60 1307 -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(411)]. 3.34/3.60 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(1307,a,1299,a)]. 3.34/3.60 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(1307,a,1305,a)]. 3.34/3.60 1308 ordere216010020id_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 3.34/3.60 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(1308,a,1300,a)]. 3.34/3.60 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(1308,a,1301,a)]. 3.34/3.60 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(1308,a,1302,a)]. 3.34/3.60 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(1308,a,1303,a)]. 3.34/3.60 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(1308,a,1304,a)]. 3.42/3.62 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(1308,a,1306,a)]. 3.42/3.62 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(1308,a,1307,a)]. 3.42/3.62 1309 -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(561)]. 3.42/3.62 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(1309,a,1299,a)]. 3.42/3.62 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(1309,a,1305,a)]. 3.42/3.62 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(1309,a,1308,a)]. 3.42/3.62 1310 -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(581)]. 3.42/3.62 1311 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | plus_plus(A,B,C) != zero_zero(A) | ti(A,B) = zero_zero(A) # label(fact_435_add__nonneg__eq__0__iff) # label(axiom). [clausify(691)]. 3.42/3.62 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | plus_plus(nat,A,B) != zero_zero(nat) | ti(nat,A) = zero_zero(nat). [resolve(1311,a,1299,a)]. 3.42/3.62 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | plus_plus(int,A,B) != zero_zero(int) | ti(int,A) = zero_zero(int). [resolve(1311,a,1305,a)]. 3.42/3.62 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | plus_plus(real,A,B) != zero_zero(real) | ti(real,A) = zero_zero(real). [resolve(1311,a,1308,a)]. 3.42/3.62 1312 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | plus_plus(A,B,C) != zero_zero(A) | ti(A,C) = zero_zero(A) # label(fact_435_add__nonneg__eq__0__iff) # label(axiom). [clausify(691)]. 3.42/3.62 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | plus_plus(nat,A,B) != zero_zero(nat) | ti(nat,B) = zero_zero(nat). [resolve(1312,a,1299,a)]. 3.42/3.62 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | plus_plus(int,A,B) != zero_zero(int) | ti(int,B) = zero_zero(int). [resolve(1312,a,1305,a)]. 3.42/3.62 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | plus_plus(real,A,B) != zero_zero(real) | ti(real,B) = zero_zero(real). [resolve(1312,a,1308,a)]. 3.48/3.72 1313 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | plus_plus(A,B,C) = zero_zero(A) | ti(A,B) != zero_zero(A) | ti(A,C) != zero_zero(A) # label(fact_435_add__nonneg__eq__0__iff) # label(axiom). [clausify(691)]. 3.48/3.72 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | plus_plus(nat,A,B) = zero_zero(nat) | ti(nat,A) != zero_zero(nat) | ti(nat,B) != zero_zero(nat). [resolve(1313,a,1299,a)]. 3.48/3.72 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | plus_plus(int,A,B) = zero_zero(int) | ti(int,A) != zero_zero(int) | ti(int,B) != zero_zero(int). [resolve(1313,a,1305,a)]. 3.48/3.72 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | plus_plus(real,A,B) = zero_zero(real) | ti(real,A) != zero_zero(real) | ti(real,B) != zero_zero(real). [resolve(1313,a,1308,a)]. 3.48/3.72 1314 -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(833)]. 3.48/3.72 1315 -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(997)]. 3.48/3.72 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(nat),B),zero_zero(nat))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,A,B)),zero_zero(nat))). [resolve(1315,a,1299,a)]. 3.48/3.72 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(1315,a,1305,a)]. 3.48/3.72 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(1315,a,1308,a)]. 3.48/3.72 1316 -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(1006)]. 3.48/3.72 1317 -zero_neq_one(A) | one_one(A) != zero_zero(A) # label(fact_164_zero__neq__one) # label(axiom). [clausify(71)]. 3.48/3.72 1318 zero_neq_one(int) # label(arity_Int_Oint___Rings_Ozero__neq__one) # label(axiom). [assumption]. 3.48/3.72 Derived: one_one(int) != zero_zero(int). [resolve(1317,a,1318,a)]. 3.48/3.72 1319 -zero_neq_one(A) | -no_zero_divisors(A) | -mult_zero(A) | -power(A) | ti(A,B) != zero_zero(A) | number_number_of(nat,C) = zero_zero(nat) | hAPP(nat,A,power_power(A,B),number_number_of(nat,C)) = zero_zero(A) # label(fact_86_power__eq__0__iff__number__of) # label(axiom). [clausify(106)]. 3.48/3.72 Derived: -no_zero_divisors(int) | -mult_zero(int) | -power(int) | ti(int,A) != zero_zero(int) | number_number_of(nat,B) = zero_zero(nat) | hAPP(nat,int,power_power(int,A),number_number_of(nat,B)) = zero_zero(int). [resolve(1319,a,1318,a)]. 3.48/3.74 1320 -zero_neq_one(A) | -no_zero_divisors(A) | -mult_zero(A) | -power(A) | ti(A,B) = zero_zero(A) | hAPP(nat,A,power_power(A,B),number_number_of(nat,C)) != zero_zero(A) # label(fact_86_power__eq__0__iff__number__of) # label(axiom). [clausify(106)]. 3.48/3.74 Derived: -no_zero_divisors(int) | -mult_zero(int) | -power(int) | ti(int,A) = zero_zero(int) | hAPP(nat,int,power_power(int,A),number_number_of(nat,B)) != zero_zero(int). [resolve(1320,a,1318,a)]. 3.48/3.74 1321 -zero_neq_one(A) | -no_zero_divisors(A) | -mult_zero(A) | -power(A) | number_number_of(nat,B) != zero_zero(nat) | hAPP(nat,A,power_power(A,C),number_number_of(nat,B)) != zero_zero(A) # label(fact_86_power__eq__0__iff__number__of) # label(axiom). [clausify(106)]. 3.48/3.74 Derived: -no_zero_divisors(int) | -mult_zero(int) | -power(int) | number_number_of(nat,A) != zero_zero(nat) | hAPP(nat,int,power_power(int,B),number_number_of(nat,A)) != zero_zero(int). [resolve(1321,a,1318,a)]. 3.48/3.74 1322 zero_neq_one(real) # label(arity_RealDef_Oreal___Rings_Ozero__neq__one) # label(axiom). [assumption]. 3.48/3.74 Derived: -no_zero_divisors(real) | -mult_zero(real) | -power(real) | ti(real,A) != zero_zero(real) | number_number_of(nat,B) = zero_zero(nat) | hAPP(nat,real,power_power(real,A),number_number_of(nat,B)) = zero_zero(real). [resolve(1322,a,1319,a)]. 3.48/3.74 Derived: -no_zero_divisors(real) | -mult_zero(real) | -power(real) | ti(real,A) = zero_zero(real) | hAPP(nat,real,power_power(real,A),number_number_of(nat,B)) != zero_zero(real). [resolve(1322,a,1320,a)]. 3.48/3.74 Derived: -no_zero_divisors(real) | -mult_zero(real) | -power(real) | number_number_of(nat,A) != zero_zero(nat) | hAPP(nat,real,power_power(real,B),number_number_of(nat,A)) != zero_zero(real). [resolve(1322,a,1321,a)]. 3.48/3.74 1323 zero_neq_one(nat) # label(arity_Nat_Onat___Rings_Ozero__neq__one) # label(axiom). [assumption]. 3.48/3.74 Derived: one_one(nat) != zero_zero(nat). [resolve(1323,a,1317,a)]. 3.48/3.74 Derived: -no_zero_divisors(nat) | -mult_zero(nat) | -power(nat) | ti(nat,A) != zero_zero(nat) | number_number_of(nat,B) = zero_zero(nat) | hAPP(nat,nat,power_power(nat,A),number_number_of(nat,B)) = zero_zero(nat). [resolve(1323,a,1319,a)]. 3.48/3.74 Derived: -no_zero_divisors(nat) | -mult_zero(nat) | -power(nat) | ti(nat,A) = zero_zero(nat) | hAPP(nat,nat,power_power(nat,A),number_number_of(nat,B)) != zero_zero(nat). [resolve(1323,a,1320,a)]. 3.48/3.74 Derived: -no_zero_divisors(nat) | -mult_zero(nat) | -power(nat) | number_number_of(nat,A) != zero_zero(nat) | hAPP(nat,nat,power_power(nat,B),number_number_of(nat,A)) != zero_zero(nat). [resolve(1323,a,1321,a)]. 3.48/3.74 1324 -power(A) | -mult_zero(A) | -zero_neq_one(A) | -no_zero_divisors(A) | hAPP(nat,A,power_power(A,B),C) != zero_zero(A) | zero_zero(nat) != C # label(fact_185_power__eq__0__iff) # label(axiom). [clausify(481)]. 3.48/3.74 Derived: -power(int) | -mult_zero(int) | -no_zero_divisors(int) | hAPP(nat,int,power_power(int,A),B) != zero_zero(int) | zero_zero(nat) != B. [resolve(1324,c,1318,a)]. 3.48/3.74 Derived: -power(real) | -mult_zero(real) | -no_zero_divisors(real) | hAPP(nat,real,power_power(real,A),B) != zero_zero(real) | zero_zero(nat) != B. [resolve(1324,c,1322,a)]. 3.48/3.74 1325 -power(A) | -mult_zero(A) | -zero_neq_one(A) | -no_zero_divisors(A) | hAPP(nat,A,power_power(A,B),C) != zero_zero(A) | ti(A,B) = zero_zero(A) # label(fact_185_power__eq__0__iff) # label(axiom). [clausify(481)]. 3.48/3.74 Derived: -power(int) | -mult_zero(int) | -no_zero_divisors(int) | hAPP(nat,int,power_power(int,A),B) != zero_zero(int) | ti(int,A) = zero_zero(int). [resolve(1325,c,1318,a)]. 3.48/3.74 Derived: -power(real) | -mult_zero(real) | -no_zero_divisors(real) | hAPP(nat,real,power_power(real,A),B) != zero_zero(real) | ti(real,A) = zero_zero(real). [resolve(1325,c,1322,a)]. 3.48/3.74 Derived: -power(nat) | -mult_zero(nat) | -no_zero_divisors(nat) | hAPP(nat,nat,power_power(nat,A),B) != zero_zero(nat) | ti(nat,A) = zero_zero(nat). [resolve(1325,c,1323,a)]. 3.48/3.74 1326 -power(A) | -mult_zero(A) | -zero_neq_one(A) | -no_zero_divisors(A) | hAPP(nat,A,power_power(A,B),C) = zero_zero(A) | zero_zero(nat) = C | ti(A,B) != zero_zero(A) # label(fact_185_power__eq__0__iff) # label(axiom). [clausify(481)]. 3.63/3.85 Derived: -power(int) | -mult_zero(int) | -no_zero_divisors(int) | hAPP(nat,int,power_power(int,A),B) = zero_zero(int) | zero_zero(nat) = B | ti(int,A) != zero_zero(int). [resolve(1326,c,1318,a)]. 3.63/3.85 Derived: -power(real) | -mult_zero(real) | -no_zero_divisors(real) | hAPP(nat,real,power_power(real,A),B) = zero_zero(real) | zero_zero(nat) = B | ti(real,A) != zero_zero(real). [resolve(1326,c,1322,a)]. 3.63/3.85 Derived: -power(nat) | -mult_zero(nat) | -no_zero_divisors(nat) | hAPP(nat,nat,power_power(nat,A),B) = zero_zero(nat) | zero_zero(nat) = B | ti(nat,A) != zero_zero(nat). [resolve(1326,c,1323,a)]. 3.63/3.85 1327 -zero_neq_one(A) | one_one(A) != zero_zero(A) # label(fact_163_one__neq__zero) # label(axiom). [clausify(848)]. 3.63/3.85 Derived: one_one(real) != zero_zero(real). [resolve(1327,a,1322,a)]. 3.63/3.85 1328 linord20386208strict(int) # label(arity_Int_Oint___Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 3.63/3.85 1329 -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(57)]. 3.63/3.85 1330 -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(135)]. 3.63/3.85 1331 -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(185)]. 3.63/3.85 1332 -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(219)]. 3.63/3.85 1333 -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(224)]. 3.63/3.85 1334 -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(227)]. 3.63/3.85 1335 -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(249)]. 3.63/3.85 1336 -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(259)]. 3.63/3.85 1337 -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(377)]. 3.66/3.86 1338 -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(430)]. 3.66/3.86 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(1328,a,1329,a)]. 3.66/3.86 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),times_times(int,A,B)),times_times(int,A,C))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)). [resolve(1328,a,1330,a)]. 3.66/3.86 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,B,A)),zero_zero(int))). [resolve(1328,a,1331,a)]. 3.66/3.86 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),times_times(int,A,C)),times_times(int,B,C))). [resolve(1328,a,1332,a)]. 3.66/3.86 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(1328,a,1334,a)]. 3.66/3.86 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),D)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(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(1328,a,1335,a)]. 3.66/3.86 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(1328,a,1336,a)]. 3.66/3.86 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(1328,a,1337,a)]. 3.66/3.86 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_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(1328,a,1338,a)]. 3.66/3.86 1339 -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(503)]. 3.66/3.86 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(1339,a,1328,a)]. 3.66/3.86 1340 -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(511)]. 3.66/3.87 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(1340,a,1328,a)]. 3.66/3.87 1341 -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(664)]. 3.66/3.87 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(1341,a,1328,a)]. 3.66/3.87 1342 linord20386208strict(nat) # label(arity_Nat_Onat___Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 3.66/3.87 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(1342,a,1329,a)]. 3.66/3.87 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(1342,a,1333,a)]. 3.66/3.87 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(1342,a,1335,a)]. 3.66/3.87 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),times_times(nat,A,B)),times_times(nat,C,B))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),C)). [resolve(1342,a,1336,a)]. 3.66/3.87 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(1342,a,1337,a)]. 3.66/3.87 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(1342,a,1338,a)]. 3.66/3.87 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),times_times(nat,A,B))). [resolve(1342,a,1339,a)]. 3.66/3.88 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(1342,a,1340,a)]. 3.66/3.88 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(1342,a,1341,a)]. 3.66/3.88 1343 linord20386208strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 3.66/3.88 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(1343,a,1329,a)]. 3.66/3.88 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(1343,a,1330,a)]. 3.66/3.88 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,B,A)),zero_zero(real))). [resolve(1343,a,1331,a)]. 3.66/3.88 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),zero_zero(real)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),times_times(real,A,C)),times_times(real,B,C))). [resolve(1343,a,1332,a)]. 3.66/3.88 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(1343,a,1333,a)]. 3.66/3.88 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(1343,a,1334,a)]. 3.66/3.88 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),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(1343,a,1335,a)]. 3.66/3.88 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(1343,a,1336,a)]. 3.66/3.88 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_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(1343,a,1338,a)]. 3.78/3.98 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),times_times(real,A,B))). [resolve(1343,a,1339,a)]. 3.78/3.98 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(1343,a,1340,a)]. 3.78/3.98 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(1343,a,1341,a)]. 3.78/3.98 1344 -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(810)]. 3.78/3.98 1345 -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(933)]. 3.78/3.98 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),times_times(int,C,A)),times_times(int,C,B))). [resolve(1345,a,1328,a)]. 3.78/3.98 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(1345,a,1342,a)]. 3.78/3.98 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),times_times(real,C,A)),times_times(real,C,B))). [resolve(1345,a,1343,a)]. 3.78/3.98 1346 -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(986)]. 3.78/3.98 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(1346,a,1328,a)]. 3.78/3.98 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(1346,a,1342,a)]. 3.78/3.98 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(1346,a,1343,a)]. 3.78/3.98 1347 -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))) | abs_abs(A,times_times(A,B,C)) = times_times(A,abs_abs(A,B),abs_abs(A,C)) # label(fact_925_abs__eq__mult) # label(axiom). [clausify(525)]. 3.86/4.10 1348 ordered_ring_abs(int) # label(arity_Int_Oint___Rings_Oordered__ring__abs) # label(axiom). [assumption]. 3.86/4.10 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))) | abs_abs(int,times_times(int,A,B)) = times_times(int,abs_abs(int,A),abs_abs(int,B)). [resolve(1347,a,1348,a)]. 3.86/4.10 1349 -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)) | abs_abs(A,times_times(A,B,C)) = times_times(A,abs_abs(A,B),abs_abs(A,C)) # label(fact_925_abs__eq__mult) # label(axiom). [clausify(525)]. 3.86/4.10 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | abs_abs(int,times_times(int,A,B)) = times_times(int,abs_abs(int,A),abs_abs(int,B)). [resolve(1349,a,1348,a)]. 3.86/4.10 1350 -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))) | abs_abs(A,times_times(A,B,C)) = times_times(A,abs_abs(A,B),abs_abs(A,C)) # label(fact_925_abs__eq__mult) # label(axiom). [clausify(525)]. 3.86/4.10 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))) | abs_abs(int,times_times(int,A,B)) = times_times(int,abs_abs(int,A),abs_abs(int,B)). [resolve(1350,a,1348,a)]. 3.86/4.10 1351 -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)) | abs_abs(A,times_times(A,B,C)) = times_times(A,abs_abs(A,B),abs_abs(A,C)) # label(fact_925_abs__eq__mult) # label(axiom). [clausify(525)]. 3.86/4.10 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | abs_abs(int,times_times(int,A,B)) = times_times(int,abs_abs(int,A),abs_abs(int,B)). [resolve(1351,a,1348,a)]. 3.86/4.10 1352 ordered_ring_abs(real) # label(arity_RealDef_Oreal___Rings_Oordered__ring__abs) # label(axiom). [assumption]. 3.86/4.10 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),zero_zero(real))) | abs_abs(real,times_times(real,A,B)) = times_times(real,abs_abs(real,A),abs_abs(real,B)). [resolve(1352,a,1347,a)]. 3.86/4.10 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | abs_abs(real,times_times(real,A,B)) = times_times(real,abs_abs(real,A),abs_abs(real,B)). [resolve(1352,a,1349,a)]. 3.86/4.10 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),zero_zero(real))) | abs_abs(real,times_times(real,A,B)) = times_times(real,abs_abs(real,A),abs_abs(real,B)). [resolve(1352,a,1350,a)]. 3.86/4.10 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | abs_abs(real,times_times(real,A,B)) = times_times(real,abs_abs(real,A),abs_abs(real,B)). [resolve(1352,a,1351,a)]. 3.86/4.10 1353 comm_semiring(int) # label(arity_Int_Oint___Rings_Ocomm__semiring) # label(axiom). [assumption]. 3.86/4.10 1354 -comm_semiring(A) | plus_plus(A,times_times(A,B,C),times_times(A,D,C)) = times_times(A,plus_plus(A,B,D),C) # label(fact_339_comm__semiring__class_Odistrib) # label(axiom). [clausify(75)]. 3.86/4.10 1355 comm_semiring(real) # label(arity_RealDef_Oreal___Rings_Ocomm__semiring) # label(axiom). [assumption]. 4.14/4.35 1356 comm_semiring(nat) # label(arity_Nat_Onat___Rings_Ocomm__semiring) # label(axiom). [assumption]. 4.14/4.35 Derived: plus_plus(nat,times_times(nat,A,B),times_times(nat,C,B)) = times_times(nat,plus_plus(nat,A,C),B). [resolve(1356,a,1354,a)]. 4.14/4.35 1357 monoid_add(real) # label(arity_RealDef_Oreal___Groups_Omonoid__add) # label(axiom). [assumption]. 4.14/4.35 1358 -monoid_add(A) | plus_plus(A,B,zero_zero(A)) = ti(A,B) # label(fact_215_add__0__right) # label(axiom). [clausify(78)]. 4.14/4.35 1359 -monoid_add(A) | plus_plus(A,zero_zero(A),B) = ti(A,B) # label(fact_219_add__0__left) # label(axiom). [clausify(115)]. 4.14/4.35 1360 -monoid_add(A) | plus_plus(A,B,C) = ti(A,plus_plus(A,B,C)) # label(tsy_c_Groups_Oplus__class_Oplus_0_res) # label(axiom). [clausify(453)]. 4.14/4.35 Derived: plus_plus(real,A,B) = ti(real,plus_plus(real,A,B)). [resolve(1360,a,1357,a)]. 4.14/4.35 1361 monoid_add(int) # label(arity_Int_Oint___Groups_Omonoid__add) # label(axiom). [assumption]. 4.14/4.35 Derived: plus_plus(int,A,B) = ti(int,plus_plus(int,A,B)). [resolve(1361,a,1360,a)]. 4.14/4.35 1362 -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(576)]. 4.14/4.35 Derived: plus_plus(real,A,ti(real,B)) = plus_plus(real,A,B). [resolve(1362,a,1357,a)]. 4.14/4.35 Derived: plus_plus(int,A,ti(int,B)) = plus_plus(int,A,B). [resolve(1362,a,1361,a)]. 4.14/4.35 1363 monoid_add(nat) # label(arity_Nat_Onat___Groups_Omonoid__add) # label(axiom). [assumption]. 4.14/4.35 Derived: plus_plus(nat,A,B) = ti(nat,plus_plus(nat,A,B)). [resolve(1363,a,1360,a)]. 4.14/4.35 Derived: plus_plus(nat,A,ti(nat,B)) = plus_plus(nat,A,B). [resolve(1363,a,1362,a)]. 4.14/4.35 1364 -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(946)]. 4.14/4.35 Derived: plus_plus(real,ti(real,A),B) = plus_plus(real,A,B). [resolve(1364,a,1357,a)]. 4.14/4.35 Derived: plus_plus(int,ti(int,A),B) = plus_plus(int,A,B). [resolve(1364,a,1361,a)]. 4.14/4.35 Derived: plus_plus(nat,ti(nat,A),B) = plus_plus(nat,A,B). [resolve(1364,a,1363,a)]. 4.14/4.35 1365 idom(int) # label(arity_Int_Oint___Rings_Oidom) # label(axiom). [assumption]. 4.14/4.35 1366 -idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),times_times(A,B,C)),times_times(A,B,D))) | ti(A,B) = zero_zero(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),D)) # label(fact_740_dvd__mult__cancel__left) # label(axiom). [clausify(81)]. 4.14/4.35 1367 -idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),times_times(A,B,C)),times_times(A,B,D))) | ti(A,B) != zero_zero(A) # label(fact_740_dvd__mult__cancel__left) # label(axiom). [clausify(81)]. 4.14/4.35 1368 -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(81)]. 4.14/4.35 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),times_times(int,A,B)),times_times(int,A,C))) | ti(int,A) = zero_zero(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),C)). [resolve(1365,a,1366,a)]. 4.14/4.35 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),times_times(int,A,B)),times_times(int,A,C))) | ti(int,A) != zero_zero(int). [resolve(1365,a,1367,a)]. 4.14/4.35 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(1365,a,1368,a)]. 4.14/4.35 1369 -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),times_times(A,B,D)),times_times(A,C,D))) # label(fact_741_dvd__mult__cancel__right) # label(axiom). [clausify(682)]. 4.14/4.35 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),times_times(int,A,C)),times_times(int,B,C))). [resolve(1369,a,1365,a)]. 4.14/4.35 1370 -idom(A) | ti(A,B) != zero_zero(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),times_times(A,C,B)),times_times(A,D,B))) # label(fact_741_dvd__mult__cancel__right) # label(axiom). [clausify(682)]. 4.23/4.45 Derived: ti(int,A) != zero_zero(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),times_times(int,B,A)),times_times(int,C,A))). [resolve(1370,a,1365,a)]. 4.23/4.45 1371 -idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | ti(A,D) = zero_zero(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),times_times(A,B,D)),times_times(A,C,D))) # label(fact_741_dvd__mult__cancel__right) # label(axiom). [clausify(682)]. 4.23/4.45 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | ti(int,C) = zero_zero(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),times_times(int,A,C)),times_times(int,B,C))). [resolve(1371,a,1365,a)]. 4.23/4.45 1372 idom(real) # label(arity_RealDef_Oreal___Rings_Oidom) # label(axiom). [assumption]. 4.23/4.45 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),times_times(real,A,B)),times_times(real,A,C))) | ti(real,A) = zero_zero(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),B),C)). [resolve(1372,a,1366,a)]. 4.23/4.45 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),times_times(real,A,B)),times_times(real,A,C))) | ti(real,A) != zero_zero(real). [resolve(1372,a,1367,a)]. 4.23/4.45 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(1372,a,1368,a)]. 4.23/4.45 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),times_times(real,A,C)),times_times(real,B,C))). [resolve(1372,a,1369,a)]. 4.23/4.45 Derived: ti(real,A) != zero_zero(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),times_times(real,B,A)),times_times(real,C,A))). [resolve(1372,a,1370,a)]. 4.23/4.45 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | ti(real,C) = zero_zero(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),times_times(real,A,C)),times_times(real,B,C))). [resolve(1372,a,1371,a)]. 4.23/4.45 1373 linord219039673up_add(real) # label(arity_RealDef_Oreal___Groups_Olinordered__ab__group__add) # label(axiom). [assumption]. 4.23/4.45 1374 -linord219039673up_add(A) | ti(A,B) != zero_zero(A) | plus_plus(A,B,B) = zero_zero(A) # label(fact_216_double__zero__sym) # label(axiom). [clausify(83)]. 4.23/4.45 1375 -linord219039673up_add(A) | ti(A,B) = zero_zero(A) | plus_plus(A,B,B) != zero_zero(A) # label(fact_216_double__zero__sym) # label(axiom). [clausify(83)]. 4.23/4.45 1376 -linord219039673up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,B,B)),zero_zero(A))) # label(fact_433_double__add__le__zero__iff__single__add__le__zero) # label(axiom). [clausify(134)]. 4.23/4.45 1377 -linord219039673up_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,B,B)),zero_zero(A))) # label(fact_433_double__add__le__zero__iff__single__add__le__zero) # label(axiom). [clausify(134)]. 4.23/4.45 Derived: ti(real,A) != zero_zero(real) | plus_plus(real,A,A) = zero_zero(real). [resolve(1373,a,1374,a)]. 4.23/4.45 Derived: ti(real,A) = zero_zero(real) | plus_plus(real,A,A) != zero_zero(real). [resolve(1373,a,1375,a)]. 4.23/4.45 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),plus_plus(real,A,A)),zero_zero(real))). [resolve(1373,a,1377,a)]. 4.23/4.45 1378 -linord219039673up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),plus_plus(A,B,B))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) # label(fact_432_zero__le__double__add__iff__zero__le__single__add) # label(axiom). [clausify(341)]. 4.23/4.45 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),plus_plus(real,A,A))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)). [resolve(1378,a,1373,a)]. 4.23/4.45 1379 -linord219039673up_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),plus_plus(A,B,B))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) # label(fact_432_zero__le__double__add__iff__zero__le__single__add) # label(axiom). [clausify(341)]. 4.36/4.57 1380 linord219039673up_add(int) # label(arity_Int_Oint___Groups_Olinordered__ab__group__add) # label(axiom). [assumption]. 4.36/4.57 Derived: ti(int,A) != zero_zero(int) | plus_plus(int,A,A) = zero_zero(int). [resolve(1380,a,1374,a)]. 4.36/4.57 Derived: ti(int,A) = zero_zero(int) | plus_plus(int,A,A) != zero_zero(int). [resolve(1380,a,1375,a)]. 4.36/4.57 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,A,A)),zero_zero(int))). [resolve(1380,a,1377,a)]. 4.36/4.57 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),plus_plus(int,A,A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)). [resolve(1380,a,1378,a)]. 4.36/4.57 1381 -linord219039673up_add(A) | ti(A,B) != zero_zero(A) | plus_plus(A,B,B) = zero_zero(A) # label(fact_65_double__eq__0__iff) # label(axiom). [clausify(425)]. 4.36/4.57 1382 -linord219039673up_add(A) | ti(A,B) = zero_zero(A) | plus_plus(A,B,B) != zero_zero(A) # label(fact_65_double__eq__0__iff) # label(axiom). [clausify(425)]. 4.36/4.57 1383 -linord219039673up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,B,B)),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) # label(fact_195_double__add__less__zero__iff__single__add__less__zero) # label(axiom). [clausify(642)]. 4.36/4.57 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(1383,a,1373,a)]. 4.36/4.57 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(1383,a,1380,a)]. 4.36/4.57 1384 -linord219039673up_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,B,B)),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) # label(fact_195_double__add__less__zero__iff__single__add__less__zero) # label(axiom). [clausify(642)]. 4.36/4.57 1385 -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(952)]. 4.36/4.57 1386 -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(952)]. 4.36/4.57 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(1386,a,1373,a)]. 4.36/4.57 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(1386,a,1380,a)]. 4.36/4.57 1387 -ordere453448008miring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),times_times(A,C,B)),zero_zero(A))) # label(fact_269_split__mult__neg__le) # label(axiom). [clausify(306)]. 4.36/4.57 1388 ordere453448008miring(int) # label(arity_Int_Oint___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 4.36/4.57 1389 ordere453448008miring(nat) # label(arity_Nat_Onat___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 4.36/4.57 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),times_times(int,B,A)),zero_zero(int))). [resolve(1387,a,1388,a)]. 4.36/4.60 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),zero_zero(nat))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),times_times(nat,B,A)),zero_zero(nat))). [resolve(1387,a,1389,a)]. 4.36/4.60 1390 -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(306)]. 4.36/4.60 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),times_times(int,A,B)),zero_zero(int))). [resolve(1390,a,1388,a)]. 4.36/4.60 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),zero_zero(nat))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),times_times(nat,A,B)),zero_zero(nat))). [resolve(1390,a,1389,a)]. 4.36/4.60 1391 ordere453448008miring(real) # label(arity_RealDef_Oreal___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 4.36/4.60 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),times_times(real,B,A)),zero_zero(real))). [resolve(1391,a,1387,a)]. 4.36/4.60 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),times_times(real,A,B)),zero_zero(real))). [resolve(1391,a,1390,a)]. 4.36/4.60 1392 -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(445)]. 4.36/4.60 1393 -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(476)]. 4.36/4.60 1394 -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(768)]. 4.36/4.60 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),times_times(int,A,B))). [resolve(1394,a,1388,a)]. 4.36/4.60 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(1394,a,1389,a)]. 4.36/4.60 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),times_times(real,A,B))). [resolve(1394,a,1391,a)]. 4.57/4.81 1395 -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(832)]. 4.57/4.81 1396 -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(139)]. 4.57/4.81 1397 order(int) # label(arity_Int_Oint___Orderings_Oorder) # label(axiom). [assumption]. 4.57/4.81 1398 -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(177)]. 4.57/4.81 Derived: ord_less(int) = ti(fun(int,fun(int,bool)),ord_less(int)). [resolve(1398,a,1397,a)]. 4.57/4.81 1399 -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(248)]. 4.57/4.81 1400 -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(290)]. 4.57/4.81 1401 -order(A) | hAPP(A,bool,B,ti(A,C)) = hAPP(A,bool,B,C) # label(tsy_c_hAPP_1_arg2) # label(axiom). [clausify(487)]. 4.57/4.81 1402 -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(497)]. 4.57/4.81 1403 -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(608)]. 4.57/4.81 Derived: ti(fun(int,fun(int,bool)),ord_less_eq(int)) = ord_less_eq(int). [resolve(1403,a,1397,a)]. 4.57/4.81 1404 order(real) # label(arity_RealDef_Oreal___Orderings_Oorder) # label(axiom). [assumption]. 4.57/4.81 Derived: ord_less(real) = ti(fun(real,fun(real,bool)),ord_less(real)). [resolve(1404,a,1398,a)]. 4.57/4.81 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(1404,a,1400,a)]. 4.57/4.81 Derived: ti(fun(real,fun(real,bool)),ord_less_eq(real)) = ord_less_eq(real). [resolve(1404,a,1403,a)]. 4.57/4.81 1405 order(bool) # label(arity_HOL_Obool___Orderings_Oorder) # label(axiom). [assumption]. 4.57/4.81 Derived: hAPP(bool,bool,ti(fun(bool,bool),A),B) = hAPP(bool,bool,A,B). [resolve(1405,a,1396,a)]. 4.57/4.81 Derived: ord_less(bool) = ti(fun(bool,fun(bool,bool)),ord_less(bool)). [resolve(1405,a,1398,a)]. 4.57/4.81 Derived: hAPP(bool,fun(bool,bool),ti(fun(bool,fun(bool,bool)),A),B) = hAPP(bool,fun(bool,bool),A,B). [resolve(1405,a,1399,a)]. 4.57/4.81 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(1405,a,1400,a)]. 4.57/4.81 Derived: hAPP(bool,bool,A,ti(bool,B)) = hAPP(bool,bool,A,B). [resolve(1405,a,1401,a)]. 4.57/4.81 Derived: hAPP(bool,fun(bool,bool),A,ti(bool,B)) = hAPP(bool,fun(bool,bool),A,B). [resolve(1405,a,1402,a)]. 4.57/4.81 Derived: ti(fun(bool,fun(bool,bool)),ord_less_eq(bool)) = ord_less_eq(bool). [resolve(1405,a,1403,a)]. 4.57/4.81 1406 -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(797)]. 4.57/4.81 Derived: ti(fun(bool,bool),hAPP(bool,fun(bool,bool),A,B)) = hAPP(bool,fun(bool,bool),A,B). [resolve(1406,a,1405,a)]. 4.57/4.81 1407 order(nat) # label(arity_Nat_Onat___Orderings_Oorder) # label(axiom). [assumption]. 4.57/4.81 Derived: ord_less(nat) = ti(fun(nat,fun(nat,bool)),ord_less(nat)). [resolve(1407,a,1398,a)]. 4.57/4.81 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(1407,a,1400,a)]. 4.57/4.81 Derived: ti(fun(nat,fun(nat,bool)),ord_less_eq(nat)) = ord_less_eq(nat). [resolve(1407,a,1403,a)]. 4.57/4.81 1408 -order(A) | ti(bool,hAPP(A,bool,B,C)) = hAPP(A,bool,B,C) # label(tsy_c_hAPP_1_res) # label(axiom). [clausify(979)]. 4.57/4.81 Derived: ti(bool,hAPP(bool,bool,A,B)) = hAPP(bool,bool,A,B). [resolve(1408,a,1405,a)]. 4.57/4.81 1409 group_add(int) # label(arity_Int_Oint___Groups_Ogroup__add) # label(axiom). [assumption]. 4.78/5.05 1410 -group_add(A) | ti(A,B) != ti(A,C) | minus_minus(A,C,B) = zero_zero(A) # label(fact_320_right__minus__eq) # label(axiom). [clausify(91)]. 4.78/5.05 1411 -group_add(A) | ti(A,B) = ti(A,C) | minus_minus(A,C,B) != zero_zero(A) # label(fact_320_right__minus__eq) # label(axiom). [clausify(91)]. 4.78/5.05 1412 -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(104)]. 4.78/5.05 1413 -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(207)]. 4.78/5.05 1414 -group_add(A) | minus_minus(A,B,C) = ti(A,minus_minus(A,B,C)) # label(tsy_c_Groups_Ominus__class_Ominus_0_res) # label(axiom). [clausify(223)]. 4.78/5.05 Derived: ti(int,A) != ti(int,B) | minus_minus(int,B,A) = zero_zero(int). [resolve(1409,a,1410,a)]. 4.78/5.05 Derived: ti(int,A) = ti(int,B) | minus_minus(int,B,A) != zero_zero(int). [resolve(1409,a,1411,a)]. 4.78/5.05 Derived: minus_minus(int,A,ti(int,B)) = minus_minus(int,A,B). [resolve(1409,a,1412,a)]. 4.78/5.05 Derived: minus_minus(int,ti(int,A),B) = minus_minus(int,A,B). [resolve(1409,a,1413,a)]. 4.78/5.05 Derived: minus_minus(int,A,B) = ti(int,minus_minus(int,A,B)). [resolve(1409,a,1414,a)]. 4.78/5.05 1415 -group_add(A) | minus_minus(A,plus_plus(A,B,C),C) = ti(A,B) # label(fact_326_add__diff__cancel) # label(axiom). [clausify(317)]. 4.78/5.05 Derived: minus_minus(int,plus_plus(int,A,B),B) = ti(int,A). [resolve(1415,a,1409,a)]. 4.78/5.05 1416 -group_add(A) | minus_minus(A,B,zero_zero(A)) = ti(A,B) # label(fact_323_diff__0__right) # label(axiom). [clausify(502)]. 4.78/5.05 Derived: minus_minus(int,A,zero_zero(int)) = ti(int,A). [resolve(1416,a,1409,a)]. 4.78/5.05 1417 -group_add(A) | minus_minus(A,B,B) = zero_zero(A) # label(fact_322_diff__self) # label(axiom). [clausify(600)]. 4.78/5.05 Derived: minus_minus(int,A,A) = zero_zero(int). [resolve(1417,a,1409,a)]. 4.78/5.05 1418 group_add(real) # label(arity_RealDef_Oreal___Groups_Ogroup__add) # label(axiom). [assumption]. 4.78/5.05 Derived: ti(real,A) != ti(real,B) | minus_minus(real,B,A) = zero_zero(real). [resolve(1418,a,1410,a)]. 4.78/5.05 Derived: ti(real,A) = ti(real,B) | minus_minus(real,B,A) != zero_zero(real). [resolve(1418,a,1411,a)]. 4.78/5.05 Derived: minus_minus(real,A,ti(real,B)) = minus_minus(real,A,B). [resolve(1418,a,1412,a)]. 4.78/5.05 Derived: minus_minus(real,ti(real,A),B) = minus_minus(real,A,B). [resolve(1418,a,1413,a)]. 4.78/5.05 Derived: minus_minus(real,A,B) = ti(real,minus_minus(real,A,B)). [resolve(1418,a,1414,a)]. 4.78/5.05 Derived: minus_minus(real,plus_plus(real,A,B),B) = ti(real,A). [resolve(1418,a,1415,a)]. 4.78/5.05 Derived: minus_minus(real,A,zero_zero(real)) = ti(real,A). [resolve(1418,a,1416,a)]. 4.78/5.05 Derived: minus_minus(real,A,A) = zero_zero(real). [resolve(1418,a,1417,a)]. 4.78/5.05 1419 -group_add(A) | plus_plus(A,minus_minus(A,B,C),C) = ti(A,B) # label(fact_327_diff__add__cancel) # label(axiom). [clausify(943)]. 4.78/5.05 Derived: plus_plus(int,minus_minus(int,A,B),B) = ti(int,A). [resolve(1419,a,1409,a)]. 4.78/5.05 Derived: plus_plus(real,minus_minus(real,A,B),B) = ti(real,A). [resolve(1419,a,1418,a)]. 4.78/5.05 1420 zero(nat) # label(arity_Nat_Onat___Groups_Ozero) # label(axiom). [assumption]. 4.78/5.05 1421 -zero(A) | ti(A,zero_zero(A)) = zero_zero(A) # label(tsy_c_Groups_Ozero__class_Ozero_res) # label(axiom). [clausify(97)]. 4.78/5.05 Derived: ti(nat,zero_zero(nat)) = zero_zero(nat). [resolve(1420,a,1421,a)]. 4.78/5.05 1422 zero(int) # label(arity_Int_Oint___Groups_Ozero) # label(axiom). [assumption]. 4.78/5.05 Derived: ti(int,zero_zero(int)) = zero_zero(int). [resolve(1422,a,1421,a)]. 4.78/5.05 1423 zero(real) # label(arity_RealDef_Oreal___Groups_Ozero) # label(axiom). [assumption]. 4.78/5.05 Derived: ti(real,zero_zero(real)) = zero_zero(real). [resolve(1423,a,1421,a)]. 4.78/5.05 1424 -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(183)]. 4.78/5.05 1425 monoid_mult(nat) # label(arity_Nat_Onat___Groups_Omonoid__mult) # label(axiom). [assumption]. 4.78/5.05 1426 -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(195)]. 4.87/5.09 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(1426,a,1425,a)]. 4.87/5.09 1427 -monoid_mult(A) | hAPP(nat,A,power_power(A,hAPP(nat,A,power_power(A,B),C)),D) = hAPP(nat,A,power_power(A,B),times_times(nat,C,D)) # label(fact_600_power__mult) # label(axiom). [clausify(221)]. 4.87/5.09 1428 -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(301)]. 4.87/5.09 1429 monoid_mult(real) # label(arity_RealDef_Oreal___Groups_Omonoid__mult) # label(axiom). [assumption]. 4.87/5.09 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(1429,a,1426,a)]. 4.87/5.09 1430 -monoid_mult(A) | ti(A,B) = times_times(A,one_one(A),B) # label(fact_352_mult__1__left) # label(axiom). [clausify(320)]. 4.87/5.09 1431 -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(336)]. 4.87/5.09 1432 -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(441)]. 4.87/5.09 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(1432,a,1425,a)]. 4.87/5.09 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(1432,a,1429,a)]. 4.87/5.09 1433 monoid_mult(int) # label(arity_Int_Oint___Groups_Omonoid__mult) # label(axiom). [assumption]. 4.87/5.09 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(1433,a,1426,a)]. 4.87/5.09 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(1433,a,1432,a)]. 4.87/5.09 1434 -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(585)]. 4.87/5.09 1435 -monoid_mult(A) | ti(A,B) = times_times(A,B,one_one(A)) # label(fact_349_mult__1__right) # label(axiom). [clausify(650)]. 4.87/5.09 1436 -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(798)]. 4.87/5.09 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(1436,a,1425,a)]. 4.87/5.09 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(1436,a,1429,a)]. 4.87/5.09 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(1436,a,1433,a)]. 4.87/5.09 1437 -monoid_mult(A) | -number(A) | times_times(A,number_number_of(A,B),number_number_of(A,B)) = hAPP(nat,A,power_power(A,number_number_of(A,B)),number_number_of(nat,bit0(bit1(pls)))) # label(fact_545_power2__eq__square__number__of) # label(axiom). [clausify(902)]. 4.87/5.09 1438 -monoid_mult(A) | one_one(A) = hAPP(nat,A,power_power(A,one_one(A)),B) # label(fact_166_power__one) # label(axiom). [clausify(922)]. 5.17/5.40 Derived: one_one(nat) = hAPP(nat,nat,power_power(nat,one_one(nat)),A). [resolve(1438,a,1425,a)]. 5.17/5.40 Derived: one_one(real) = hAPP(nat,real,power_power(real,one_one(real)),A). [resolve(1438,a,1429,a)]. 5.17/5.40 Derived: one_one(int) = hAPP(nat,int,power_power(int,one_one(int)),A). [resolve(1438,a,1433,a)]. 5.17/5.40 1439 -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(285)]. 5.17/5.40 1440 ordere779506340up_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 5.17/5.40 1441 ordere779506340up_add(int) # label(arity_Int_Oint___Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 5.17/5.40 1442 -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(517)]. 5.17/5.40 1443 -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(528)]. 5.17/5.40 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(1443,a,1440,a)]. 5.17/5.40 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(1443,a,1441,a)]. 5.17/5.40 1444 ordere779506340up_add(nat) # label(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 5.17/5.40 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(1444,a,1443,a)]. 5.17/5.40 1445 -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(399)]. 5.17/5.40 1446 semiring_char_0(real) # label(arity_RealDef_Oreal___Nat_Osemiring__char__0) # label(axiom). [assumption]. 5.17/5.40 Derived: hAPP(nat,real,semiring_1_of_nat(real),A) != hAPP(nat,real,semiring_1_of_nat(real),B) | A = B. [resolve(1445,a,1446,a)]. 5.17/5.40 1447 -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(399)]. 5.17/5.40 Derived: hAPP(nat,real,semiring_1_of_nat(real),A) = hAPP(nat,real,semiring_1_of_nat(real),B) | A != B. [resolve(1447,a,1446,a)]. 5.17/5.40 1448 semiring_char_0(int) # label(arity_Int_Oint___Nat_Osemiring__char__0) # label(axiom). [assumption]. 5.17/5.40 1449 semiring_char_0(nat) # label(arity_Nat_Onat___Nat_Osemiring__char__0) # label(axiom). [assumption]. 5.17/5.40 Derived: hAPP(nat,nat,semiring_1_of_nat(nat),A) != hAPP(nat,nat,semiring_1_of_nat(nat),B) | A = B. [resolve(1449,a,1445,a)]. 5.17/5.40 Derived: hAPP(nat,nat,semiring_1_of_nat(nat),A) = hAPP(nat,nat,semiring_1_of_nat(nat),B) | A != B. [resolve(1449,a,1447,a)]. 5.17/5.40 1450 linord581940658strict(int) # label(arity_Int_Oint___Rings_Olinordered__ring__strict) # label(axiom). [assumption]. 5.17/5.40 1451 -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(118)]. 5.17/5.40 1452 -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(118)]. 5.17/5.41 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(1450,a,1452,a)]. 5.17/5.41 1453 linord581940658strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__ring__strict) # label(axiom). [assumption]. 5.17/5.41 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(1453,a,1452,a)]. 5.17/5.41 1454 -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),times_times(A,B,C)),times_times(A,B,D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),C)) # label(fact_308_mult__le__cancel__left__neg) # label(axiom). [clausify(166)]. 5.17/5.41 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),times_times(int,A,B)),times_times(int,A,C))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),B)). [resolve(1454,a,1450,a)]. 5.17/5.41 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),times_times(real,A,B)),times_times(real,A,C))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),B)). [resolve(1454,a,1453,a)]. 5.17/5.41 1455 -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),times_times(A,B,C)),times_times(A,B,D))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),C)) # label(fact_308_mult__le__cancel__left__neg) # label(axiom). [clausify(166)]. 5.17/5.41 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),times_times(int,A,B)),times_times(int,A,C))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),B)). [resolve(1455,a,1450,a)]. 5.17/5.41 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),times_times(real,A,B)),times_times(real,A,C))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),B)). [resolve(1455,a,1453,a)]. 5.17/5.41 1456 -linord581940658strict(A) | plus_plus(A,times_times(A,B,B),times_times(A,C,C)) != zero_zero(A) | ti(A,B) = zero_zero(A) # label(fact_428_sum__squares__eq__zero__iff) # label(axiom). [clausify(168)]. 5.17/5.41 Derived: plus_plus(int,times_times(int,A,A),times_times(int,B,B)) != zero_zero(int) | ti(int,A) = zero_zero(int). [resolve(1456,a,1450,a)]. 5.17/5.41 Derived: plus_plus(real,times_times(real,A,A),times_times(real,B,B)) != zero_zero(real) | ti(real,A) = zero_zero(real). [resolve(1456,a,1453,a)]. 5.17/5.41 1457 -linord581940658strict(A) | plus_plus(A,times_times(A,B,B),times_times(A,C,C)) != zero_zero(A) | ti(A,C) = zero_zero(A) # label(fact_428_sum__squares__eq__zero__iff) # label(axiom). [clausify(168)]. 5.17/5.41 Derived: plus_plus(int,times_times(int,A,A),times_times(int,B,B)) != zero_zero(int) | ti(int,B) = zero_zero(int). [resolve(1457,a,1450,a)]. 5.17/5.41 Derived: plus_plus(real,times_times(real,A,A),times_times(real,B,B)) != zero_zero(real) | ti(real,B) = zero_zero(real). [resolve(1457,a,1453,a)]. 5.17/5.41 1458 -linord581940658strict(A) | plus_plus(A,times_times(A,B,B),times_times(A,C,C)) = zero_zero(A) | ti(A,B) != zero_zero(A) | ti(A,C) != zero_zero(A) # label(fact_428_sum__squares__eq__zero__iff) # label(axiom). [clausify(168)]. 5.17/5.41 Derived: plus_plus(int,times_times(int,A,A),times_times(int,B,B)) = zero_zero(int) | ti(int,A) != zero_zero(int) | ti(int,B) != zero_zero(int). [resolve(1458,a,1450,a)]. 5.23/5.41 Derived: plus_plus(real,times_times(real,A,A),times_times(real,B,B)) = zero_zero(real) | ti(real,A) != zero_zero(real) | ti(real,B) != zero_zero(real). [resolve(1458,a,1453,a)]. 5.23/5.41 1459 -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(170)]. 5.23/5.41 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(1459,a,1450,a)]. 5.23/5.41 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(1459,a,1453,a)]. 5.23/5.41 1460 -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(170)]. 5.23/5.41 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(1460,a,1450,a)]. 5.23/5.41 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(1460,a,1453,a)]. 5.23/5.41 1461 -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(175)]. 5.23/5.41 1462 -linord581940658strict(A) | ti(A,B) != zero_zero(A) | ti(A,C) != zero_zero(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_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(225)]. 5.23/5.41 Derived: ti(int,A) != zero_zero(int) | ti(int,B) != zero_zero(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,times_times(int,A,A),times_times(int,B,B))),zero_zero(int))). [resolve(1462,a,1450,a)]. 5.23/5.41 Derived: ti(real,A) != zero_zero(real) | ti(real,B) != zero_zero(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),plus_plus(real,times_times(real,A,A),times_times(real,B,B))),zero_zero(real))). [resolve(1462,a,1453,a)]. 5.23/5.41 1463 -linord581940658strict(A) | ti(A,B) = zero_zero(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,times_times(A,B,B),times_times(A,C,C))),zero_zero(A))) # label(fact_311_sum__squares__le__zero__iff) # label(axiom). [clausify(225)]. 5.23/5.41 Derived: ti(int,A) = zero_zero(int) | -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(1463,a,1450,a)]. 5.23/5.41 Derived: ti(real,A) = zero_zero(real) | -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(1463,a,1453,a)]. 5.23/5.41 1464 -linord581940658strict(A) | ti(A,B) = zero_zero(A) | -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(225)]. 5.23/5.43 Derived: ti(int,A) = zero_zero(int) | -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(1464,a,1450,a)]. 5.23/5.43 Derived: ti(real,A) = zero_zero(real) | -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(1464,a,1453,a)]. 5.23/5.43 1465 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),times_times(A,D,C)),times_times(A,D,B))) # label(fact_411_mult__less__cancel__left__disj) # label(axiom). [clausify(606)]. 5.23/5.43 1466 -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(606)]. 5.23/5.43 1467 -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(606)]. 5.23/5.43 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(1467,a,1450,a)]. 5.23/5.43 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(1467,a,1453,a)]. 5.23/5.43 1468 -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(606)]. 5.23/5.43 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(1468,a,1450,a)]. 5.23/5.43 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(1468,a,1453,a)]. 5.23/5.43 1469 -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(606)]. 5.23/5.43 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(1469,a,1450,a)]. 5.23/5.43 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(1469,a,1453,a)]. 5.23/5.43 1470 -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(606)]. 5.23/5.44 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(1470,a,1450,a)]. 5.23/5.44 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(1470,a,1453,a)]. 5.23/5.44 1471 -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),times_times(A,B,C)),times_times(A,B,D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) # label(fact_309_mult__le__cancel__left__pos) # label(axiom). [clausify(617)]. 5.23/5.44 1472 -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),times_times(A,B,C)),times_times(A,B,D))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) # label(fact_309_mult__le__cancel__left__pos) # label(axiom). [clausify(617)]. 5.23/5.44 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),times_times(int,A,B)),times_times(int,A,C))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)). [resolve(1472,a,1450,a)]. 5.23/5.44 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),times_times(real,A,B)),times_times(real,A,C))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),C)). [resolve(1472,a,1453,a)]. 5.23/5.44 1473 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),plus_plus(A,times_times(A,B,B),times_times(A,C,C)))) | ti(A,B) != zero_zero(A) | ti(A,C) != zero_zero(A) # label(fact_486_sum__squares__gt__zero__iff) # label(axiom). [clausify(681)]. 5.23/5.44 Derived: -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)))) | ti(int,A) != zero_zero(int) | ti(int,B) != zero_zero(int). [resolve(1473,a,1450,a)]. 5.23/5.44 Derived: -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)))) | ti(real,A) != zero_zero(real) | ti(real,B) != zero_zero(real). [resolve(1473,a,1453,a)]. 5.23/5.44 1474 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),plus_plus(A,times_times(A,B,B),times_times(A,C,C)))) | ti(A,B) = zero_zero(A) # label(fact_486_sum__squares__gt__zero__iff) # label(axiom). [clausify(681)]. 5.23/5.44 Derived: 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)))) | ti(int,A) = zero_zero(int). [resolve(1474,a,1450,a)]. 5.23/5.44 Derived: 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)))) | ti(real,A) = zero_zero(real). [resolve(1474,a,1453,a)]. 5.23/5.44 1475 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),plus_plus(A,times_times(A,B,B),times_times(A,C,C)))) | ti(A,C) = zero_zero(A) # label(fact_486_sum__squares__gt__zero__iff) # label(axiom). [clausify(681)]. 5.23/5.44 Derived: 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)))) | ti(int,B) = zero_zero(int). [resolve(1475,a,1450,a)]. 5.23/5.44 Derived: 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)))) | ti(real,B) = zero_zero(real). [resolve(1475,a,1453,a)]. 5.23/5.44 1476 -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(739)]. 5.23/5.44 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),times_times(int,B,C)),times_times(int,A,C))). [resolve(1476,a,1450,a)]. 5.23/5.44 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),times_times(real,B,C)),times_times(real,A,C))). [resolve(1476,a,1453,a)]. 5.23/5.44 1477 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),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_290_mult__le__0__iff) # label(axiom). [clausify(770)]. 5.23/5.44 1478 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),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_290_mult__le__0__iff) # label(axiom). [clausify(770)]. 5.23/5.44 1479 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),times_times(A,B,C)),zero_zero(A))) # label(fact_290_mult__le__0__iff) # label(axiom). [clausify(770)]. 5.23/5.44 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),times_times(int,A,B)),zero_zero(int))). [resolve(1479,a,1450,a)]. 5.23/5.44 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),times_times(real,A,B)),zero_zero(real))). [resolve(1479,a,1453,a)]. 5.23/5.44 1480 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),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_290_mult__le__0__iff) # label(axiom). [clausify(770)]. 5.23/5.44 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),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),times_times(int,A,B)),zero_zero(int))). [resolve(1480,a,1450,a)]. 5.23/5.44 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),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),times_times(real,A,B)),zero_zero(real))). [resolve(1480,a,1453,a)]. 5.23/5.44 1481 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),times_times(A,C,B)),zero_zero(A))) # label(fact_290_mult__le__0__iff) # label(axiom). [clausify(770)]. 5.23/5.44 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),times_times(int,B,A)),zero_zero(int))). [resolve(1481,a,1450,a)]. 5.23/5.45 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),times_times(real,B,A)),zero_zero(real))). [resolve(1481,a,1453,a)]. 5.23/5.45 1482 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),times_times(A,C,B)),zero_zero(A))) # label(fact_290_mult__le__0__iff) # label(axiom). [clausify(770)]. 5.23/5.45 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),times_times(int,B,A)),zero_zero(int))). [resolve(1482,a,1450,a)]. 5.23/5.45 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),times_times(real,B,A)),zero_zero(real))). [resolve(1482,a,1453,a)]. 5.23/5.45 1483 -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(818)]. 5.23/5.45 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(1483,a,1450,a)]. 5.23/5.45 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(1483,a,1453,a)]. 5.23/5.45 1484 -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(847)]. 5.23/5.45 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(1484,a,1450,a)]. 5.23/5.45 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(1484,a,1453,a)]. 5.23/5.45 1485 -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(847)]. 5.23/5.45 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(1485,a,1450,a)]. 5.23/5.45 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(1485,a,1453,a)]. 5.26/5.45 1486 -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(847)]. 5.26/5.45 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(1486,a,1450,a)]. 5.26/5.45 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(1486,a,1453,a)]. 5.26/5.45 1487 -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(847)]. 5.26/5.45 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(1487,a,1450,a)]. 5.26/5.45 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(1487,a,1453,a)]. 5.26/5.45 1488 -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(847)]. 5.26/5.45 1489 -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),D),B)) | -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(847)]. 5.26/5.45 1490 -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(872)]. 5.26/5.45 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(1490,a,1450,a)]. 5.26/5.45 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(1490,a,1453,a)]. 5.26/5.45 1491 -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(872)]. 5.26/5.45 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)),A)). [resolve(1491,a,1450,a)]. 5.26/5.45 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)),A)). [resolve(1491,a,1453,a)]. 5.35/5.55 1492 -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(872)]. 5.35/5.55 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)),B)). [resolve(1492,a,1450,a)]. 5.35/5.55 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)),B)). [resolve(1492,a,1453,a)]. 5.35/5.55 1493 -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(872)]. 5.35/5.55 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(1493,a,1450,a)]. 5.35/5.55 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(1493,a,1453,a)]. 5.35/5.55 1494 -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),C),zero_zero(A))) # label(fact_291_zero__le__mult__iff) # label(axiom). [clausify(872)]. 5.35/5.55 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),B),zero_zero(int))). [resolve(1494,a,1450,a)]. 5.35/5.55 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),B),zero_zero(real))). [resolve(1494,a,1453,a)]. 5.35/5.55 1495 -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)),C)) | -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(872)]. 5.35/5.55 1496 cancel_semigroup_add(int) # label(arity_Int_Oint___Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 5.35/5.55 1497 -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(140)]. 5.35/5.55 1498 -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(140)]. 5.35/5.55 1499 -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(309)]. 5.47/5.68 Derived: ti(int,A) != ti(int,B) | plus_plus(int,C,A) = plus_plus(int,C,B). [resolve(1496,a,1497,a)]. 5.47/5.68 1500 cancel_semigroup_add(nat) # label(arity_Nat_Onat___Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 5.47/5.68 Derived: ti(nat,A) != ti(nat,B) | plus_plus(nat,C,A) = plus_plus(nat,C,B). [resolve(1500,a,1497,a)]. 5.47/5.68 1501 -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(626)]. 5.47/5.68 1502 -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(659)]. 5.47/5.68 Derived: plus_plus(int,A,B) != plus_plus(int,C,B) | ti(int,A) = ti(int,C). [resolve(1502,a,1496,a)]. 5.47/5.68 Derived: plus_plus(nat,A,B) != plus_plus(nat,C,B) | ti(nat,A) = ti(nat,C). [resolve(1502,a,1500,a)]. 5.47/5.68 1503 -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(672)]. 5.47/5.68 1504 cancel_semigroup_add(real) # label(arity_RealDef_Oreal___Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 5.47/5.68 Derived: ti(real,A) != ti(real,B) | plus_plus(real,C,A) = plus_plus(real,C,B). [resolve(1504,a,1497,a)]. 5.47/5.68 Derived: plus_plus(real,A,B) != plus_plus(real,C,B) | ti(real,A) = ti(real,C). [resolve(1504,a,1502,a)]. 5.47/5.68 1505 -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(940)]. 5.47/5.68 1506 -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(940)]. 5.47/5.68 Derived: plus_plus(int,A,B) = plus_plus(int,C,B) | ti(int,A) != ti(int,C). [resolve(1506,a,1496,a)]. 5.47/5.68 Derived: plus_plus(nat,A,B) = plus_plus(nat,C,B) | ti(nat,A) != ti(nat,C). [resolve(1506,a,1500,a)]. 5.47/5.68 Derived: plus_plus(real,A,B) = plus_plus(real,C,B) | ti(real,A) != ti(real,C). [resolve(1506,a,1504,a)]. 5.47/5.68 1507 -cancel_semigroup_add(A) | plus_plus(A,B,C) = ti(A,plus_plus(A,B,C)) # label(tsy_c_Groups_Oplus__class_Oplus_2_res) # label(axiom). [clausify(990)]. 5.47/5.68 1508 ordere223160158up_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 5.47/5.68 1509 -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(148)]. 5.47/5.68 1510 -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(311)]. 5.47/5.68 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(1508,a,1509,a)]. 5.47/5.68 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(1508,a,1510,a)]. 5.47/5.68 1511 ordere223160158up_add(nat) # label(arity_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 5.47/5.68 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(1511,a,1510,a)]. 5.47/5.68 1512 -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(523)]. 5.55/5.80 1513 -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(611)]. 5.55/5.80 1514 -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(718)]. 5.55/5.80 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(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(1514,a,1508,a)]. 5.55/5.80 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(1514,a,1511,a)]. 5.55/5.80 1515 ordere223160158up_add(int) # label(arity_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 5.55/5.80 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),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(1515,a,1509,a)]. 5.55/5.80 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),D)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,A,C)),plus_plus(int,B,D))). [resolve(1515,a,1510,a)]. 5.55/5.80 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(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(1515,a,1514,a)]. 5.55/5.80 1516 semiring_1(nat) # label(arity_Nat_Onat___Rings_Osemiring__1) # label(axiom). [assumption]. 5.55/5.80 1517 -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(157)]. 5.55/5.80 1518 -semiring_1(A) | hAPP(nat,A,semiring_1_of_nat(A),zero_zero(nat)) = zero_zero(A) # label(fact_188_of__nat__0) # label(axiom). [clausify(335)]. 5.55/5.80 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(1516,a,1517,a)]. 5.55/5.80 Derived: hAPP(nat,nat,semiring_1_of_nat(nat),zero_zero(nat)) = zero_zero(nat). [resolve(1516,a,1518,a)]. 5.55/5.80 1519 semiring_1(int) # label(arity_Int_Oint___Rings_Osemiring__1) # label(axiom). [assumption]. 5.55/5.80 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(1519,a,1517,a)]. 5.55/5.80 1520 -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(392)]. 5.55/5.80 1521 -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(633)]. 5.55/5.80 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(1521,a,1516,a)]. 5.70/5.92 1522 semiring_1(real) # label(arity_RealDef_Oreal___Rings_Osemiring__1) # label(axiom). [assumption]. 5.70/5.92 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(1522,a,1517,a)]. 5.70/5.92 Derived: hAPP(nat,real,semiring_1_of_nat(real),zero_zero(nat)) = zero_zero(real). [resolve(1522,a,1518,a)]. 5.70/5.92 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(1522,a,1521,a)]. 5.70/5.92 1523 -semiring_1(A) | hAPP(nat,A,power_power(A,zero_zero(A)),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(A) # label(fact_4_zero__power2) # label(axiom). [clausify(758)]. 5.70/5.92 Derived: hAPP(nat,nat,power_power(nat,zero_zero(nat)),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(nat). [resolve(1523,a,1516,a)]. 5.70/5.92 Derived: hAPP(nat,int,power_power(int,zero_zero(int)),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(int). [resolve(1523,a,1519,a)]. 5.70/5.92 Derived: hAPP(nat,real,power_power(real,zero_zero(real)),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(real). [resolve(1523,a,1522,a)]. 5.70/5.92 1524 -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(786)]. 5.70/5.92 Derived: semiring_1_of_nat(nat) = ti(fun(nat,nat),semiring_1_of_nat(nat)). [resolve(1524,a,1516,a)]. 5.70/5.92 Derived: semiring_1_of_nat(int) = ti(fun(nat,int),semiring_1_of_nat(int)). [resolve(1524,a,1519,a)]. 5.70/5.92 Derived: semiring_1_of_nat(real) = ti(fun(nat,real),semiring_1_of_nat(real)). [resolve(1524,a,1522,a)]. 5.70/5.92 1525 -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(964)]. 5.70/5.92 Derived: one_one(nat) = hAPP(nat,nat,semiring_1_of_nat(nat),one_one(nat)). [resolve(1525,a,1516,a)]. 5.70/5.92 Derived: one_one(int) = hAPP(nat,int,semiring_1_of_nat(int),one_one(nat)). [resolve(1525,a,1519,a)]. 5.70/5.92 Derived: one_one(real) = hAPP(nat,real,semiring_1_of_nat(real),one_one(nat)). [resolve(1525,a,1522,a)]. 5.70/5.92 1526 -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(1007)]. 5.70/5.92 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(1526,a,1516,a)]. 5.70/5.92 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(1526,a,1519,a)]. 5.70/5.92 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(1526,a,1522,a)]. 5.70/5.92 1527 -semiring_1(A) | hAPP(nat,A,power_power(A,hAPP(nat,A,semiring_1_of_nat(A),B)),C) = hAPP(nat,A,semiring_1_of_nat(A),hAPP(nat,nat,power_power(nat,B),C)) # label(fact_172_of__nat__power) # label(axiom). [clausify(1009)]. 5.70/5.92 Derived: hAPP(nat,nat,power_power(nat,hAPP(nat,nat,semiring_1_of_nat(nat),A)),B) = hAPP(nat,nat,semiring_1_of_nat(nat),hAPP(nat,nat,power_power(nat,A),B)). [resolve(1527,a,1516,a)]. 5.70/5.92 Derived: hAPP(nat,int,power_power(int,hAPP(nat,int,semiring_1_of_nat(int),A)),B) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,power_power(nat,A),B)). [resolve(1527,a,1519,a)]. 5.70/5.92 Derived: hAPP(nat,real,power_power(real,hAPP(nat,real,semiring_1_of_nat(real),A)),B) = hAPP(nat,real,semiring_1_of_nat(real),hAPP(nat,nat,power_power(nat,A),B)). [resolve(1527,a,1522,a)]. 5.70/5.92 1528 -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(730)]. 6.05/6.26 1529 ordere1490568538miring(real) # label(arity_RealDef_Oreal___Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 6.05/6.26 1530 ordere1490568538miring(nat) # label(arity_Nat_Onat___Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 6.05/6.26 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(1528,a,1530,a)]. 6.05/6.26 1531 ordere1490568538miring(int) # label(arity_Int_Oint___Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 6.05/6.26 1532 ab_semigroup_mult(nat) # label(arity_Nat_Onat___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 6.05/6.26 1533 -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(171)]. 6.05/6.26 1534 -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(398)]. 6.05/6.26 1535 ab_semigroup_mult(real) # label(arity_RealDef_Oreal___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 6.05/6.26 1536 ab_semigroup_mult(int) # label(arity_Int_Oint___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 6.05/6.26 1537 -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(841)]. 6.05/6.26 1538 -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(913)]. 6.05/6.26 1539 linord626643107strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring__1__strict) # label(axiom). [assumption]. 6.05/6.26 1540 -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(174)]. 6.05/6.26 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),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(1539,a,1540,a)]. 6.05/6.26 1541 linord626643107strict(int) # label(arity_Int_Oint___Rings_Olinordered__semiring__1__strict) # label(axiom). [assumption]. 6.05/6.26 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),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(1541,a,1540,a)]. 6.05/6.26 1542 -one(A) | one_one(A) = ti(A,one_one(A)) # label(tsy_c_Groups_Oone__class_Oone_res) # label(axiom). [clausify(500)]. 6.05/6.26 1543 one(int) # label(arity_Int_Oint___Groups_Oone) # label(axiom). [assumption]. 6.05/6.26 1544 one(nat) # label(arity_Nat_Onat___Groups_Oone) # label(axiom). [assumption]. 6.05/6.26 Derived: one_one(int) = ti(int,one_one(int)). [resolve(1542,a,1543,a)]. 6.05/6.26 Derived: one_one(nat) = ti(nat,one_one(nat)). [resolve(1542,a,1544,a)]. 6.05/6.26 1545 one(real) # label(arity_RealDef_Oreal___Groups_Oone) # label(axiom). [assumption]. 6.26/6.49 Derived: one_one(real) = ti(real,one_one(real)). [resolve(1545,a,1542,a)]. 6.26/6.49 1546 linordered_ring(real) # label(arity_RealDef_Oreal___Rings_Olinordered__ring) # label(axiom). [assumption]. 6.26/6.49 1547 -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(182)]. 6.26/6.49 1548 -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(396)]. 6.26/6.49 1549 -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(547)]. 6.26/6.49 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),times_times(real,A,A)),zero_zero(real))). [resolve(1546,a,1547,a)]. 6.26/6.49 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),times_times(real,A,A))). [resolve(1546,a,1548,a)]. 6.26/6.49 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(1546,a,1549,a)]. 6.26/6.49 1550 linordered_ring(int) # label(arity_Int_Oint___Rings_Olinordered__ring) # label(axiom). [assumption]. 6.26/6.49 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),times_times(int,A,A)),zero_zero(int))). [resolve(1550,a,1547,a)]. 6.26/6.49 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),times_times(int,A,A))). [resolve(1550,a,1548,a)]. 6.26/6.49 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(1550,a,1549,a)]. 6.26/6.49 1551 -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(886)]. 6.26/6.49 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(1551,a,1546,a)]. 6.26/6.49 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(1551,a,1550,a)]. 6.26/6.49 1552 ring_11004092258visors(real) # label(arity_RealDef_Oreal___Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 6.26/6.49 1553 -ring_11004092258visors(A) | hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))) != zero_zero(A) | ti(A,B) = zero_zero(A) # label(fact_5_zero__eq__power2) # label(axiom). [clausify(192)]. 6.26/6.49 1554 -ring_11004092258visors(A) | hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(A) | ti(A,B) != zero_zero(A) # label(fact_5_zero__eq__power2) # label(axiom). [clausify(192)]. 6.26/6.49 1555 -ring_11004092258visors(A) | ti(A,B) = zero_zero(A) | hAPP(nat,A,power_power(A,B),C) != zero_zero(A) # label(fact_165_field__power__not__zero) # label(axiom). [clausify(406)]. 6.26/6.49 Derived: hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))) != zero_zero(real) | ti(real,A) = zero_zero(real). [resolve(1552,a,1553,a)]. 6.26/6.49 Derived: hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(real) | ti(real,A) != zero_zero(real). [resolve(1552,a,1554,a)]. 6.26/6.49 Derived: ti(real,A) = zero_zero(real) | hAPP(nat,real,power_power(real,A),B) != zero_zero(real). [resolve(1552,a,1555,a)]. 6.26/6.49 1556 ring_11004092258visors(int) # label(arity_Int_Oint___Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 6.26/6.49 Derived: hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))) != zero_zero(int) | ti(int,A) = zero_zero(int). [resolve(1556,a,1553,a)]. 6.26/6.49 Derived: hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(int) | ti(int,A) != zero_zero(int). [resolve(1556,a,1554,a)]. 6.26/6.49 Derived: ti(int,A) = zero_zero(int) | hAPP(nat,int,power_power(int,A),B) != zero_zero(int). [resolve(1556,a,1555,a)]. 6.48/6.72 1557 -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(258)]. 6.48/6.72 1558 linord893533164strict(int) # label(arity_Int_Oint___Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 6.48/6.72 1559 linord893533164strict(nat) # label(arity_Nat_Onat___Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 6.48/6.72 1560 linord893533164strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 6.48/6.72 1561 -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(508)]. 6.48/6.72 1562 ordered_ring(real) # label(arity_RealDef_Oreal___Rings_Oordered__ring) # label(axiom). [assumption]. 6.48/6.72 1563 -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,C,B))) # label(fact_270_split__mult__pos__le) # label(axiom). [clausify(551)]. 6.48/6.72 1564 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),times_times(A,B,C))) # label(fact_270_split__mult__pos__le) # label(axiom). [clausify(551)]. 6.48/6.72 1565 -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(692)]. 6.48/6.72 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(1565,a,1562,a)]. 6.48/6.72 1566 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,times_times(A,B,C),D)),plus_plus(A,times_times(A,E,C),F))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,times_times(A,minus_minus(A,B,E),C),D)),F)) # label(fact_295_less__add__iff1) # label(axiom). [clausify(763)]. 6.48/6.72 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),plus_plus(real,times_times(real,A,B),C)),plus_plus(real,times_times(real,D,B),E))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),plus_plus(real,times_times(real,minus_minus(real,A,D),B),C)),E)). [resolve(1566,a,1562,a)]. 6.48/6.72 1567 -ordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,times_times(A,B,C),D)),plus_plus(A,times_times(A,E,C),F))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),plus_plus(A,times_times(A,minus_minus(A,B,E),C),D)),F)) # label(fact_295_less__add__iff1) # label(axiom). [clausify(763)]. 6.48/6.72 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),plus_plus(real,times_times(real,A,B),C)),plus_plus(real,times_times(real,D,B),E))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),plus_plus(real,times_times(real,minus_minus(real,A,D),B),C)),E)). [resolve(1567,a,1562,a)]. 6.48/6.72 1568 ordered_ring(int) # label(arity_Int_Oint___Rings_Oordered__ring) # label(axiom). [assumption]. 6.48/6.72 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(1568,a,1565,a)]. 6.48/6.73 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,times_times(int,A,B),C)),plus_plus(int,times_times(int,D,B),E))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,times_times(int,minus_minus(int,A,D),B),C)),E)). [resolve(1568,a,1566,a)]. 6.48/6.73 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,times_times(int,A,B),C)),plus_plus(int,times_times(int,D,B),E))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),plus_plus(int,times_times(int,minus_minus(int,A,D),B),C)),E)). [resolve(1568,a,1567,a)]. 6.48/6.73 1569 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),plus_plus(A,times_times(A,minus_minus(A,C,D),E),F))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,times_times(A,D,E),B)),plus_plus(A,times_times(A,C,E),F))) # label(fact_287_le__add__iff2) # label(axiom). [clausify(926)]. 6.48/6.73 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),plus_plus(real,times_times(real,minus_minus(real,B,C),D),E))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),plus_plus(real,times_times(real,C,D),A)),plus_plus(real,times_times(real,B,D),E))). [resolve(1569,a,1562,a)]. 6.48/6.73 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),plus_plus(int,times_times(int,minus_minus(int,B,C),D),E))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,times_times(int,C,D),A)),plus_plus(int,times_times(int,B,D),E))). [resolve(1569,a,1568,a)]. 6.48/6.73 1570 -ordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),plus_plus(A,times_times(A,minus_minus(A,C,D),E),F))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),plus_plus(A,times_times(A,D,E),B)),plus_plus(A,times_times(A,C,E),F))) # label(fact_287_le__add__iff2) # label(axiom). [clausify(926)]. 6.48/6.73 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),plus_plus(real,times_times(real,minus_minus(real,B,C),D),E))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),plus_plus(real,times_times(real,C,D),A)),plus_plus(real,times_times(real,B,D),E))). [resolve(1570,a,1562,a)]. 6.48/6.73 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),plus_plus(int,times_times(int,minus_minus(int,B,C),D),E))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),plus_plus(int,times_times(int,C,D),A)),plus_plus(int,times_times(int,B,D),E))). [resolve(1570,a,1568,a)]. 6.48/6.73 1571 -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(938)]. 6.48/6.73 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(1571,a,1562,a)]. 6.48/6.73 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(1571,a,1568,a)]. 6.48/6.73 1572 -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(938)]. 6.48/6.73 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(1572,a,1562,a)]. 6.48/6.73 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(1572,a,1568,a)]. 6.64/6.83 1573 -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(968)]. 6.64/6.83 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),times_times(real,B,C)),times_times(real,A,C))). [resolve(1573,a,1562,a)]. 6.64/6.83 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(1573,a,1568,a)]. 6.64/6.83 1574 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(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_eq(A),plus_plus(A,times_times(A,B,D),E)),plus_plus(A,times_times(A,C,D),F))) # label(fact_285_le__add__iff1) # label(axiom). [clausify(980)]. 6.64/6.83 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(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_eq(real),plus_plus(real,times_times(real,A,C),D)),plus_plus(real,times_times(real,B,C),E))). [resolve(1574,a,1562,a)]. 6.64/6.83 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(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_eq(int),plus_plus(int,times_times(int,A,C),D)),plus_plus(int,times_times(int,B,C),E))). [resolve(1574,a,1568,a)]. 6.64/6.83 1575 -ordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(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_eq(A),plus_plus(A,times_times(A,B,D),E)),plus_plus(A,times_times(A,C,D),F))) # label(fact_285_le__add__iff1) # label(axiom). [clausify(980)]. 6.64/6.83 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(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_eq(real),plus_plus(real,times_times(real,A,C),D)),plus_plus(real,times_times(real,B,C),E))). [resolve(1575,a,1562,a)]. 6.64/6.83 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(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_eq(int),plus_plus(int,times_times(int,A,C),D)),plus_plus(int,times_times(int,B,C),E))). [resolve(1575,a,1568,a)]. 6.64/6.83 1576 -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(334)]. 6.64/6.83 1577 ab_group_add(real) # label(arity_RealDef_Oreal___Groups_Oab__group__add) # label(axiom). [assumption]. 6.64/6.83 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(1576,a,1577,a)]. 6.64/6.83 1578 ab_group_add(int) # label(arity_Int_Oint___Groups_Oab__group__add) # label(axiom). [assumption]. 6.64/6.83 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(1578,a,1576,a)]. 6.64/6.83 1579 -ab_group_add(A) | minus_minus(A,B,C) != zero_zero(A) | ti(A,C) = ti(A,B) # label(fact_321_eq__iff__diff__eq__0) # label(axiom). [clausify(720)]. 6.64/6.83 1580 -ab_group_add(A) | minus_minus(A,B,C) = zero_zero(A) | ti(A,C) != ti(A,B) # label(fact_321_eq__iff__diff__eq__0) # label(axiom). [clausify(720)]. 6.87/7.05 1581 -ab_group_add(A) | minus_minus(A,B,C) != minus_minus(A,D,E) | ti(A,C) != ti(A,B) | ti(A,E) = ti(A,D) # label(fact_248_diff__eq__diff__eq) # label(axiom). [clausify(820)]. 6.87/7.05 Derived: minus_minus(real,A,B) != minus_minus(real,C,D) | ti(real,B) != ti(real,A) | ti(real,D) = ti(real,C). [resolve(1581,a,1577,a)]. 6.87/7.05 Derived: minus_minus(int,A,B) != minus_minus(int,C,D) | ti(int,B) != ti(int,A) | ti(int,D) = ti(int,C). [resolve(1581,a,1578,a)]. 6.87/7.05 1582 -ab_group_add(A) | minus_minus(A,B,C) != minus_minus(A,D,E) | ti(A,C) = ti(A,B) | ti(A,E) != ti(A,D) # label(fact_248_diff__eq__diff__eq) # label(axiom). [clausify(820)]. 6.87/7.05 Derived: minus_minus(real,A,B) != minus_minus(real,C,D) | ti(real,B) = ti(real,A) | ti(real,D) != ti(real,C). [resolve(1582,a,1577,a)]. 6.87/7.05 Derived: minus_minus(int,A,B) != minus_minus(int,C,D) | ti(int,B) = ti(int,A) | ti(int,D) != ti(int,C). [resolve(1582,a,1578,a)]. 6.87/7.05 1583 -number(A) | -linorder(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,B)),number_number_of(A,C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,C)),number_number_of(A,B))) # label(fact_448_le__number__of__eq__not__less) # label(axiom). [clausify(871)]. 6.87/7.05 1584 linorder(int) # label(arity_Int_Oint___Orderings_Olinorder) # label(axiom). [assumption]. 6.87/7.05 1585 linorder(real) # label(arity_RealDef_Oreal___Orderings_Olinorder) # label(axiom). [assumption]. 6.87/7.05 1586 linorder(nat) # label(arity_Nat_Onat___Orderings_Olinorder) # label(axiom). [assumption]. 6.87/7.05 Derived: -number(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,A)),number_number_of(int,B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,B)),number_number_of(int,A))). [resolve(1583,b,1584,a)]. 6.87/7.05 Derived: -number(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,A)),number_number_of(real,B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,B)),number_number_of(real,A))). [resolve(1583,b,1585,a)]. 6.87/7.05 Derived: -number(nat) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),number_number_of(nat,A)),number_number_of(nat,B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),number_number_of(nat,B)),number_number_of(nat,A))). [resolve(1583,b,1586,a)]. 6.87/7.05 1587 -number(A) | -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,B)),number_number_of(A,C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,C)),number_number_of(A,B))) # label(fact_448_le__number__of__eq__not__less) # label(axiom). [clausify(871)]. 6.87/7.05 Derived: -number(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,A)),number_number_of(int,B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,B)),number_number_of(int,A))). [resolve(1587,b,1584,a)]. 6.87/7.05 Derived: -number(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,A)),number_number_of(real,B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,B)),number_number_of(real,A))). [resolve(1587,b,1585,a)]. 6.87/7.05 Derived: -number(nat) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),number_number_of(nat,A)),number_number_of(nat,B))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),number_number_of(nat,B)),number_number_of(nat,A))). [resolve(1587,b,1586,a)]. 6.87/7.05 1588 ab_semigroup_add(int) # label(arity_Int_Oint___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 6.87/7.05 1589 -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(371)]. 6.87/7.05 1590 -ab_semigroup_add(A) | plus_plus(A,B,C) = ti(A,plus_plus(A,B,C)) # label(tsy_c_Groups_Oplus__class_Oplus_1_res) # label(axiom). [clausify(510)]. 6.87/7.05 1591 -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(539)]. 7.19/7.39 1592 ab_semigroup_add(real) # label(arity_RealDef_Oreal___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 7.19/7.39 1593 -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(735)]. 7.19/7.39 1594 ab_semigroup_add(nat) # label(arity_Nat_Onat___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 7.19/7.39 1595 -ring_n68954251visors(A) | times_times(A,B,C) != zero_zero(A) | ti(A,C) = zero_zero(A) | ti(A,B) = zero_zero(A) # label(fact_330_mult__eq__0__iff) # label(axiom). [clausify(934)]. 7.19/7.39 1596 ring_n68954251visors(real) # label(arity_RealDef_Oreal___Rings_Oring__no__zero__divisors) # label(axiom). [assumption]. 7.19/7.39 1597 ring_n68954251visors(int) # label(arity_Int_Oint___Rings_Oring__no__zero__divisors) # label(axiom). [assumption]. 7.19/7.39 Derived: times_times(real,A,B) != zero_zero(real) | ti(real,B) = zero_zero(real) | ti(real,A) = zero_zero(real). [resolve(1595,a,1596,a)]. 7.19/7.39 Derived: times_times(int,A,B) != zero_zero(int) | ti(int,B) = zero_zero(int) | ti(int,A) = zero_zero(int). [resolve(1595,a,1597,a)]. 7.19/7.39 1598 -ring_n68954251visors(A) | times_times(A,B,C) = zero_zero(A) | ti(A,C) != zero_zero(A) # label(fact_330_mult__eq__0__iff) # label(axiom). [clausify(934)]. 7.19/7.39 Derived: times_times(real,A,B) = zero_zero(real) | ti(real,B) != zero_zero(real). [resolve(1598,a,1596,a)]. 7.19/7.39 Derived: times_times(int,A,B) = zero_zero(int) | ti(int,B) != zero_zero(int). [resolve(1598,a,1597,a)]. 7.19/7.39 1599 -ring_n68954251visors(A) | times_times(A,B,C) = zero_zero(A) | ti(A,B) != zero_zero(A) # label(fact_330_mult__eq__0__iff) # label(axiom). [clausify(934)]. 7.19/7.39 Derived: times_times(real,A,B) = zero_zero(real) | ti(real,A) != zero_zero(real). [resolve(1599,a,1596,a)]. 7.19/7.39 Derived: times_times(int,A,B) = zero_zero(int) | ti(int,A) != zero_zero(int). [resolve(1599,a,1597,a)]. 7.19/7.39 1600 linordered_semiring(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring) # label(axiom). [assumption]. 7.19/7.39 1601 -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(372)]. 7.19/7.39 1602 -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(566)]. 7.19/7.39 1603 linordered_semiring(int) # label(arity_Int_Oint___Rings_Olinordered__semiring) # label(axiom). [assumption]. 7.19/7.39 1604 linordered_semiring(nat) # label(arity_Nat_Onat___Rings_Olinordered__semiring) # label(axiom). [assumption]. 7.19/7.39 1605 ordered_ab_group_add(int) # label(arity_Int_Oint___Groups_Oordered__ab__group__add) # label(axiom). [assumption]. 7.19/7.39 1606 -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(462)]. 7.19/7.39 1607 -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(462)]. 7.19/7.39 1608 -ordered_ab_group_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),minus_minus(A,B,C)),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_238_le__iff__diff__le__0) # label(axiom). [clausify(521)]. 7.19/7.39 1609 -ordered_ab_group_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),minus_minus(A,B,C)),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_238_le__iff__diff__le__0) # label(axiom). [clausify(521)]. 7.19/7.39 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),minus_minus(int,A,B)),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(1605,a,1608,a)]. 7.19/7.40 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),minus_minus(int,A,B)),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(1605,a,1609,a)]. 7.19/7.40 1610 -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(694)]. 7.19/7.40 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(1610,a,1605,a)]. 7.19/7.40 1611 -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(694)]. 7.19/7.40 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(1611,a,1605,a)]. 7.19/7.40 1612 ordered_ab_group_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__group__add) # label(axiom). [assumption]. 7.19/7.40 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(1612,a,1606,a)]. 7.19/7.40 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(1612,a,1607,a)]. 7.19/7.40 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),minus_minus(real,A,B)),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)). [resolve(1612,a,1608,a)]. 7.19/7.40 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),minus_minus(real,A,B)),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)). [resolve(1612,a,1609,a)]. 7.19/7.40 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(1612,a,1610,a)]. 7.19/7.40 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(1612,a,1611,a)]. 7.19/7.40 1613 -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(858)]. 7.19/7.40 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(1613,a,1605,a)]. 7.19/7.40 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(1613,a,1612,a)]. 7.19/7.40 1614 -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(858)]. 7.19/7.40 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(1614,a,1605,a)]. 7.19/7.40 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(1614,a,1612,a)]. 7.50/7.74 1615 -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(843)]. 7.50/7.74 1616 comm_ring_1(real) # label(arity_RealDef_Oreal___Rings_Ocomm__ring__1) # label(axiom). [assumption]. 7.50/7.74 1617 comm_ring_1(int) # label(arity_Int_Oint___Rings_Ocomm__ring__1) # label(axiom). [assumption]. 7.50/7.74 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),minus_minus(real,B,C))). [resolve(1615,a,1616,a)]. 7.50/7.74 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),minus_minus(int,B,C))). [resolve(1615,a,1617,a)]. 7.50/7.74 1618 -ring_1(A) | minus_minus(A,times_times(A,B,B),one_one(A)) = times_times(A,plus_plus(A,B,one_one(A)),minus_minus(A,B,one_one(A))) # label(fact_297_real__squared__diff__one__factored) # label(axiom). [clausify(827)]. 7.50/7.74 1619 ring_1(real) # label(arity_RealDef_Oreal___Rings_Oring__1) # label(axiom). [assumption]. 7.50/7.74 Derived: minus_minus(real,times_times(real,A,A),one_one(real)) = times_times(real,plus_plus(real,A,one_one(real)),minus_minus(real,A,one_one(real))). [resolve(1618,a,1619,a)]. 7.50/7.74 1620 -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(853)]. 7.50/7.74 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(1620,a,1619,a)]. 7.50/7.74 1621 ring_1(int) # label(arity_Int_Oint___Rings_Oring__1) # label(axiom). [assumption]. 7.50/7.74 Derived: minus_minus(int,times_times(int,A,A),one_one(int)) = times_times(int,plus_plus(int,A,one_one(int)),minus_minus(int,A,one_one(int))). [resolve(1621,a,1618,a)]. 7.50/7.74 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | minus_minus(int,hAPP(nat,int,semiring_1_of_nat(int),B),hAPP(nat,int,semiring_1_of_nat(int),A)) = hAPP(nat,int,semiring_1_of_nat(int),minus_minus(nat,B,A)). [resolve(1621,a,1620,a)]. 7.50/7.74 1622 -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(911)]. 7.50/7.74 1623 linord1278240602ring_1(int) # label(arity_Int_Oint___Rings_Olinordered__semiring__1) # label(axiom). [assumption]. 7.50/7.74 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(1622,a,1623,a)]. 7.50/7.74 1624 linord1278240602ring_1(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring__1) # label(axiom). [assumption]. 7.50/7.74 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(1624,a,1622,a)]. 9.51/9.72 9.51/9.72 ============================== end predicate elimination ============= 9.51/9.72 9.51/9.72 Auto_denials: (non-Horn, no changes). 9.51/9.72 9.51/9.72 Term ordering decisions: 9.51/9.72 Function symbol KB weights: int=1. nat=1. bool=1. real=1. pls=1. min=1. m=1. zprime=1. n=1. s=1. t=1. s1=1. twoSqu658283162sum2sq=1. fFalse=1. fTrue=1. tn=1. x=1. y=1. m1=1. c1=1. c2=1. c3=1. c4=1. c5=1. c6=1. c7=1. c8=1. c9=1. c10=1. c11=1. c12=1. fun=1. number_number_of=1. ti=1. power_power=1. abs_abs=1. zcong=1. legendre=1. f11=1. f14=1. f15=1. f17=1. f18=1. f21=1. f23=1. f27=1. f28=1. zero_zero=1. ord_less_eq=1. ord_less=1. one_one=1. dvd_dvd=1. bit1=1. bit0=1. semiring_1_of_nat=1. nat_1=1. succ=1. quadRes=1. undefined=1. f1=1. f2=1. f4=1. f9=1. f10=1. f12=1. f13=1. f16=1. f29=1. f30=1. times_times=1. plus_plus=1. minus_minus=1. f3=1. f5=1. f6=1. f7=1. f8=1. f19=1. f20=1. f22=1. f24=1. f25=1. f26=1. f31=1. hAPP=1. if=1. 9.51/9.72 9.51/9.72 ============================== end of process initial clauses ======== 9.51/9.72 9.51/9.72 ============================== CLAUSES FOR SEARCH ==================== 9.51/9.72 9.51/9.72 ============================== end of clauses for search ============= 9.51/9.72 9.51/9.72 ============================== SEARCH ================================ 9.51/9.72 9.51/9.72 % Starting search at 6.42 seconds. 9.51/9.72 9.51/9.72 Low Water (keep): wt=73.000, iters=3606 9.51/9.72 9.51/9.72 Low Water (keep): wt=65.000, iters=3431 9.51/9.72 9.51/9.72 Low Water (keep): wt=64.000, iters=3408 9.51/9.72 9.51/9.72 Low Water (keep): wt=63.000, iters=3408 9.51/9.72 9.51/9.72 Low Water (keep): wt=60.000, iters=3375 9.51/9.72 9.51/9.72 Low Water (keep): wt=58.000, iters=3359 9.51/9.72 9.51/9.72 Low Water (keep): wt=57.000, iters=3337 9.51/9.72 9.51/9.72 Low Water (keep): wt=56.000, iters=3414 9.51/9.72 9.51/9.72 Low Water (keep): wt=55.000, iters=3390 9.51/9.72 9.51/9.72 Low Water (keep): wt=53.000, iters=3423 9.51/9.72 9.51/9.72 Low Water (keep): wt=52.000, iters=3406 9.51/9.72 9.51/9.72 Low Water (keep): wt=51.000, iters=3363 9.51/9.72 9.51/9.72 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 36 (0.00 of 6.93 sec). 9.51/9.72 9.51/9.72 Low Water (keep): wt=50.000, iters=3350 9.51/9.72 9.51/9.72 Low Water (keep): wt=49.000, iters=3436 9.51/9.72 9.51/9.72 Low Water (keep): wt=48.000, iters=3353 9.51/9.72 9.51/9.72 Low Water (keep): wt=47.000, iters=3432 9.51/9.72 9.51/9.72 Low Water (keep): wt=46.000, iters=3374 9.51/9.72 9.51/9.72 Low Water (keep): wt=45.000, iters=3415 9.51/9.72 9.51/9.72 Low Water (keep): wt=44.000, iters=3348 9.51/9.72 9.51/9.72 Low Water (keep): wt=43.000, iters=3405 9.51/9.72 9.51/9.72 Low Water (keep): wt=42.000, iters=3367 9.51/9.72 9.51/9.72 Low Water (keep): wt=41.000, iters=3340 9.51/9.72 9.51/9.72 Low Water (keep): wt=40.000, iters=3349 9.51/9.72 9.51/9.72 Low Water (keep): wt=39.000, iters=3352 9.51/9.72 9.51/9.72 Low Water (keep): wt=38.000, iters=3387 9.51/9.72 9.51/9.72 Low Water (keep): wt=37.000, iters=3396 9.51/9.72 9.51/9.72 Low Water (keep): wt=36.000, iters=3357 9.51/9.72 9.51/9.72 Low Water (keep): wt=35.000, iters=3361 9.51/9.72 9.51/9.72 Low Water (keep): wt=34.000, iters=3409 9.51/9.72 9.51/9.72 Low Water (keep): wt=33.000, iters=3412 9.51/9.72 9.51/9.72 Low Water (keep): wt=32.000, iters=3334 9.51/9.72 9.51/9.72 Low Water (keep): wt=31.000, iters=3359 9.51/9.72 9.51/9.72 Low Water (keep): wt=30.000, iters=3343 9.51/9.72 9.51/9.72 Low Water (keep): wt=29.000, iters=3404 9.51/9.72 9.51/9.72 Low Water (keep): wt=28.000, iters=3346 9.51/9.72 9.51/9.72 Low Water (keep): wt=27.000, iters=3350 9.51/9.72 9.51/9.72 ============================== PROOF ================================= 9.51/9.72 % SZS status Theorem 9.51/9.72 % SZS output start Refutation 9.51/9.72 9.51/9.72 % Proof 1 at 7.99 (+ 0.04) seconds. 9.51/9.72 % Length of proof is 68. 9.51/9.72 % Level of proof is 7. 9.51/9.72 % Maximum clause weight is 32.000. 9.51/9.72 % Given clauses 1326. 9.51/9.72 9.51/9.72 205 (all Z1 all Z2 all Z3 plus_plus(int,Z1,plus_plus(int,Z2,Z3)) = plus_plus(int,plus_plus(int,Z1,Z2),Z3)) # label(fact_45_zadd__assoc) # label(axiom) # label(non_clause). [assumption]. 9.51/9.72 254 (all B_1_1 nat_1(B_1_1) = nat_1(ti(int,B_1_1))) # label(tsy_c_Int_Onat_arg1) # label(axiom) # label(non_clause). [assumption]. 9.51/9.72 325 (all Z plus_plus(int,Z,zero_zero(int)) = ti(int,Z)) # label(fact_79_zadd__0__right) # label(axiom) # label(non_clause). [assumption]. 9.51/9.72 347 (all K_1 succ(K_1) = plus_plus(int,K_1,one_one(int))) # label(fact_519_succ__def) # label(axiom) # label(non_clause). [assumption]. 9.51/9.72 453 (all B_1_1 all B_2_1 all X_a (monoid_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_0_res) # label(axiom) # label(non_clause). [assumption]. 9.51/9.72 465 (all K_1 bit1(K_1) = succ(bit0(K_1))) # label(fact_514_succ__Bit0) # label(axiom) # label(non_clause). [assumption]. 9.51/9.72 469 (all K_1 minus_minus(int,K_1,min) = succ(K_1)) # label(fact_777_diff__bin__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.51/9.72 481 (all X_a (power(X_a) & mult_zero(X_a) & zero_neq_one(X_a) & no_zero_divisors(X_a) -> (all A_2 all Na (hAPP(nat,X_a,power_power(X_a,A_2),Na) = zero_zero(X_a) <-> Na != zero_zero(nat) & zero_zero(X_a) = ti(X_a,A_2))))) # label(fact_185_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.51/9.72 500 (all X_a (one(X_a) -> one_one(X_a) = ti(X_a,one_one(X_a)))) # label(tsy_c_Groups_Oone__class_Oone_res) # label(axiom) # label(non_clause). [assumption]. 9.51/9.72 600 (all X_a (group_add(X_a) -> (all A_1 minus_minus(X_a,A_1,A_1) = zero_zero(X_a)))) # label(fact_322_diff__self) # label(axiom) # label(non_clause). [assumption]. 9.51/9.72 762 (all V_1 number_number_of(nat,V_1) = nat_1(number_number_of(int,V_1))) # label(fact_427_nat__number__of__def) # label(axiom) # label(non_clause). [assumption]. 9.51/9.72 779 (all Z all W plus_plus(int,Z,W) = plus_plus(int,W,Z)) # label(fact_47_zadd__commute) # label(axiom) # label(non_clause). [assumption]. 9.51/9.72 844 (all K_1 bit0(K_1) = plus_plus(int,K_1,K_1)) # label(fact_78_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 9.51/9.72 905 (all Z_1 all Wa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_1),Wa)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),Wa)) & ti(int,Z_1) != ti(int,Wa))) # label(fact_380_zless__le) # label(axiom) # label(non_clause). [assumption]. 9.51/9.72 943 (all X_a (group_add(X_a) -> (all A_1 all B_1 plus_plus(X_a,minus_minus(X_a,A_1,B_1),B_1) = ti(X_a,A_1)))) # label(fact_327_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 9.51/9.72 992 (all K_1 number_number_of(int,K_1) = ti(int,K_1)) # label(fact_120_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 9.51/9.72 1011 -(zero_zero(int) != hAPP(nat,int,power_power(int,plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n))),number_number_of(nat,bit0(bit1(pls))))) # label(conj_0) # label(negated_conjecture) # label(non_clause). [assumption]. 9.51/9.72 1318 zero_neq_one(int) # label(arity_Int_Oint___Rings_Ozero__neq__one) # label(axiom). [assumption]. 9.51/9.72 1325 -power(A) | -mult_zero(A) | -zero_neq_one(A) | -no_zero_divisors(A) | hAPP(nat,A,power_power(A,B),C) != zero_zero(A) | ti(A,B) = zero_zero(A) # label(fact_185_power__eq__0__iff) # label(axiom). [clausify(481)]. 9.51/9.72 1360 -monoid_add(A) | plus_plus(A,B,C) = ti(A,plus_plus(A,B,C)) # label(tsy_c_Groups_Oplus__class_Oplus_0_res) # label(axiom). [clausify(453)]. 9.51/9.72 1361 monoid_add(int) # label(arity_Int_Oint___Groups_Omonoid__add) # label(axiom). [assumption]. 9.51/9.72 1409 group_add(int) # label(arity_Int_Oint___Groups_Ogroup__add) # label(axiom). [assumption]. 9.51/9.72 1417 -group_add(A) | minus_minus(A,B,B) = zero_zero(A) # label(fact_322_diff__self) # label(axiom). [clausify(600)]. 9.51/9.72 1419 -group_add(A) | plus_plus(A,minus_minus(A,B,C),C) = ti(A,B) # label(fact_327_diff__add__cancel) # label(axiom). [clausify(943)]. 9.51/9.72 1542 -one(A) | one_one(A) = ti(A,one_one(A)) # label(tsy_c_Groups_Oone__class_Oone_res) # label(axiom). [clausify(500)]. 9.51/9.72 1543 one(int) # label(arity_Int_Oint___Groups_Oone) # label(axiom). [assumption]. 9.51/9.72 1745 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n)))) # label(fact_0_n1pos) # label(axiom). [assumption]. 9.51/9.72 1824 succ(min) = pls # label(fact_776_succ__Min) # label(axiom). [assumption]. 9.51/9.72 1825 pls = succ(min). [copy(1824),flip(a)]. 9.51/9.72 1857 plus_plus(int,plus_plus(int,A,B),C) = plus_plus(int,A,plus_plus(int,B,C)) # label(fact_45_zadd__assoc) # label(axiom). [clausify(205)]. 9.51/9.72 1923 nat_1(ti(int,A)) = nat_1(A) # label(tsy_c_Int_Onat_arg1) # label(axiom). [clausify(254)]. 9.51/9.72 1991 mult_zero(int) # label(arity_Int_Oint___Rings_Omult__zero) # label(axiom). [assumption]. 9.51/9.72 2013 plus_plus(int,A,zero_zero(int)) = ti(int,A) # label(fact_79_zadd__0__right) # label(axiom). [clausify(325)]. 9.51/9.72 2032 succ(A) = plus_plus(int,A,one_one(int)) # label(fact_519_succ__def) # label(axiom). [clausify(347)]. 9.51/9.72 2133 bit1(pls) = succ(pls) # label(fact_513_succ__Pls) # label(axiom). [assumption]. 9.51/9.72 2134 plus_plus(int,min,plus_plus(int,one_one(int),one_one(int))) = bit1(plus_plus(int,min,one_one(int))). [copy(2133),rewrite([1825(1),2032(2),1825(7),2032(8),2032(12),1857(15)]),flip(a)]. 9.51/9.72 2174 succ(bit0(A)) = bit1(A) # label(fact_514_succ__Bit0) # label(axiom). [clausify(465)]. 9.51/9.72 2175 bit1(A) = plus_plus(int,bit0(A),one_one(int)). [copy(2174),rewrite([2032(2)]),flip(a)]. 9.51/9.72 2184 succ(A) = minus_minus(int,A,min) # label(fact_777_diff__bin__simps_I2_J) # label(axiom). [clausify(469)]. 9.51/9.72 2185 plus_plus(int,A,one_one(int)) = minus_minus(int,A,min). [copy(2184),rewrite([2032(1)])]. 9.51/9.72 2341 zero_zero(int) = pls # label(fact_73_Pls__def) # label(axiom). [assumption]. 9.51/9.72 2342 minus_minus(int,min,min) = zero_zero(int). [copy(2341),rewrite([1825(3),2032(4),2185(7)]),flip(a)]. 9.51/9.72 2542 nat_1(number_number_of(int,A)) = number_number_of(nat,A) # label(fact_427_nat__number__of__def) # label(axiom). [clausify(762)]. 9.51/9.72 2560 plus_plus(int,A,B) = plus_plus(int,B,A) # label(fact_47_zadd__commute) # label(axiom). [clausify(779)]. 9.51/9.72 2632 no_zero_divisors(int) # label(arity_Int_Oint___Rings_Ono__zero__divisors) # label(axiom). [assumption]. 9.51/9.72 2660 bit0(A) = plus_plus(int,A,A) # label(fact_78_Bit0__def) # label(axiom). [clausify(844)]. 9.51/9.72 2683 power(int) # label(arity_Int_Oint___Power_Opower) # label(axiom). [assumption]. 9.51/9.72 2738 -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | ti(int,B) != ti(int,A) # label(fact_380_zless__le) # label(axiom). [clausify(905)]. 9.51/9.72 2801 zero_zero(int) = number_number_of(int,pls) # label(fact_19_zero__is__num__zero) # label(axiom). [assumption]. 9.51/9.72 2802 number_number_of(int,zero_zero(int)) = zero_zero(int). [copy(2801),rewrite([1825(4),2032(5),2185(8),2342(7)]),flip(a)]. 9.51/9.72 2835 ti(int,A) = number_number_of(int,A) # label(fact_120_number__of__is__id) # label(axiom). [clausify(992)]. 9.51/9.72 2865 hAPP(nat,int,power_power(int,plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n))),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(int) # label(conj_0) # label(negated_conjecture). [clausify(1011)]. 9.51/9.72 2866 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,plus_plus(int,minus_minus(int,zero_zero(int),min),minus_minus(int,zero_zero(int),min)))) = zero_zero(int). [copy(2865),rewrite([1825(16),2032(17),2185(20),2342(19),2175(18),2660(19),2013(22),2835(20),2802(20),2185(21),2660(21)])]. 9.51/9.72 3649 -power(int) | -mult_zero(int) | -no_zero_divisors(int) | hAPP(nat,int,power_power(int,A),B) != zero_zero(int) | ti(int,A) = zero_zero(int). [resolve(1325,c,1318,a)]. 9.51/9.72 3650 hAPP(nat,int,power_power(int,A),B) != zero_zero(int) | number_number_of(int,A) = zero_zero(int). [copy(3649),rewrite([2835(16)]),unit_del(a,2683),unit_del(b,1991),unit_del(c,2632)]. 9.51/9.72 3723 plus_plus(int,A,B) = ti(int,plus_plus(int,A,B)). [resolve(1361,a,1360,a)]. 9.51/9.72 3724 number_number_of(int,plus_plus(int,A,B)) = plus_plus(int,A,B). [copy(3723),rewrite([2835(6)]),flip(a)]. 9.51/9.72 3805 minus_minus(int,A,A) = zero_zero(int). [resolve(1417,a,1409,a)]. 9.51/9.72 3815 plus_plus(int,minus_minus(int,A,B),B) = ti(int,A). [resolve(1419,a,1409,a)]. 9.51/9.72 3816 plus_plus(int,A,minus_minus(int,B,A)) = number_number_of(int,B). [copy(3815),rewrite([2560(4),2835(6)])]. 9.51/9.72 3994 one_one(int) = ti(int,one_one(int)). [resolve(1542,a,1543,a)]. 9.51/9.72 3995 number_number_of(int,one_one(int)) = one_one(int). [copy(3994),rewrite([2835(6)]),flip(a)]. 9.51/9.72 4199 minus_minus(int,zero_zero(int),min) = one_one(int). [back_rewrite(2134),rewrite([2185(8),3816(8),3995(4),2185(7),3805(6),2175(5),2660(6),2013(9),2835(7),2802(7),2185(8)]),flip(a)]. 9.51/9.72 4362 -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | number_number_of(int,B) != number_number_of(int,A). [back_rewrite(2738),rewrite([2835(13),2835(15)])]. 9.51/9.72 4407 number_number_of(nat,A) = nat_1(A). [back_rewrite(1923),rewrite([2835(2),2542(3)])]. 9.51/9.72 4637 hAPP(nat,int,power_power(int,plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n))),nat_1(minus_minus(int,one_one(int),min))) = zero_zero(int). [back_rewrite(2866),rewrite([4199(21),4199(23),2185(21),4407(21)])]. 9.51/9.72 11225 plus_plus(int,one_one(int),hAPP(nat,int,semiring_1_of_nat(int),n)) != zero_zero(int). [resolve(4362,a,1745,a),rewrite([3724(12),2802(14)])]. 9.51/9.72 12302 $F. [resolve(4637,a,3650,a),rewrite([3724(12)]),unit_del(a,11225)]. 9.51/9.72 9.51/9.72 % SZS output end Refutation 9.51/9.72 ============================== end of proof ========================== 9.51/9.72 9.51/9.72 ============================== STATISTICS ============================ 9.51/9.72 9.51/9.72 Given=1326. Generated=40498. Kept=9848. proofs=1. 9.51/9.72 Usable=1160. Sos=6329. Demods=934. Limbo=0, Disabled=4954. Hints=0. 9.51/9.72 Megabytes=33.73. 9.51/9.72 User_CPU=7.99, System_CPU=0.04, Wall_clock=8. 9.51/9.72 9.51/9.72 ============================== end of statistics ===================== 9.51/9.72 9.51/9.72 ============================== end of search ========================= 9.51/9.72 9.51/9.72 THEOREM PROVED 9.51/9.72 % SZS status Theorem 9.51/9.72 9.51/9.72 Exiting with 1 proof. 9.51/9.72 9.51/9.72 Process 2138 exit (max_proofs) Mon Jul 3 05:30:08 2023 9.51/9.72 Prover9 interrupted 9.51/9.73 EOF