0.12/0.13 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.12/0.13 % Command : tptp2X_and_run_prover9 %d %s 0.13/0.34 % Computer : n011.cluster.edu 0.13/0.34 % Model : x86_64 x86_64 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.13/0.34 % Memory : 8042.1875MB 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64 0.13/0.34 % CPULimit : 180 0.13/0.34 % DateTime : Thu Aug 29 14:43:12 EDT 2019 0.13/0.35 % CPUTime : 7.53/7.57 ============================== Prover9 =============================== 7.53/7.57 Prover9 (32) version 2009-11A, November 2009. 7.53/7.57 Process 20206 was started by sandbox on n011.cluster.edu, 7.53/7.57 Thu Aug 29 14:43:18 2019 7.53/7.57 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 180 -f /tmp/Prover9_19870_n011.cluster.edu". 7.53/7.57 ============================== end of head =========================== 7.53/7.57 7.53/7.57 ============================== INPUT ================================= 7.53/7.57 7.53/7.57 % Reading from file /tmp/Prover9_19870_n011.cluster.edu 7.53/7.57 7.53/7.57 set(prolog_style_variables). 7.53/7.57 set(auto2). 7.53/7.57 % set(auto2) -> set(auto). 7.53/7.57 % set(auto) -> set(auto_inference). 7.53/7.57 % set(auto) -> set(auto_setup). 7.53/7.57 % set(auto_setup) -> set(predicate_elim). 7.53/7.57 % set(auto_setup) -> assign(eq_defs, unfold). 7.53/7.57 % set(auto) -> set(auto_limits). 7.53/7.57 % set(auto_limits) -> assign(max_weight, "100.000"). 7.53/7.57 % set(auto_limits) -> assign(sos_limit, 20000). 7.53/7.57 % set(auto) -> set(auto_denials). 7.53/7.57 % set(auto) -> set(auto_process). 7.53/7.57 % set(auto2) -> assign(new_constants, 1). 7.53/7.57 % set(auto2) -> assign(fold_denial_max, 3). 7.53/7.57 % set(auto2) -> assign(max_weight, "200.000"). 7.53/7.57 % set(auto2) -> assign(max_hours, 1). 7.53/7.57 % assign(max_hours, 1) -> assign(max_seconds, 3600). 7.53/7.57 % set(auto2) -> assign(max_seconds, 0). 7.53/7.57 % set(auto2) -> assign(max_minutes, 5). 7.53/7.57 % assign(max_minutes, 5) -> assign(max_seconds, 300). 7.53/7.57 % set(auto2) -> set(sort_initial_sos). 7.53/7.57 % set(auto2) -> assign(sos_limit, -1). 7.53/7.57 % set(auto2) -> assign(lrs_ticks, 3000). 7.53/7.57 % set(auto2) -> assign(max_megs, 400). 7.53/7.57 % set(auto2) -> assign(stats, some). 7.53/7.57 % set(auto2) -> clear(echo_input). 7.53/7.57 % set(auto2) -> set(quiet). 7.53/7.57 % set(auto2) -> clear(print_initial_clauses). 7.53/7.57 % set(auto2) -> clear(print_given). 7.53/7.57 assign(lrs_ticks,-1). 7.53/7.57 assign(sos_limit,10000). 7.53/7.57 assign(order,kbo). 7.53/7.57 set(lex_order_vars). 7.53/7.57 clear(print_given). 7.53/7.57 7.53/7.57 % formulas(sos). % not echoed (5752 formulas) 7.53/7.57 7.53/7.57 ============================== end of input ========================== 7.53/7.57 7.53/7.57 % From the command line: assign(max_seconds, 180). 7.53/7.57 7.53/7.57 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 7.53/7.57 7.53/7.57 % Formulas that are not ordinary clauses: 7.53/7.57 1 (all X_a (number_ring(X_a) -> (all W hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),hAPP(int,int,uminus_uminus(int),W))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),one_one(X_a)),hAPP(int,X_a,number_number_of(X_a),W))))) # label(fact_2104_diff__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 2 (all Nat_4 hAPP(nat,nat,suc,Nat_4) != zero_zero(nat)) # label(fact_1313_nat_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 3 (all X_a (archim1804426504_field(X_a) -> (all X exists Z_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_2)),X)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z_2),one_one(int))))))))) # label(fact_3642_floor__exists) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 4 (all N hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),N) = hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)))),pi)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) # label(fact_2366_sin__cos__npi) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 5 (all X_a (real_normed_vector(X_a) -> (all Xa all X_2 all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),X_2),A_3),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),X_2)),Xa)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_3),Xa)),sequentially)))))) # label(fact_4764_scaleR__left_OLIMSEQ) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 6 (all X hAPP(real,real,sin,X) = hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),X))) # label(fact_2301_sin__cos__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 7 (all X all Y all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Y),Z_1))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),Z_1)),Y))))) # label(fact_1099_div__prop1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 8 (all K_4 hAPP(int,int,succ,hAPP(int,int,bit1,K_4)) = hAPP(int,int,bit0,hAPP(int,int,succ,K_4))) # label(fact_285_succ__Bit1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 9 (all X all N (-hBOOL(hAPP(nat,bool,even_odd_even(nat),N)) -> hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)),N) = X)) # label(fact_2679_odd__real__root__pow) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 10 (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),zero_zero(X_a)),A_1)) -> one_one(X_a) = hAPP(X_a,X_a,sgn_sgn(X_a),A_1))))) # label(fact_2303_sgn__pos) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 11 (all X_a (linordered_field(X_a) -> (all Z_1 all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),Y)),X)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)))))))) # label(fact_1892_mult__imp__le__div__pos) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 12 (all N hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),hAPP(nat,nat,fact_fact(nat),N))) # label(fact_2553_fact__plus__one__nat) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 13 (all X_a (real_normed_field(X_a) -> (all Ya all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_2)) -> hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),X_2)),Ya))))))) # label(fact_4129_divide_OCauchy) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 14 (all Na (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Na)) -> hAPP(int,nat,nat_1,Na) = hAPP(fun(int,bool),nat,finite_card(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less_eq(int)),Na)))))) # label(fact_3855_card__bdd__int__set__l__le) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 15 (all X_a (real_normed_algebra(X_a) -> (all Xa_2 all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_2),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_2),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y))))) # label(fact_1244_mult__right_Oadd) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 16 (all X_a (linordered_idom(X_a) -> (all Xa all Na (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),Na)),zero_zero(X_a))) <-> (zero_zero(X_a) = ti(X_a,Xa) & hBOOL(hAPP(nat,bool,even_odd_even(nat),Na)) | -hBOOL(hAPP(nat,bool,even_odd_even(nat),Na)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),zero_zero(X_a)))) & Na != zero_zero(nat))))) # label(fact_2508_power__le__zero__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 17 (all X_a (archim1804426504_field(X_a) -> (all X exists N_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1)))))) # label(fact_2835_ex__less__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 18 (all X_a (division_ring(X_a) -> (all B all A_1 (zero_zero(X_a) != ti(X_a,A_1) -> (ti(X_a,B) != zero_zero(X_a) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B))),hAPP(X_a,X_a,inverse_inverse(X_a),B))))))) # label(fact_2613_division__ring__inverse__add) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 19 (all K_4 all I_1 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_2)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),K_4))))) # label(fact_137_add__less__mono1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 20 (all B all A_1 (ti(int,A_1) != zero_zero(int) | zero_zero(int) != ti(int,B) -> one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B))))) # label(fact_3483_div__zgcd__relprime) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 21 (all K_4 all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K_4)) -> hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),hAPP(nat,nat,suc,zero_zero(nat)))) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),K_4),M)),K_4)))) # label(fact_1509_int__power__div__base) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 22 (all N all X (X = hAPP(int,real,real_1(int),hAPP(real,int,archim2085082626_floor(real),X)) -> hAPP(real,int,archim2085082626_floor(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N)) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(real,int,archim2085082626_floor(real),X)),N))) # label(fact_3395_floor__power) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 23 (all A_1 zero_zero(complex) = hAPP(real,complex,hAPP(real,fun(real,complex),rcis,zero_zero(real)),A_1)) # label(fact_3193_rcis__zero__mod) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 24 (all X_a (linord1117847801e_zero(X_a) -> (all B_2 all C all A_3 ((-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),C),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)),B_2))) & (-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_eq(X_a),zero_zero(X_a)),A_3)))) & (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)),A_3)))))) # label(fact_1822_divide__le__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 25 (all K hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less_eq(nat)),K))))) # label(fact_3844_finite__Collect__le__nat) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 26 (all Xa all Ya (hAPP(product_prod(int,int),int,product_snd(int,int),Ya) != zero_zero(int) & hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(product_prod(int,int),int,product_fst(int,int),Xa)),hAPP(product_prod(int,int),int,product_snd(int,int),Ya)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(product_prod(int,int),int,product_fst(int,int),Ya)),hAPP(product_prod(int,int),int,product_snd(int,int),Xa)) & zero_zero(int) != hAPP(product_prod(int,int),int,product_snd(int,int),Xa) <-> hBOOL(hAPP(fun(product_prod(product_prod(int,int),product_prod(int,int)),bool),bool,hAPP(product_prod(product_prod(int,int),product_prod(int,int)),fun(fun(product_prod(product_prod(int,int),product_prod(int,int)),bool),bool),member(product_prod(product_prod(int,int),product_prod(int,int))),hAPP(product_prod(int,int),product_prod(product_prod(int,int),product_prod(int,int)),hAPP(product_prod(int,int),fun(product_prod(int,int),product_prod(product_prod(int,int),product_prod(int,int))),product_Pair(product_prod(int,int),product_prod(int,int)),Xa),Ya)),ratrel)))) # label(fact_3754_ratrel__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 27 (all X_a (real_normed_vector(X_a) -> (all X exists K_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2)) & (all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),X_1),X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),X_1)),K_2)))))))) # label(fact_4779_scaleR__left_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 28 (all M hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),M)))) # label(fact_613_le__square) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 29 (all X_a (archim1727834104eiling(X_a) -> (all Xa all Va (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archimedean_ceiling(X_a),Xa)),hAPP(int,int,number_number_of(int),Va))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,number_number_of(X_a),Va)),one_one(X_a)))))))) # label(fact_3336_ceiling__less__number__of) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 30 (all N all M M = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),M)),N)) # label(fact_811_diff__add__inverse) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 31 (all X all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archimedean_ceiling(real),X)),A_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(int,real,real_1(int),A_1))))) # label(fact_3394_ceiling__le__real) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 32 (all A_3 all Xa (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> hAPP(int,int,uminus_uminus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_3),hAPP(int,nat,nat_1,Xa))) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,uminus_uminus(int),A_3)),hAPP(int,nat,nat_1,Xa))))) # label(fact_2101_neg__odd__power) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 33 (all U all Ma all Na all J all I (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),I)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),Ma)),Na)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I),U)),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J),U)),Na)))))) # label(fact_869_nat__le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 34 (all X_a all Xa all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa),A_2)) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa),B_1))))) # label(fact_2206_in__mono) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 35 (all X_a (semiri456707255roduct(X_a) -> (all C_1 all D_3 all A_1 all B all R_3 (ti(X_a,R_3) != zero_zero(X_a) -> (ti(X_a,A_1) = ti(X_a,B) & ti(X_a,C_1) != ti(X_a,D_3) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),R_3),C_1)) != hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),R_3),D_3))))))) # label(fact_458_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 36 (all Q_2 all P_1 ((all A_4 all B_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_4)) -> (hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_4),B_4) = one_one(int) -> hBOOL(hAPP(rat,bool,P_1,hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_4),B_4)))))) -> hBOOL(hAPP(rat,bool,P_1,Q_2)))) # label(fact_4439_Rat__induct) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 37 (all N hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),pi),hAPP(nat,real,real_1(nat),N))) = zero_zero(real)) # label(fact_2409_sin__npi2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 38 (all X_a (real_normed_vector(X_a) -> (all X_2 ((all E (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E)) -> (exists M_3 all M_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),M_2)) -> (all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,X_2,M_2)),hAPP(nat,X_a,X_2,N_1)))),E)))))))) <-> hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_2)))))) # label(fact_3340_Cauchy__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 39 (all X all Y all Z_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Z_1)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z_1),X)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z_1),Y)))))) # label(fact_632_real__mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 40 (all X_a (real_normed_vector(X_a) -> (all X_2 all K_3 ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,X_2,N_1))),K_3))) -> hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2)))))) # label(fact_3525_BseqI_H) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 41 (all Diff all F all Na all H (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) & (all M_2 all T_4 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_4),H)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T_4)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),Na)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_2)),T_4),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),T_4))))) & F = hAPP(nat,fun(real,real),Diff,zero_zero(nat)) -> (exists T_4 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),T_4)) & hAPP(real,real,F,H) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),H))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,Na),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),H),Na))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),T_4),H)))))) # label(fact_4306_Maclaurin__objl) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 42 (all Xa all P_1 ((all A_4 ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_4)) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_4),one_one(int))))) -> hBOOL(hAPP(int,bool,P_1,A_4)))) -> hBOOL(hAPP(int,bool,P_1,Xa)))) # label(fact_1202_d22set__induct__old) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 43 (all X_a (order(X_a) -> (all F ((all X_1 all Y_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,F,X_1)),hAPP(nat,X_a,F,Y_1))))) -> hBOOL(hAPP(fun(nat,X_a),bool,hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,X_a),bool),mono(nat,X_a),dvd_dvd(nat)),F)))))) # label(fact_4819_dvd_OmonoI) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 44 (all X all Y (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_eq(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Y),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),Y))),X)))) # label(fact_1121_zdiv__leq__prop) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 45 (all X_a all X_b (comm_monoid_mult(X_b) -> (all B_2 all A_3 all S (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S)) -> (-hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),S)) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),fequal(X_a),A_3))),B_2)),one_one(X_b))),S) = one_one(X_b)) & (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),S)) -> hAPP(X_a,X_b,B_2,A_3) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),fequal(X_a),A_3))),B_2)),one_one(X_b))),S)))))) # label(fact_3983_setprod__delta_H) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 46 (all N hAPP(real,int,archim2085082626_floor(real),hAPP(real,real,uminus_uminus(real),hAPP(nat,real,real_1(nat),N))) = hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),N))) # label(fact_2904_floor__minus__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 47 (all X all Y hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,X),Y))),hAPP(nat,int,semiring_1_of_nat(int),X))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,X),Y))),hAPP(nat,int,semiring_1_of_nat(int),Y)))) # label(fact_3237_bezw__aux) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 48 (all X_a (real_algebra_1(X_a) -> (all R_3 hAPP(real,X_a,of_real(X_a),R_3) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),one_one(X_a))))) # label(fact_4705_of__real__def) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 49 (all M all N hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),hAPP(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))),hAPP(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))))),xzgcda,M),N),M),N),one_one(int)),zero_zero(int)),zero_zero(int)),one_one(int)) = hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),xzgcd,M),N)) # label(fact_3109_xzgcd__def) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 50 (all X all Y hAPP(complex,real,re,hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(complex,real,re,X)),hAPP(complex,real,re,Y))) # label(fact_3031_Re_Odiff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 51 (all X_a (real_normed_field(X_a) -> (all F all Xa all D_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D_1)) -> (zero_zero(X_a) != hAPP(X_a,X_a,F,Xa) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),inverse_inverse(X_a)),F)),Xa),hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),D_1),hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,F,Xa)),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))))))))))))) # label(fact_3930_DERIV__inverse__fun) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 52 (all X all P_5 (hBOOL(hAPP(nat,bool,prime,P_5)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),P_5)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,X),P_5)))))) # label(fact_3095_prime__coprime__lt) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 53 (all X_a (ordere236663937imp_le(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),B))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)))))) # label(fact_209_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 54 (all X_a all X_b (ordered_ab_group_add(X_b) -> (all I all B_1 all F all Sb (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),Sb)) -> ((all I_2 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I_2),Sb)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(X_a,X_b,F,I_2))))) -> (B_1 = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),Sb) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I),Sb)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,I)),B_1))))))))) # label(fact_4326_setsum__nonneg__leq__bound) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 55 (all I all M_1 (hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),zero_zero(nat)),M_1)) -> zero_zero(nat) != hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),combs(nat,bool,bool),hAPP(fun(nat,bool),fun(nat,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(fun(nat,bool),bool)),fun(fun(nat,bool),fun(nat,bool)),combc(nat,fun(nat,bool),bool),member(nat)),M_1))),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),hAPP(nat,nat,suc,I))))))) # label(fact_3903_card__less) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 56 (all P_4 all A_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_3)) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,A_3),P_4))))) # label(fact_1204_wset__mem__mem) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 57 (all X_a (ring_1(X_a) -> (all B_2 all A_3 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),ring_1_Ints(X_a))) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),ring_1_Ints(X_a))) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_2)),ring_1_Ints(X_a)))))))) # label(fact_3823_Ints__add) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 58 (all X_a (ring_1(X_a) -> (all B_2 all A_3 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),ring_1_Ints(X_a))) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),ring_1_Ints(X_a))) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2)),ring_1_Ints(X_a)))))))) # label(fact_3824_Ints__diff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 59 (all Ma all Na (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),Ma)),hAPP(int,real,number_number_of(real),Na))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Ma),hAPP(int,int,number_number_of(int),Na))))) # label(fact_3412_number__of__less__real__of__int__iff2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.57 60 (all X_b all X_a all F all A_3 all B_2 (hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),F,A_3),B_2)) -> hBOOL(hAPP(product_prod(X_a,X_b),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),F),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_3),B_2))))) # label(fact_4066_splitI) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 61 (all X_a all X_b (comm_monoid_add(X_b) & ordere779506340up_add(X_b) -> (all F all A_2 ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(X_a,X_b,F,X_1))))) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2))))))) # label(fact_4336_setsum__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 62 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),Xa)),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,fun(nat,real)),fun(real,real),hAPP(fun(fun(nat,real),real),fun(fun(real,fun(nat,real)),fun(real,real)),combb(fun(nat,real),real,real),suminf(real)),hAPP(fun(real,fun(nat,real)),fun(real,fun(nat,real)),hAPP(fun(fun(nat,real),fun(nat,real)),fun(fun(real,fun(nat,real)),fun(real,fun(nat,real))),combb(fun(nat,real),fun(nat,real),real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min))))),hAPP(fun(real,fun(nat,real)),fun(real,fun(nat,real)),hAPP(fun(fun(nat,real),fun(nat,real)),fun(fun(real,fun(nat,real)),fun(real,fun(nat,real))),combb(fun(nat,real),fun(nat,real),real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),inverse_divide(real),one_one(real))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat))))))),hAPP(fun(nat,nat),fun(real,fun(nat,real)),hAPP(fun(real,fun(fun(nat,nat),fun(nat,real))),fun(fun(nat,nat),fun(real,fun(nat,real))),combc(real,fun(nat,nat),fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(real,fun(fun(nat,nat),fun(nat,real))),hAPP(fun(fun(nat,real),fun(fun(nat,nat),fun(nat,real))),fun(fun(real,fun(nat,real)),fun(real,fun(fun(nat,nat),fun(nat,real)))),combb(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),real),combb(nat,real,nat)),power_power(real))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat))))))),Xa),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),Xa)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))))) # label(fact_3994_DERIV__arctan__series) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 63 (all X_a (real_normed_field(X_a) -> (all Na all F all Xa all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(nat,fun(X_a,X_a),hAPP(fun(X_a,fun(nat,X_a)),fun(nat,fun(X_a,X_a)),combc(X_a,nat,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(nat,X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(nat,X_a))),combb(X_a,fun(nat,X_a),X_a),power_power(X_a)),F)),Na)),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),D),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,F,Xa)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,zero_zero(nat)))))))))))) # label(fact_3918_DERIV__power) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 64 (all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> N = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))))) # label(fact_1464_Suc__diff__1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 65 (all A_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),one_one(nat)),A_3)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_3),one_one(nat))) <-> one_one(nat) != A_3)) # label(fact_1258_gcd__lcm__complete__lattice__nat_Obot__less) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 66 (all T_1 comple44059174attice(filter(T_1))) # label(arity_Limits_Ofilter___Complete__Lattice_Ocomplete__lattice) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 67 (all X_a all X_b ti(fun(fun(X_a,X_b),fun(fun(X_a,bool),bool)),nat_tr1645093318rphism(X_a,X_b)) = nat_tr1645093318rphism(X_a,X_b)) # label(tsy_c_Nat__Transfer_Otransfer__morphism_res) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 68 (all X_a (linordered_field(X_a) -> (all A_1 all B (ti(X_a,B) != zero_zero(X_a) -> hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)))))) # label(fact_1858_nonzero__abs__divide) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 69 (all X_b all X_a (ord(X_b) & order(X_a) -> (all C all F all B_2 all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,B_2)),A_3)) -> (hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),C),B_2)) -> ((all X_1 all Y_1 (hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Y_1),X_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,Y_1)),hAPP(X_b,X_a,F,X_1))))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,C)),A_3)))))))) # label(fact_2840_xt6) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 70 (all M all N ((-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)) & (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) -> M = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)))) # label(fact_1536_mod__if) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 71 (all X_a (number_semiring(X_a) -> (all Z_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z_1),Z_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Z_1)))) # label(fact_546_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 72 (all Xa hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),sin)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),Xa),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,Xa)),hAPP(real,real,sin,Xa))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,Xa)),hAPP(real,real,sin,Xa)))))) # label(fact_3939_DERIV__sin__realpow2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 73 (all X_b all X_a all B_2 all F all Xa all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa),A_2)) -> (hAPP(X_a,X_b,F,Xa) = ti(X_b,B_2) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),B_2),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),F),A_2)))))) # label(fact_3661_rev__image__eqI) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 74 (all F all G ((exists N_2 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(nat,real,F,N_1))),hAPP(nat,real,G,N_1))))) -> (hBOOL(hAPP(fun(nat,real),bool,summable(real),G)) -> hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),abs_abs(real)),F)))))) # label(fact_4045_summable__rabs__comparison__test) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 75 (all X hAPP(complex,real,re,X) = hAPP(complex,real,re,hAPP(complex,complex,cnj,X))) # label(fact_3020_complex__Re__cnj) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 76 (all X_1 hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),cos_coeff)),hAPP(real,fun(nat,real),power_power(real),X_1))) = hAPP(real,real,cos,X_1)) # label(fact_3999_lemma__cos__ext) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 77 (all X_a (linordered_idom(X_a) -> (all B all D_3 all A_1 all C_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),C_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,abs_abs(X_a),B)),D_3)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),D_3)))))))) # label(fact_992_abs__mult__less) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 78 (all X_a (ring(X_a) -> (all A_1 all B hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B)))) # label(fact_1907_minus__mult__left) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 79 (all X_a all X_b (comm_semiring_1(X_b) -> (all F all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> ((exists X_1 (hAPP(X_a,X_b,F,X_1) = zero_zero(X_b) & hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2)))) -> zero_zero(X_b) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2)))))) # label(fact_3976_setprod__zero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 80 (all P all Q (hBOOL(P) | -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q)))) # label(help_fconj_2_1_U) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 81 (all X_a (ord(X_a) -> hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,X_a)),max(X_a),ord_less_eq(X_a)) = ord_max(X_a))) # label(fact_4982_max__ord__max) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 82 (all X_b all X_a (topological_space(X_a) & real_algebra_1(X_b) & real_normed_vector(X_b) -> (all G all L all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,real),fun(X_a,X_b),hAPP(fun(real,X_b),fun(fun(X_a,real),fun(X_a,X_b)),combb(real,X_b,X_a),of_real(X_b)),G)),hAPP(real,X_b,of_real(X_b),L)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4620_LIM__of__real) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 83 (all X_a (real_vector(X_a) -> (all X all Y all Xa_2 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),X),Xa_2)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Y),Xa_2)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)),Xa_2)))) # label(fact_4695_scaleR__left_Oadd) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 84 (all X_a (number_ring(X_a) & ring_char_0(X_a) -> (all Wa (hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,Wa)))) <-> hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),Wa))))))) # label(fact_2157_iszero__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 85 (all X_a (banach(X_a) & real_normed_field(X_a) -> (all X all Y hAPP(X_a,X_a,exp(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,exp(X_a),X)),hAPP(X_a,X_a,exp(X_a),Y))))) # label(fact_1748_exp__diff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 86 (all X_a (linord1117847801e_zero(X_a) -> (all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B_2)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_3)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),B_2))))))) # label(fact_1845_zero__less__divide__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 87 (all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),hAPP(real,real,cos,X)))) # label(fact_2289_cos__ge__minus__one) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 88 (all X_a (ring_div(X_a) -> (all A_1 all B all A_6 (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_6),B) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_6)),B))))) # label(fact_1945_mod__minus__cong) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 89 (all X_a all C all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1))) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),A_2)))) # label(fact_2242_DiffD1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 90 (all X_b all X_a all A_3 all B_2 all A_8 all B_7 (ti(X_a,A_3) = ti(X_a,A_8) & ti(X_b,B_7) = ti(X_b,B_2) <-> hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_8),B_7) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_3),B_2))) # label(fact_3154_Pair__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 91 (all R_3 all P_5 all Q_3 (hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_5),Q_3) = hAPP(rat,product_prod(int,int),quotient_of,R_3) -> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),P_5),Q_3) = one_one(int))) # label(fact_4430_quotient__of__coprime) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 92 (all Z_1 hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),Z_1),hAPP(complex,complex,cnj,Z_1))) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,norm_norm(complex),Z_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) # label(fact_2952_complex__mod__mult__cnj) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 93 (all X_a (real_normed_vector(X_a) -> (all X_2 all L_2 ((all R_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1)) -> (exists No all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),No),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,X_2,N_1)),L_2))),R_1)))))) <-> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),L_2),sequentially)))))) # label(fact_4544_LIMSEQ__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 94 (all X_a (linorder(X_a) -> (all Xa all Ya (hAPP(X_a,fun(X_a,bool),ord_atMost(X_a),Xa) = hAPP(X_a,fun(X_a,bool),ord_atMost(X_a),Ya) <-> ti(X_a,Xa) = ti(X_a,Ya))))) # label(fact_4965_atMost__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 95 (all Ma all Na hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),Ma),hAPP(nat,nat,suc,Na)) = hAPP(nat,nat,hAPP(fun(nat,nat),fun(nat,nat),hAPP(nat,fun(fun(nat,nat),fun(nat,nat)),nat_case(nat),zero_zero(nat)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),suc),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),ord_min(nat)),Na))),Ma)) # label(fact_4857_min__Suc2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 96 (all X_a (real_algebra_1(X_a) -> (all X hAPP(real,X_a,of_real(X_a),hAPP(real,real,uminus_uminus(real),X)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(real,X_a,of_real(X_a),X))))) # label(fact_2806_of__real__minus) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 97 (all A_3 all B_2 all P_1 ((all A_4 all B_4 all C_4 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B_4),C_4)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_4),B_4)) & hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),B_4),C_4))) & hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_4),B_4))) -> hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_4),C_4))))) -> ((all X_1 exists D_2 ((all A_4 all B_4 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),B_4)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B_4),A_4)),D_2)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_4),X_1)) -> hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_4),B_4))))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_2)))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2)) -> hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_3),B_2))))))) # label(fact_3235_lemma__BOLZANO) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 98 (all X_a (semiring_div(X_a) -> (all A_1 all B all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C_1))),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C_1)))) # label(fact_1283_mod__mult__right__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 99 (all K all F all Na ((all I_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),Na)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_2),one_one(nat)))),hAPP(nat,int,F,I_2)))),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_2 (ti(int,K) = hAPP(nat,int,F,I_2) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),Na)))))))) # label(fact_1066_int__val__lemma) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 100 (all X_a (ring_1(X_a) -> (all K ((zero_zero(int) = ti(int,K) -> hAPP(int,X_a,ring_1_of_int(X_a),K) = zero_zero(X_a)) & (ti(int,K) != zero_zero(int) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),zero_zero(int))) -> hAPP(int,X_a,ring_1_of_int(X_a),K) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,uminus_uminus(int),K)))) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),zero_zero(int))) -> hAPP(int,X_a,ring_1_of_int(X_a),K) = hAPP(product_prod(int,int),X_a,hAPP(fun(int,fun(int,X_a)),fun(product_prod(int,int),X_a),product_prod_case(int,int,X_a),hAPP(fun(int,X_a),fun(int,fun(int,X_a)),hAPP(fun(int,fun(X_a,fun(int,X_a))),fun(fun(int,X_a),fun(int,fun(int,X_a))),combs(int,X_a,fun(int,X_a)),hAPP(fun(int,fun(int,fun(X_a,X_a))),fun(int,fun(X_a,fun(int,X_a))),hAPP(fun(fun(int,fun(X_a,X_a)),fun(X_a,fun(int,X_a))),fun(fun(int,fun(int,fun(X_a,X_a))),fun(int,fun(X_a,fun(int,X_a)))),combb(fun(int,fun(X_a,X_a)),fun(X_a,fun(int,X_a)),int),combc(int,X_a,X_a)),hAPP(fun(int,X_a),fun(int,fun(int,fun(X_a,X_a))),hAPP(fun(X_a,fun(int,fun(X_a,X_a))),fun(fun(int,X_a),fun(int,fun(int,fun(X_a,X_a)))),combb(X_a,fun(int,fun(X_a,X_a)),int),hAPP(fun(int,fun(X_a,fun(X_a,X_a))),fun(X_a,fun(int,fun(X_a,X_a))),combc(int,X_a,fun(X_a,X_a)),hAPP(fun(int,bool),fun(int,fun(X_a,fun(X_a,X_a))),hAPP(fun(bool,fun(X_a,fun(X_a,X_a))),fun(fun(int,bool),fun(int,fun(X_a,fun(X_a,X_a)))),combb(bool,fun(X_a,fun(X_a,X_a)),int),if(X_a)),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),fequal(int)),zero_zero(int))))),hAPP(fun(int,X_a),fun(int,X_a),hAPP(fun(int,fun(X_a,X_a)),fun(fun(int,X_a),fun(int,X_a)),combs(int,X_a,X_a),hAPP(fun(int,X_a),fun(int,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(int,X_a),fun(int,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),int),plus_plus(X_a)),ring_1_of_int(X_a))),ring_1_of_int(X_a))))),hAPP(X_a,fun(int,X_a),hAPP(fun(int,fun(X_a,X_a)),fun(X_a,fun(int,X_a)),combc(int,X_a,X_a),hAPP(fun(int,X_a),fun(int,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(int,X_a),fun(int,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),int),plus_plus(X_a)),hAPP(fun(int,X_a),fun(int,X_a),hAPP(fun(int,fun(X_a,X_a)),fun(fun(int,X_a),fun(int,X_a)),combs(int,X_a,X_a),hAPP(fun(int,X_a),fun(int,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(int,X_a),fun(int,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),int),plus_plus(X_a)),ring_1_of_int(X_a))),ring_1_of_int(X_a)))),one_one(X_a)))),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,K),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))))) # label(fact_4005_of__int__num) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 101 (all T_2 all T_1 (comple44059174attice(T_1) -> comple44059174attice(fun(T_2,T_1)))) # label(arity_fun___Complete__Lattice_Ocomplete__lattice) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 102 (all K_4 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),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_4),N))))) # label(fact_861_dvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 103 (all X_a (topological_space(X_a) -> (all F all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,real),fun(X_a,real),hAPP(fun(real,real),fun(fun(X_a,real),fun(X_a,real)),combb(real,real,X_a),abs_abs(real)),F)),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4618_LIM__rabs__zero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 104 (all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),M)) -> hAPP(nat,nat,suc,hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),M),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),M),one_one(int))))) # label(fact_1516_inv__is__inv__aux) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 105 (all M all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,suc,N))))) # label(fact_1559_mod__Suc__eq__mod__add3) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 106 (all X hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,sin,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,cos,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) # label(fact_2298_sin__squared__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 107 (all Xa all Ya (-hBOOL(hAPP(nat,bool,even_odd_even(nat),Ya)) & -hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa)) | hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa)) & hBOOL(hAPP(nat,bool,even_odd_even(nat),Ya)) <-> hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Xa),Ya))))) # label(fact_2470_even__sum__nat) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 108 (all X_b (monoid_mult(X_b) & number(X_b) -> (all W hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),W)),hAPP(int,X_b,number_number_of(X_b),W)) = hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),hAPP(int,X_b,number_number_of(X_b),W)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) # label(fact_551_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 109 (all Xa all A_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archimedean_ceiling(real),Xa)),A_3)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),hAPP(int,real,real_1(int),A_3))))) # label(fact_3392_ceiling__le__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 110 (all X_a all F all A_2 hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co302546470etprod(X_a,nat),F),A_2) = hAPP(int,nat,nat_1,hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co302546470etprod(X_a,int),hAPP(fun(X_a,nat),fun(X_a,int),hAPP(fun(nat,int),fun(fun(X_a,nat),fun(X_a,int)),combb(nat,int,X_a),semiring_1_of_nat(int)),F)),A_2))) # label(fact_3970_transfer__nat__int__sum__prod2_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 111 (all A_1 all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int)))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(int,int,hAPP(int,fun(int,int),inv,P_5),A_1))))))) # label(fact_1176_inv__g__1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 112 (all X_a (comm_semiring_1(X_a) -> (all M all A_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),M),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),M)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),one_one(X_a))),M)))) # label(fact_463_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 113 (all Na (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Na)) -> hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(int,bool),nat,finite_card(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less_eq(int)),Na))))) = ti(int,Na))) # label(fact_3906_int__card__bdd__int__set__l__le) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 114 (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),M),N)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),N))))) # label(fact_1469_div__geq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 115 (all A_1 all B all Q_3 all R_3 (hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_3)),R_3) = ti(int,A_1) -> ((-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),R_3),zero_zero(int))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_3))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_3),B)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_3))) -> hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_3),R_3)))))) # label(fact_1640_divmod__int__relI) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 116 (all X_a (real_normed_vector(X_a) -> (all R_3 all X hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,sgn_sgn(real),R_3)),hAPP(X_a,X_a,sgn_sgn(X_a),X)) = hAPP(X_a,X_a,sgn_sgn(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),X))))) # label(fact_4692_sgn__scaleR) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 117 (all F all A_3 all B_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),B_2)) -> ((all X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),B_2)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),X_1)) -> (exists Y_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),Y_1)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y_1),zero_zero(real))))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,B_2)),hAPP(real,real,F,A_3)))))) # label(fact_2864_DERIV__neg__imp__decreasing) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 118 (all X_a (metric_space(X_a) -> (all E_2 all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_2)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E_2)) -> (exists M_3 all M_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),M_2)) -> (all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),hAPP(nat,X_a,X_2,M_2)),hAPP(nat,X_a,X_2,N_1))),E_2))))))))))) # label(fact_3547_metric__CauchyD) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 119 (all X_a all F all A_2 hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),hAPP(fun(X_a,nat),fun(X_a,int),hAPP(fun(nat,int),fun(fun(X_a,nat),fun(X_a,int)),combb(nat,int,X_a),semiring_1_of_nat(int)),F)),A_2) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),A_2))) # label(fact_4278_int__setsum) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 120 (all X_a ti(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),powp(X_a)) = powp(X_a)) # label(tsy_c_Predicate_OPowp_res) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 121 (all X_a all X_b ti(fun(X_a,fun(X_b,X_a)),combk(X_a,X_b)) = combk(X_a,X_b)) # label(tsy_c_COMBK_res) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 122 (all X_a (field_inverse_zero(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),A_1) = hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B))))) # label(fact_2579_inverse__divide) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 123 (all X_b all X_a all A_1 all B all A_6 all B_5 (hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_1),B) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_6),B_5) -> -(ti(X_a,A_1) = ti(X_a,A_6) -> ti(X_b,B) != ti(X_b,B_5)))) # label(fact_3155_Pair__inject) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 124 (all X_a (real_normed_field(X_a) & banach(X_a) -> (all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,exp(X_a),X)),hAPP(X_a,X_a,exp(X_a),Y)) = hAPP(X_a,X_a,exp(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y))))) # label(fact_1774_mult__exp__exp) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 125 (all X_a (semiring_div(X_a) -> (all Ma all K all Na (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K),Na)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K),Ma)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),Ma),Na)))))))) # label(fact_1305_dvd__mod__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 126 (all X all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5)) -> (hBOOL(hAPP(int,bool,zprime,P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),X))),X),P_5)))))) # label(fact_1186_MultInv__prop4) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 127 (all X_a (ring_1(X_a) -> (all N hAPP(nat,X_a,semiring_1_of_nat(X_a),N) = hAPP(int,X_a,ring_1_of_int(X_a),hAPP(nat,int,semiring_1_of_nat(int),N))))) # label(fact_3611_of__int__of__nat__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 128 (all C (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),C)),one_one(real))) -> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(real,fun(nat,real),power_power(real),C)),zero_zero(real)),sequentially)))) # label(fact_4463_LIMSEQ__rabs__realpow__zero2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 129 (all A_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A_3)) <-> zero_zero(nat) = A_3)) # label(fact_1252_gcd__lcm__complete__lattice__nat_Otop__unique) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 130 (all X -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,exp(real),X)),zero_zero(real)))) # label(fact_1766_not__exp__le__zero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 131 (all Na hAPP(nat,nat,fact_fact(nat),Na) = hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co302546470etprod(nat,nat),combi(nat)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),one_one(nat)),Na))) # label(fact_4565_fact__altdef__nat) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 132 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)))))))) # label(fact_199_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 133 (all N hAPP(nat,complex,semiring_1_of_nat(complex),N) = hAPP(complex,complex,cnj,hAPP(nat,complex,semiring_1_of_nat(complex),N))) # label(fact_2962_complex__cnj__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 134 (all Q_3 all R_3 hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),Q_3),R_3) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),Q_3),hAPP(rat,rat,uminus_uminus(rat),R_3))) # label(fact_3733_diff__rat__def) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 135 (all X_b all X_a (comm_monoid_mult(X_a) -> (all F all G all A_2 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),F),A_2)),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),G),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,X_a)),combs(X_b,X_a,X_a),hAPP(fun(X_b,X_a),fun(X_b,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_b),times_times(X_a)),F)),G)),A_2)))) # label(fact_3973_setprod__timesf) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 136 (all N all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N))) # label(fact_1538_le__mod__geq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 137 (all Ya all Xa all A_3 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A_3)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Ya)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),log,A_3),Xa)),hAPP(real,real,hAPP(real,fun(real,real),log,A_3),Ya))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya))))))) # label(fact_2031_log__less__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 138 -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))),hAPP(int,int,number_number_of(int),min))) -> hAPP(int,int,hAPP(int,fun(int,int),legendre,hAPP(int,int,number_number_of(int),min)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))) != one_one(int) # label(fact_697__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]. 7.53/7.58 139 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),K)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),hAPP(int,int,bit1,K))))) # label(fact_674_rel__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 140 (all X_a (comm_ring_1(X_a) -> (all X all N (hBOOL(hAPP(nat,bool,even_odd_even(nat),N)) -> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),N))))) # label(fact_2479_Parity_Opower__minus__even) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 141 (all Na all Diff all F all H (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H)) -> (hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F -> ((all M_2 all T_4 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T_4)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_4),H)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),Na)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_2)),T_4),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),T_4))))) -> (exists T_4 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),T_4)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_4),H)) & hAPP(real,real,F,H) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),H))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,Na),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),H),Na))))))))) # label(fact_4309_Maclaurin2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 142 (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_141_nat__less__cases) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 143 (all K_4 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),min) = hAPP(int,int,pred,K_4)) # label(fact_2932_add__Min__right) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 144 (all X_a (number_ring(X_a) & field(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls)))))) # label(fact_1782_divide__numeral__1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 145 (all D_1 all A_3 all B_2 (hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_1),B_2) = one_one(nat) & one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_1),A_3) <-> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_3),B_2)) = one_one(nat))) # label(fact_3292_coprime__mul__eq__nat) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 146 (all C all D_1 all A_3 all B_2 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),Ma)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,C),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_3),D_1)),Ma)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,C),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_2),D_1)),Ma))))) # label(fact_897_zcong__zmult__prop1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 147 (all X_a (linorder(X_a) -> (all X all Y all Z_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z_1))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Z_1))))))) # label(fact_4980_min__max_Odistrib__sup__le) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 148 (all B_2 all Q_2 all Ra hAPP(product_prod(int,int),product_prod(int,int),hAPP(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),hAPP(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),if(product_prod(int,int)),hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Ra),B_2))),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Q_2)),one_one(int))),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Ra),B_2))),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Q_2)),Ra)) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,B_2),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_2),Ra))) # label(fact_3840_adjust__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 149 (all Va all V_1 ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Va),V_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Va),pls))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(int,nat,number_number_of(nat),Va)),hAPP(int,nat,number_number_of(nat),V_1))))) # label(fact_498_le__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 150 (all X_a (ring_1(X_a) -> (all Z hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)),ring_1_Ints(X_a)))))) # label(fact_3833_Ints__of__int) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 151 (all X_b all X_a all R_2 all K all T_3 all S (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),T_3)) -> ((all X_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),T_3)) -> hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),S))),hAPP(X_b,fun(X_a,bool),hAPP(fun(X_a,fun(X_b,bool)),fun(X_b,fun(X_a,bool)),combc(X_a,X_b,bool),R_2),X_1)))) = K)) -> hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,nat),hAPP(fun(fun(X_b,bool),nat),fun(fun(X_a,fun(X_b,bool)),fun(X_a,nat)),combb(fun(X_b,bool),nat,X_a),finite_card(X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_a),collect(X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_a),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),T_3)))),R_2)))),S) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),hAPP(fun(X_b,bool),nat,finite_card(X_b),T_3)))))) # label(fact_4325_setsum__multicount) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 152 (all A_3 all B_2 (hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B_2),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_3),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) <-> hAPP(int,int,abs_abs(int),B_2) = hAPP(int,int,abs_abs(int),A_3))) # label(fact_1053_power2__eq__iff__abs__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 153 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit0,K)),pls)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),pls)))) # label(fact_259_rel__simps_I27_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 154 (all A_3 all B_2 ((all D_2 (D_2 = one_one(nat) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_2),A_3)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_2),B_2)))) <-> one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_3),B_2))) # label(fact_3290_coprime__nat) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 155 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),hAPP(nat,nat,suc,N))) -> (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) -> N = M))) # label(fact_1326_less__SucE) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 156 (all P_1 all A_3 all B_2 all Na hAPP(product_prod(real,real),product_prod(real,real),hAPP(fun(real,fun(real,product_prod(real,real))),fun(product_prod(real,real),product_prod(real,real)),product_prod_case(real,real,product_prod(real,real)),hAPP(fun(real,fun(real,product_prod(real,real))),fun(real,fun(real,product_prod(real,real))),hAPP(fun(real,fun(fun(real,product_prod(real,real)),fun(real,product_prod(real,real)))),fun(fun(real,fun(real,product_prod(real,real))),fun(real,fun(real,product_prod(real,real)))),combs(real,fun(real,product_prod(real,real)),fun(real,product_prod(real,real))),hAPP(fun(real,fun(real,fun(product_prod(real,real),product_prod(real,real)))),fun(real,fun(fun(real,product_prod(real,real)),fun(real,product_prod(real,real)))),hAPP(fun(fun(real,fun(product_prod(real,real),product_prod(real,real))),fun(fun(real,product_prod(real,real)),fun(real,product_prod(real,real)))),fun(fun(real,fun(real,fun(product_prod(real,real),product_prod(real,real)))),fun(real,fun(fun(real,product_prod(real,real)),fun(real,product_prod(real,real))))),combb(fun(real,fun(product_prod(real,real),product_prod(real,real))),fun(fun(real,product_prod(real,real)),fun(real,product_prod(real,real))),real),combs(real,product_prod(real,real),product_prod(real,real))),hAPP(fun(real,fun(real,product_prod(real,real))),fun(real,fun(real,fun(product_prod(real,real),product_prod(real,real)))),hAPP(fun(real,fun(fun(real,product_prod(real,real)),fun(real,fun(product_prod(real,real),product_prod(real,real))))),fun(fun(real,fun(real,product_prod(real,real))),fun(real,fun(real,fun(product_prod(real,real),product_prod(real,real))))),combs(real,fun(real,product_prod(real,real)),fun(real,fun(product_prod(real,real),product_prod(real,real)))),hAPP(fun(real,fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real))))),fun(real,fun(fun(real,product_prod(real,real)),fun(real,fun(product_prod(real,real),product_prod(real,real))))),hAPP(fun(fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real)))),fun(fun(real,product_prod(real,real)),fun(real,fun(product_prod(real,real),product_prod(real,real))))),fun(fun(real,fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real))))),fun(real,fun(fun(real,product_prod(real,real)),fun(real,fun(product_prod(real,real),product_prod(real,real)))))),combb(fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real)))),fun(fun(real,product_prod(real,real)),fun(real,fun(product_prod(real,real),product_prod(real,real)))),real),combs(real,product_prod(real,real),fun(product_prod(real,real),product_prod(real,real)))),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real))))),hAPP(fun(fun(real,bool),fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real))))),fun(fun(real,fun(real,bool)),fun(real,fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real)))))),combb(fun(real,bool),fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real)))),real),hAPP(fun(bool,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real)))),fun(fun(real,bool),fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real))))),combb(bool,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real))),real),if(product_prod(real,real)))),hAPP(fun(real,fun(real,product_prod(real,real))),fun(real,fun(real,bool)),hAPP(fun(fun(real,product_prod(real,real)),fun(real,bool)),fun(fun(real,fun(real,product_prod(real,real))),fun(real,fun(real,bool))),combb(fun(real,product_prod(real,real)),fun(real,bool),real),hAPP(fun(product_prod(real,real),bool),fun(fun(real,product_prod(real,real)),fun(real,bool)),combb(product_prod(real,real),bool,real),P_1)),hAPP(fun(real,fun(real,real)),fun(real,fun(real,product_prod(real,real))),hAPP(fun(real,fun(fun(real,real),fun(real,product_prod(real,real)))),fun(fun(real,fun(real,real)),fun(real,fun(real,product_prod(real,real)))),combs(real,fun(real,real),fun(real,product_prod(real,real))),hAPP(fun(real,fun(real,product_prod(real,real))),fun(real,fun(fun(real,real),fun(real,product_prod(real,real)))),hAPP(fun(fun(real,product_prod(real,real)),fun(fun(real,real),fun(real,product_prod(real,real)))),fun(fun(real,fun(real,product_prod(real,real))),fun(real,fun(fun(real,real),fun(real,product_prod(real,real))))),combb(fun(real,product_prod(real,real)),fun(fun(real,real),fun(real,product_prod(real,real))),real),combb(real,product_prod(real,real),real)),product_Pair(real,real))),hAPP(real,fun(real,fun(real,real)),hAPP(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real))),combc(real,real,fun(real,real)),hAPP(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real))),hAPP(fun(fun(real,fun(real,real)),fun(real,fun(real,real))),fun(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real)))),combb(fun(real,fun(real,real)),fun(real,fun(real,real)),real),combc(real,real,real)),hAPP(fun(real,fun(real,real)),fun(real,fun(real,fun(real,real))),hAPP(fun(fun(real,real),fun(real,fun(real,real))),fun(fun(real,fun(real,real)),fun(real,fun(real,fun(real,real)))),combb(fun(real,real),fun(real,fun(real,real)),real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real))),plus_plus(real)))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))),hAPP(fun(real,real),fun(real,fun(real,product_prod(real,real))),hAPP(fun(real,fun(fun(real,real),fun(real,product_prod(real,real)))),fun(fun(real,real),fun(real,fun(real,product_prod(real,real)))),combc(real,fun(real,real),fun(real,product_prod(real,real))),hAPP(fun(real,fun(real,fun(real,product_prod(real,real)))),fun(real,fun(fun(real,real),fun(real,product_prod(real,real)))),hAPP(fun(fun(real,fun(real,product_prod(real,real))),fun(fun(real,real),fun(real,product_prod(real,real)))),fun(fun(real,fun(real,fun(real,product_prod(real,real)))),fun(real,fun(fun(real,real),fun(real,product_prod(real,real))))),combb(fun(real,fun(real,product_prod(real,real))),fun(fun(real,real),fun(real,product_prod(real,real))),real),combs(real,real,product_prod(real,real))),hAPP(fun(real,fun(real,real)),fun(real,fun(real,fun(real,product_prod(real,real)))),hAPP(fun(fun(real,real),fun(real,fun(real,product_prod(real,real)))),fun(fun(real,fun(real,real)),fun(real,fun(real,fun(real,product_prod(real,real))))),combb(fun(real,real),fun(real,fun(real,product_prod(real,real))),real),hAPP(fun(real,fun(real,product_prod(real,real))),fun(fun(real,real),fun(real,fun(real,product_prod(real,real)))),combb(real,fun(real,product_prod(real,real)),real),product_Pair(real,real))),hAPP(real,fun(real,fun(real,real)),hAPP(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real))),combc(real,real,fun(real,real)),hAPP(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real))),hAPP(fun(fun(real,fun(real,real)),fun(real,fun(real,real))),fun(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real)))),combb(fun(real,fun(real,real)),fun(real,fun(real,real)),real),combc(real,real,real)),hAPP(fun(real,fun(real,real)),fun(real,fun(real,fun(real,real))),hAPP(fun(fun(real,real),fun(real,fun(real,real))),fun(fun(real,fun(real,real)),fun(real,fun(real,fun(real,real)))),combb(fun(real,real),fun(real,fun(real,real)),real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real))),plus_plus(real)))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),combi(real))))),hAPP(fun(real,fun(real,real)),fun(real,fun(real,product_prod(real,real))),hAPP(fun(real,fun(fun(real,real),fun(real,product_prod(real,real)))),fun(fun(real,fun(real,real)),fun(real,fun(real,product_prod(real,real)))),combs(real,fun(real,real),fun(real,product_prod(real,real))),hAPP(fun(real,fun(real,product_prod(real,real))),fun(real,fun(fun(real,real),fun(real,product_prod(real,real)))),hAPP(fun(fun(real,product_prod(real,real)),fun(fun(real,real),fun(real,product_prod(real,real)))),fun(fun(real,fun(real,product_prod(real,real))),fun(real,fun(fun(real,real),fun(real,product_prod(real,real))))),combb(fun(real,product_prod(real,real)),fun(fun(real,real),fun(real,product_prod(real,real))),real),combb(real,product_prod(real,real),real)),product_Pair(real,real))),hAPP(real,fun(real,fun(real,real)),hAPP(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real))),combc(real,real,fun(real,real)),hAPP(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real))),hAPP(fun(fun(real,fun(real,real)),fun(real,fun(real,real))),fun(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real)))),combb(fun(real,fun(real,real)),fun(real,fun(real,real)),real),combc(real,real,real)),hAPP(fun(real,fun(real,real)),fun(real,fun(real,fun(real,real))),hAPP(fun(fun(real,real),fun(real,fun(real,real))),fun(fun(real,fun(real,real)),fun(real,fun(real,fun(real,real)))),combb(fun(real,real),fun(real,fun(real,real)),real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real))),plus_plus(real)))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),Na)) = hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),hAPP(nat,nat,suc,Na))) # label(fact_4080_Bolzano__bisect_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 157 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X)),hAPP(real,real,exp(real),X))))) # label(fact_1799_exp__ge__add__one__self__aux) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 158 (all N hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N))))) # label(fact_2394_real__of__nat__Suc__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 159 (all Xa all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),zero_zero(real))) -> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),hAPP(nat,fun(real,real),root,Na)),Xa))))) # label(fact_4940_isCont__root__neg) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 160 (all X all A_1 all B hAPP(real,real,hAPP(real,fun(real,real),powr,hAPP(real,real,hAPP(real,fun(real,real),powr,X),B)),A_1) = hAPP(real,real,hAPP(real,fun(real,real),powr,hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),B)) # label(fact_3354_powr__powr__swap) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 161 (all X_a (real_algebra(X_a) -> (all X all A_1 all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),Y)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y))))) # label(fact_4719_mult__scaleR__right) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 162 (all X_a (real_normed_vector(X_a) -> (all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2)) <-> (exists N_2 all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,X_2,N_1))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_2))))))))) # label(fact_3520_Bseq__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 163 (all X_a (number_ring(X_a) -> one_one(X_a) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls)))) # label(fact_13_numeral__1__eq__1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 164 (all X_a (field(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,inverse_inverse(X_a),B))))) # label(fact_2600_field__divide__inverse) # label(axiom) # label(non_clause). [assumption]. 7.53/7.58 165 (all X_b all X_c all X_a (semiring_0(X_a) -> (all F all A_2 all G all B_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2)),hAPP(fun(X_c,bool),X_a,hAPP(fun(X_c,X_a),fun(fun(X_c,bool),X_a),big_co1399186613setsum(X_c,X_a),G),B_1)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_c,bool),fun(X_b,X_a),hAPP(fun(X_b,fun(fun(X_c,bool),X_a)),fun(fun(X_c,bool),fun(X_b,X_a)),combc(X_b,fun(X_c,bool),X_a),hAPP(fun(X_b,fun(X_c,X_a)),fun(X_b,fun(fun(X_c,bool),X_a)),hAPP(fun(fun(X_c,X_a),fun(fun(X_c,bool),X_a)),fun(fun(X_b,fun(X_c,X_a)),fun(X_b,fun(fun(X_c,bool),X_a))),combb(fun(X_c,X_a),fun(fun(X_c,bool),X_a),X_b),big_co1399186613setsum(X_c,X_a)),hAPP(fun(X_c,X_a),fun(X_b,fun(X_c,X_a)),hAPP(fun(X_b,fun(fun(X_c,X_a),fun(X_c,X_a))),fun(fun(X_c,X_a),fun(X_b,fun(X_c,X_a))),combc(X_b,fun(X_c,X_a),fun(X_c,X_a)),hAPP(fun(X_b,fun(X_a,X_a)),fun(X_b,fun(fun(X_c,X_a),fun(X_c,X_a))),hAPP(fun(fun(X_a,X_a),fun(fun(X_c,X_a),fun(X_c,X_a))),fun(fun(X_b,fun(X_a,X_a)),fun(X_b,fun(fun(X_c,X_a),fun(X_c,X_a)))),combb(fun(X_a,X_a),fun(fun(X_c,X_a),fun(X_c,X_a)),X_b),combb(X_a,X_a,X_c)),hAPP(fun(X_b,X_a),fun(X_b,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_b),times_times(X_a)),F))),G))),B_1)),A_2)))) # label(fact_4240_setsum__product) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 166 (all X_a (real_normed_algebra(X_a) -> (all X exists K_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2)) & (all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_1)),K_2)))))))) # label(fact_2107_mult__right_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 167 (all X_a (metric_space(X_a) -> lim(X_a) = ti(fun(fun(nat,X_a),X_a),lim(X_a)))) # label(tsy_c_SEQ_Olim_res) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 168 (all A_3 all B_2 (hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),posDivAlg_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_3),B_2))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),B_2)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),zero_zero(int))) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_3),B_2) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A_3)) & (-(hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),B_2)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),zero_zero(int)))) -> hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,B_2),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_3),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B_2))) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_3),B_2)))) # label(fact_1660_posDivAlg_Opsimps) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 169 (all U hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(nat,fun(nat,bool),ord_atMost(nat),U)) = hAPP(nat,nat,suc,U)) # label(fact_4966_card__atMost) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 170 (all X_a (archim1727834104eiling(X_a) -> one_one(int) = hAPP(X_a,int,archimedean_ceiling(X_a),one_one(X_a)))) # label(fact_3307_ceiling__one) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 171 (all X_a all X_b (comm_monoid_add(X_b) -> (all F all B_1 all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),A_2))),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),B_1)))) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),B_1))),F)),zero_zero(X_b))),A_2))))) # label(fact_4215_setsum__restrict__set_H) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 172 (all X_a (ord(X_a) -> (all C_1 all A_1 all B (ti(X_a,B) = ti(X_a,A_1) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),C_1))))))) # label(fact_508_ord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 173 (all X_a (real_normed_algebra(X_a) -> (all Ya all X_2 all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),X_2),A_3)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),X_2)),Ya)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),Ya))))))) # label(fact_4110_mult__left_Osums) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 174 (all X_a (boolean_algebra(X_a) -> (all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Xa)),hAPP(X_a,X_a,uminus_uminus(X_a),Ya))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ya),Xa)))))) # label(fact_2984_compl__le__compl__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 175 (all P_4 (hBOOL(hAPP(int,bool,zprime,P_4)) -> hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),P_4) = hAPP(int,fun(int,bool),d22set,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) # label(fact_1212_d22set__eq__wset) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 176 (all X_a (semiring_1(X_a) -> hBOOL(hAPP(X_a,bool,iszero(X_a),zero_zero(X_a))))) # label(fact_2145_iszero__0) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 177 (all K_4 all L_4 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit0,K_4)),hAPP(int,int,bit0,L_4)) = hAPP(int,int,bit0,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),L_4))) # label(fact_80_add__Bit0__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 178 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y)) -> (exists X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),hAPP(real,real,tan,X_1))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1)))))) # label(fact_2460_lemma__tan__total) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 179 (all X_a (semiring_div(X_a) -> (all A_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),zero_zero(X_a)),A_1) = zero_zero(X_a)))) # label(fact_1073_div__0) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 180 (all A_1 all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,ln,X)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),A_1)))))) # label(fact_3345_ln__powr__bound) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 181 (all X_a (divisi14063676e_zero(X_a) -> (all A_3 all B_2 (hAPP(X_a,X_a,inverse_inverse(X_a),B_2) = hAPP(X_a,X_a,inverse_inverse(X_a),A_3) <-> ti(X_a,B_2) = ti(X_a,A_3))))) # label(fact_2569_inverse__eq__iff__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 182 (all A_3 all B_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),B_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),A_3)) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),d22set,A_3)))))) # label(fact_1225_d22set__mem) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 183 (all X_a all X_b (ord(X_b) -> (all F all G ((all X_1 hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,G,X_1)))) <-> hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G)))))) # label(fact_514_le__fun__def) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 184 (all K all Ma all Na (hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Na) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Ma) <-> K = zero_zero(nat) | Na = Ma)) # label(fact_605_mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 185 (all Xa all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,sqrt,Xa)),hAPP(real,real,sqrt,Ya))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya)))) # label(fact_2125_real__sqrt__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 186 (all X all Y hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y)),X))) # label(fact_4374_gcd__dvd1__int) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 187 (all X_a (linorder(X_a) -> (all X all 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_eq(X_a),Y),X)))))) # label(fact_566_leI) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 188 (all X_a (comm_ring_1(X_a) -> (all X (-hBOOL(hAPP(nat,bool,even_odd_even(nat),X)) -> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),X) = hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)))))) # label(fact_2492_minus__one__odd__power) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 189 (all X_a (real_normed_field(X_a) -> (all C all F all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),A_3)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),F)),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),C))))))) # label(fact_4112_sums__divide) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 190 (all M all N hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,suc,M)),N)) # label(fact_1337_add__Suc) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 191 (all A_1 all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),hAPP(real,int,archim2085082626_floor(real),X))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),A_1)),X)))) # label(fact_3388_real__le__floor) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 192 (all C_1 all D_3 all A_1 all B all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,C_1),D_3),M)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),C_1)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),D_3)),M))))) # label(fact_899_zcong__zadd) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 193 (all X_a (real_normed_algebra(X_a) -> (all Ya all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2)) -> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),X_2)),Ya))))))) # label(fact_4029_mult__left_Osummable) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 194 (all V hAPP(int,real,number_number_of(real),V) = hAPP(int,real,real_1(int),hAPP(int,int,number_number_of(int),V))) # label(fact_3346_real__number__of) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 195 (all K_4 all M all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_4),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_4),N))) # label(fact_813_Nat_Odiff__cancel) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 196 (all Ma (one_one(nat) = Ma <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),one_one(nat))))) # label(fact_828_nat__dvd__1__iff__1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 197 (all X_a (topological_space(X_a) -> (all L all X_2 all Xa (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),Xa),sequentially)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),L)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),X_2),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),L))),Xa),sequentially))))))) # label(fact_4467_LIMSEQ__linear) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 198 (all X_a all F1 all F2 ti(X_a,F1) = hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(X_a,fun(fun(nat,X_a),fun(nat,X_a)),nat_case(X_a),F1),F2),zero_zero(nat))) # label(fact_4139_nat__case__0) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 199 (all X_a (linordered_semidom(X_a) -> (all A_1 all N all B (hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),hAPP(nat,nat,suc,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)) -> ti(X_a,A_1) = ti(X_a,B))))))) # label(fact_1424_power__inject__base) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 200 (all B all A_1 all X (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),X)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),A_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),X)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B))))) # label(fact_3270_gcd__semilattice__nat_Oless__infI1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 201 (all X_b all X_a all Q_1 all P_1 ((all X_1 all Y_1 (hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,X_1),Y_1)) -> hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),Q_1,X_1),Y_1)))) -> hBOOL(hAPP(fun(X_a,fun(X_b,bool)),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),bool),ord_less_eq(fun(X_a,fun(X_b,bool))),P_1),Q_1)))) # label(fact_4369_predicate2I) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 202 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(real,real,sqrt,X))))) # label(fact_2147_real__sqrt__ge__one) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 203 (all X_a (comm_semiring(X_a) -> (all A_1 all B all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1))))) # label(fact_380_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 204 (all Z_1 hAPP(complex,real,re,hAPP(int,complex,ring_1_of_int(complex),Z_1)) = hAPP(int,real,ring_1_of_int(real),Z_1)) # label(fact_3602_complex__Re__of__int) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 205 (all X_a (linorder(X_a) -> (all Xa all Ya (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya)) -> (ti(X_a,Xa) = ti(X_a,Ya) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya))))))) # label(fact_564_linorder__antisym__conv1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 206 (all A_3 all P_4 (hBOOL(hAPP(int,bool,zprime,P_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),zero_zero(int)),P_4)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_4),A_3)) -> (all X_1 (hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),X_1),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,A_3),P_4))) -> hAPP(fun(int,bool),nat,finite_card(int),X_1) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))))) # label(fact_1683_SetS__elems__card) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 207 (all M all N all K_4 (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,N),K_4) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),N))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),M))))) # label(fact_3487_zrelprime__zdvd__zmult) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 208 (all W all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W),Z_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W),one_one(int))),Z_1)))) # label(fact_287_zless__imp__add1__zle) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 209 (all U all Ma all Na all I all J (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J)) -> (hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I),U)),Ma) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J),U)),Na) <-> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),I)),U)),Na) = Ma))) # label(fact_864_nat__eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 210 (all X exists N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(nat,real,real_1(nat),N_1)))) # label(fact_2724_reals__Archimedean2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 211 (all B all A_1 hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),A_1)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),A_1)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),A_1)) # label(fact_1812_real__average__minus__second) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 212 (all N hAPP(complex,real,im,hAPP(nat,complex,semiring_1_of_nat(complex),N)) = zero_zero(real)) # label(fact_3007_complex__Im__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 213 (all X_a (comm_semiring_1(X_a) -> (all X all Y all Z_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Y),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Z_1))))) # label(fact_382_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 214 (all X_a all A_2 all B_1 (ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)))) # label(fact_2208_equalityD1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 215 (all F all L all C (hBOOL(hAPP(filter(real),bool,hAPP(real,fun(filter(real),bool),hAPP(fun(real,real),fun(real,fun(filter(real),bool)),tendsto(real,real),F),L),hAPP(real,filter(real),at(real),C))) -> (L != zero_zero(real) -> (exists R_1 ((all X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),C),X_1))),R_1)) & C != X_1 -> hAPP(real,real,F,X_1) != zero_zero(real))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1))))))) # label(fact_4654_LIM__fun__not__zero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 216 (all X_a (linordered_idom(X_a) -> (all A_3 all Na (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_3),Na))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3)) | hBOOL(hAPP(nat,bool,even_odd_even(nat),Na)))))) # label(fact_2488_zero__le__power__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 217 (all X_a (real_algebra_1(X_a) & real_normed_vector(X_a) -> (all X_2 (hBOOL(hAPP(fun(nat,real),bool,summable(real),X_2)) -> hAPP(real,X_a,of_real(X_a),hAPP(fun(nat,real),real,suminf(real),X_2)) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,real),fun(nat,X_a),hAPP(fun(real,X_a),fun(fun(nat,real),fun(nat,X_a)),combb(real,X_a,nat),of_real(X_a)),X_2)))))) # label(fact_4047_of__real_Osuminf) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 218 (all X_a (linordered_idom(X_a) -> (all Xa all Na (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Xa)) & -hBOOL(hAPP(nat,bool,even_odd_even(nat),Na)) | hBOOL(hAPP(nat,bool,even_odd_even(nat),Na)) <-> 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,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),Na))))))) # label(fact_2487_zero__le__power__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 219 (all M one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,one_one(int)),M)) # label(fact_3459_zgcd__1__left) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 220 (all X_a (comm_monoid_add(X_a) & t2_space(X_a) -> (all F all Xa (hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),Xa)) <-> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) & hAPP(fun(nat,X_a),X_a,suminf(X_a),F) = ti(X_a,Xa))))) # label(fact_4120_sums__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 221 (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),hAPP(nat,nat,suc,Na))))) # label(fact_1391_less__Suc__eq__le) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 222 (all Xa (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd)) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xa),one_one(int))),zEven)))) # label(fact_1727_odd__minus__one__even) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 223 (all X_a (real_normed_vector(X_a) -> (all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2)) -> (exists K_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2)) & (all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,X_2,N_1))),K_2))))))))) # label(fact_3522_BseqD) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 224 (all A_1 all C_1 all B all D_3 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_1),C_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),D_3)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),C_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),D_3),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))))) # label(fact_2174_real__sqrt__sum__squares__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 225 (all X_a (real_algebra_1(X_a) -> (all Xa (hAPP(real,X_a,of_real(X_a),Xa) = zero_zero(X_a) <-> Xa = zero_zero(real))))) # label(fact_2794_of__real__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 226 (all X_a (comm_ring_1(X_a) -> (all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(X_a,X_a,uminus_uminus(X_a),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y)))) # label(fact_1969_comm__ring__1__class_Onormalizing__ring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 227 (all X all I_1 hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,real_1(int),I_1)),pi))) = hAPP(real,real,tan,X)) # label(fact_3424_tan__periodic__int) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 228 (all X_a (real_vector(X_a) -> (all X all Y all A_1 (zero_zero(real) != A_1 -> (hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),Y) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X) -> ti(X_a,X) = ti(X_a,Y)))))) # label(fact_4729_scaleR__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 229 (all U_1 all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),U_1),U_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),X)))) # label(fact_1988_real__minus__mult__self__le) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 230 (all X_c all X_a all X_b all C all P_4 all Q_2 (ti(product_prod(X_a,X_b),Q_2) = ti(product_prod(X_a,X_b),P_4) -> hAPP(product_prod(X_a,X_b),X_c,hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),C),P_4) = hAPP(product_prod(X_a,X_b),X_c,hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),C),Q_2))) # label(fact_4020_split__weak__cong) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 231 (all Xa (hAPP(real,real,cos,Xa) != zero_zero(real) -> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),tan),Xa)))) # label(fact_4906_isCont__tan) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 232 (all X_a (semiring_1(X_a) -> (all M hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,suc,M)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),M))))) # label(fact_1437_of__nat__Suc) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 233 (all A_1 all Q_3 all R_3 (hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),A_1),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_3),R_3))) -> (ti(int,A_1) != zero_zero(int) -> ti(int,R_3) = zero_zero(int)))) # label(fact_1614_self__remainder) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 234 (all M all R_3 all S_3 all T_5 all N (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N)) -> (hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),xzgcd,M),N) = hAPP(product_prod(int,int),product_prod(int,product_prod(int,int)),hAPP(int,fun(product_prod(int,int),product_prod(int,product_prod(int,int))),product_Pair(int,product_prod(int,int)),R_3),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),S_3),T_5)) -> hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),S_3),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),T_5),N)) = ti(int,R_3)))) # label(fact_3107_xzgcd__linear) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 235 (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_171_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 236 (all K all L (hAPP(int,int,bit0,K) = hAPP(int,int,bit0,L) <-> ti(int,K) = ti(int,L))) # label(fact_43_rel__simps_I48_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 237 (all Y all X all A_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1)) -> (one_one(real) != A_1 -> (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)) -> hAPP(real,real,hAPP(real,fun(real,real),log,A_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),log,A_1),X)),hAPP(real,real,hAPP(real,fun(real,real),log,A_1),Y))))))) # label(fact_2043_log__divide) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 238 (all X_b all X_a (topological_space(X_a) & topological_space(X_b) -> (all L all F all G all A_3 ((all X_1 (ti(X_a,X_1) != ti(X_a,A_3) -> hAPP(X_a,X_b,G,X_1) = hAPP(X_a,X_b,F,X_1))) -> (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) <-> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3)))))))) # label(fact_4791_LIM__equal) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 239 (all X_b all X_c all X_a all P_1 all Z hAPP(X_c,X_a,hAPP(X_b,fun(X_c,X_a),P_1,hAPP(product_prod(X_b,X_c),X_b,product_fst(X_b,X_c),Z)),hAPP(product_prod(X_b,X_c),X_c,product_snd(X_b,X_c),Z)) = hAPP(product_prod(X_b,X_c),X_a,hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),product_prod_case(X_b,X_c,X_a),P_1),Z)) # label(fact_4043_split__beta) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 240 (all A_1 all B hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)) # label(fact_3766_minus__rat__cancel) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 241 (all X all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),hAPP(int,int,number_number_of(int),min)),P_5)) -> -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),one_one(int)),P_5))))) # label(fact_792_zcong__neg__1__impl__ne__1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 242 (all X_a (semiring_div(X_a) -> (all A_1 all B ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B)),B))))) # label(fact_1410_semiring__div__class_Omod__div__equality_H) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 243 (all X_a (number_ring(X_a) -> hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) = hAPP(int,X_a,number_number_of(X_a),min))) # label(fact_2016_arith__simps_I31_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 244 (all X_a (order(X_a) -> (all X_2 ((all M_2 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_2),N_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,X_2,N_1)),hAPP(nat,X_a,X_2,M_2))))) -> hBOOL(hAPP(fun(nat,X_a),bool,monoseq(X_a),X_2)))))) # label(fact_4157_monoI2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 245 (all X_b all X_a (ordere142940540dd_abs(X_a) -> (all F all A_2 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),abs_abs(X_a)),F)),A_2)))))) # label(fact_4193_setsum__abs__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 246 (all X_a (field_char_0(X_a) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),zero_zero(X_a)),field_char_0_Rats(X_a))))) # label(fact_3707_Rats__0) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 247 (all X all Y all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Y),Z_1)),Z_1))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),Z_1)),Y))))) # label(fact_1124_div__prop2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 248 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),B)))))) # label(fact_1599_negDivAlg__correct) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 249 (all K hAPP(nat,fun(nat,bool),ord_lessThan(nat),hAPP(nat,nat,suc,K)) = hAPP(nat,fun(nat,bool),ord_atMost(nat),K)) # label(fact_4964_lessThan__Suc__atMost) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 250 (all X all Y hAPP(int,int,hAPP(int,fun(int,int),div_div(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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),X),Y))) # label(fact_1162_Divides_Otransfer__int__nat__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 251 (all R_3 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_3),zero_zero(real))) -> (exists N_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),N_1)),R_3)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_3),hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N_1),one_one(int))))))))) # label(fact_3448_reals__Archimedean__6c__int) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 252 (all K_4 all I_1 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_2),K_4)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),K_4))))) # label(fact_354_le__trans) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 253 (all X_a (field_inverse_zero(X_a) -> (all X all Y all Z_1 all W hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Z_1),W)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Z_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),W))))) # label(fact_1828_times__divide__times__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 254 (all X hAPP(real,real,exp(real),X) = hAPP(real,real,abs_abs(real),hAPP(real,real,exp(real),X))) # label(fact_1758_abs__exp__cancel) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 255 (all X_a (order(X_a) -> (all A_1 all B (ti(X_a,A_1) != ti(X_a,B) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))))))) # label(fact_563_order__neq__le__trans) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 256 (all X_b all X_a (topological_space(X_a) & real_normed_vector(X_b) -> (all G all M_1 all F all L_2 all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),M_1),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),plus_plus(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),L_2),M_1)),hAPP(X_a,filter(X_a),at(X_a),A_3)))))))) # label(fact_4575_LIM__add) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 257 (all X all A_1 hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)) = hAPP(real,real,hAPP(real,fun(real,real),powr,X),hAPP(real,real,uminus_uminus(real),A_1))) # label(fact_3384_powr__minus) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 258 (all K_4 all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),M)))) # label(fact_3267_dvd__gcd__D1__nat) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 259 (all Xa all F ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,F,N_1)))) -> ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N_1))),Xa))) -> hBOOL(hAPP(fun(nat,real),bool,summable(real),F))))) # label(fact_4333_pos__summable) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 260 (all A_3 all B_2 (zero_zero(real) = B_2 & zero_zero(real) = A_3 <-> hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_3),B_2) = zero_zero(complex))) # label(fact_2757_Complex__eq__0) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 261 (all N all P_5 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_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),N))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),N)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),M)))))) # label(fact_922_zprime__zdvd__zmult) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 262 (all X_b all X_a (real_normed_vector(X_a) & topological_space(X_b) -> (all F all L_2 all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_a),fun(X_a,X_b),hAPP(fun(X_a,X_b),fun(fun(X_a,X_a),fun(X_a,X_b)),combb(X_a,X_b,X_a),F),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3))),L_2),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a)))))))) # label(fact_4605_LIM__offset__zero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 263 (all X_a (archim1804426504_field(X_a) -> (all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X)) -> (exists N_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1)),X)))))))) # label(fact_2667_ex__less__of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 264 (all X_a (ordered_ab_group_add(X_a) -> (all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),hAPP(X_a,X_a,uminus_uminus(X_a),B_2))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),hAPP(X_a,X_a,uminus_uminus(X_a),A_3))))))) # label(fact_1930_less__minus__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 265 (all X_a (linord20386208strict(X_a) -> (all A_1 all C_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(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_1),B))))))) # label(fact_478_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 266 (all X_a (semiring_0(X_a) & dvd(X_a) -> (all P_1 all L ((exists X_1 (hBOOL(hAPP(X_a,bool,P_1,X_1)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),L),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),zero_zero(X_a)))))) <-> (exists X_1 hBOOL(hAPP(X_a,bool,P_1,hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),L),X_1)))))))) # label(fact_771_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 267 (all M zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),M)) # label(fact_805_diff__self__eq__0) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 268 (all X_a all X_b (real_normed_vector(X_b) -> (all X_2 ((exists N_2 all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_a,X_b,X_2,N_1))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_2))))) <-> (exists K_2 ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_a,X_b,X_2,N_1))),K_2))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2)))))))) # label(fact_2418_lemma__NBseq__def2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 269 (all A_3 all A_2 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Ma)) -> (hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),Ma)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),A_2)) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),hAPP(fun(int,bool),fun(int,fun(int,int)),rRset2norRR,A_2),Ma),A_3)),hAPP(int,fun(int,bool),norRRset,Ma))))))) # label(fact_3648_RRset2norRR__correct2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 270 (all X_a (comm_monoid_add(X_a) & comm_monoid_mult(X_a) -> (all F all A_2 (hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2)) -> hAPP(fun(int,bool),X_a,hAPP(fun(int,X_a),fun(fun(int,bool),X_a),big_co1399186613setsum(int,X_a),F),A_2) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,int),fun(nat,X_a),hAPP(fun(int,X_a),fun(fun(nat,int),fun(nat,X_a)),combb(int,X_a,nat),F),semiring_1_of_nat(int))),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2)))))) # label(fact_4293_transfer__int__nat__sum__prod_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 271 (all P_1 all I all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I),K)) -> (hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K),one_one(int)))) -> ((all I_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_2),K)) -> (hBOOL(hAPP(int,bool,P_1,I_2)) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),I_2),one_one(int))))))) -> hBOOL(hAPP(int,bool,P_1,I)))))) # label(fact_1201_int__less__induct) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 272 (all Xa all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya)) | Xa = Ya <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya)))) # label(fact_352_less__eq__real__def) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 273 (all X_a (real_normed_algebra(X_a) -> (all C all F (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) -> hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),times_times(X_a),C)),F)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),hAPP(fun(nat,X_a),X_a,suminf(X_a),F)))))) # label(fact_4016_suminf__mult) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 274 (all B_1 all P_2 all P_1 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D)) -> ((exists Z_2 all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),Z_2)) -> (hBOOL(hAPP(int,bool,P_2,X_1)) <-> hBOOL(hAPP(int,bool,P_1,X_1))))) -> ((all X_1 ((all Xa_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D))) -> (all Xb (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1)) -> hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa_1) != ti(int,X_1))))) -> (hBOOL(hAPP(int,bool,P_1,X_1)) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),D)))))) -> ((all X_1 all K_1 (hBOOL(hAPP(int,bool,P_2,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_1),D)))) <-> hBOOL(hAPP(int,bool,P_2,X_1)))) -> ((exists X_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D))) & hBOOL(hAPP(int,bool,P_2,X_1)))) | (exists X_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D))) & (exists Xa_1 (hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xa_1),X_1))) & hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),B_1)))))) <-> (exists X1 hBOOL(hAPP(int,bool,P_1,X1))))))))) # label(fact_4957_cpmi) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 275 (all X all Y hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))),one_one(real)))) # label(fact_2184_cos__x__y__le__one) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 276 (all T all A ti(T,A) = ti(T,ti(T,A))) # label(help_ti_idem) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 277 (all A_3 all A_2 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Ma)) -> (hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),Ma)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),A_2)) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),hAPP(fun(int,bool),fun(int,fun(int,int)),rRset2norRR,A_2),Ma),A_3)),hAPP(int,fun(int,bool),norRRset,Ma))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),hAPP(int,int,hAPP(int,fun(int,int),hAPP(fun(int,bool),fun(int,fun(int,int)),rRset2norRR,A_2),Ma),A_3)),Ma)))))) # label(fact_3647_RRset2norRR__correct) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 278 (all Xa all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Ya),hAPP(real,real,uminus_uminus(real),Xa))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa),Ya)),zero_zero(real))))) # label(fact_2023_real__add__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 279 (all Xa hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),sin)),sin)),Xa),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,Xa)),hAPP(real,real,sin,Xa))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,Xa)),hAPP(real,real,sin,Xa)))))) # label(fact_3861_DERIV__sin__sin__mult) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 280 (all X_a all G all A_3 all A_2 (hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),A_3),A_2)) -> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,complex),fun(X_a,real),hAPP(fun(complex,real),fun(fun(X_a,complex),fun(X_a,real)),combb(complex,real,X_a),re),G)),hAPP(complex,real,re,A_3)),A_2)))) # label(fact_4534_Re_Otendsto) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 281 (all X_a (order(X_a) -> (all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (ti(X_a,A_1) != ti(X_a,B) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))))))) # label(fact_557_order__le__neq__trans) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 282 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real))) -> (exists X_1 ((all Y_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y_1)) & Y = hAPP(real,real,cos,Y_1) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y_1),pi)) -> Y_1 = X_1)) & Y = hAPP(real,real,cos,X_1) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),pi)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X_1))))))) # label(fact_2348_cos__total) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 283 (all X_a all X_b all F all A_2 hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),bool),nat_tr1645093318rphism(X_a,X_b),F),A_2))) # label(fact_1335_transfer__morphism__def) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 284 (all Na hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),Na)))))) # label(fact_3895_bdd__int__set__l__l__finite) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 285 (all Xa all Ya (Xa = Ya <-> hAPP(real,real,sqrt,Xa) = hAPP(real,real,sqrt,Ya))) # label(fact_2123_real__sqrt__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 286 (all Q_3 all R_3 hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),Q_3),R_3) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),Q_3),hAPP(rat,rat,inverse_inverse(rat),R_3))) # label(fact_3725_divide__rat__def) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 287 (all X_a (topological_space(X_a) & comm_monoid_add(X_a) -> ti(fun(fun(nat,X_a),bool),summable(X_a)) = summable(X_a))) # label(tsy_c_Series_Osummable_res) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 288 (all A_1 hAPP(int,rat,hAPP(int,fun(int,rat),fract,zero_zero(int)),one_one(int)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),zero_zero(int))) # label(fact_3759_eq__rat_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 289 (all X_a (real_normed_vector(X_a) -> (all X_2 all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),X_2)),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially)))))) # label(fact_4470_LIMSEQ__minus__cancel) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 290 (all X_a (order(X_a) -> (all C_1 all A_1 all B (ti(X_a,A_1) = ti(X_a,B) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),A_1))))))) # label(fact_521_xt1_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 291 (all X_a (linordered_semidom(X_a) -> (all A_1 all N all N_4 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N_4)) -> (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_4)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N))))))))) # label(fact_113_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 292 (all Xa all Ya 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_eq(real),Xa),Ya)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z),Xa)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z),Ya)))))) # label(fact_635_real__mult__le__cancel__iff2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 293 (all X (hAPP(real,real,cos,X) = one_one(real) -> hAPP(real,real,sin,X) = zero_zero(real))) # label(fact_2258_cos__one__sin__zero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 294 (all N all B all A_1 all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),P_5),N)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),P_5),N)),B)))))) # label(fact_783_zprime__power__zdvd__cancel__left) # label(axiom) # label(non_clause). [assumption]. 7.53/7.59 295 (all Va (hAPP(int,nat,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_273_eq__number__of__0) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 296 (all B_2 hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,B_2) = hAPP(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_prod_case(int,int,product_prod(int,int)),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int)))),combs(int,fun(int,product_prod(int,int)),fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),hAPP(fun(fun(int,fun(product_prod(int,int),product_prod(int,int))),fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int))))),combb(fun(int,fun(product_prod(int,int),product_prod(int,int))),fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int))),int),combs(int,product_prod(int,int),product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),hAPP(fun(fun(int,product_prod(int,int)),fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int))))),combb(fun(int,product_prod(int,int)),fun(int,fun(product_prod(int,int),product_prod(int,int))),int),hAPP(fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),fun(fun(int,product_prod(int,int)),fun(int,fun(product_prod(int,int),product_prod(int,int)))),combs(int,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),hAPP(fun(int,bool),fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),hAPP(fun(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),fun(fun(int,bool),fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))),combb(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),int),if(product_prod(int,int))),hAPP(fun(int,int),fun(int,bool),hAPP(fun(int,bool),fun(fun(int,int),fun(int,bool)),combb(int,bool,int),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int))),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),minus_minus(int)),B_2))))),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(fun(int,int),fun(int,product_prod(int,int)))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combc(int,fun(int,int),fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(fun(int,int),fun(int,product_prod(int,int)))),hAPP(fun(fun(int,product_prod(int,int)),fun(fun(int,int),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(fun(int,int),fun(int,product_prod(int,int))))),combb(fun(int,product_prod(int,int)),fun(fun(int,int),fun(int,product_prod(int,int))),int),combb(int,product_prod(int,int),int)),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int)),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),hAPP(fun(int,int),fun(int,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(int,int),fun(int,fun(int,int))),combb(int,fun(int,int),int),plus_plus(int)),hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(int))))),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),minus_minus(int)),B_2))))),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int)),hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) # label(fact_4062_adjust__def) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 297 (all X_a (ring_1(X_a) -> (all N hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)))))) # label(fact_2045_power__minus1__odd) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 298 (all I_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I_1)) -> hAPP(list(int),nat,size_size(list(int)),hAPP(int,list(int),quickc666637781d_zero,I_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,nat,nat_1,I_1))),one_one(nat)))) # label(fact_3228_length__around__zero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 299 (all N all K_4 all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),N)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),N)))))) # label(fact_4403_gcd__greatest__int) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 300 (all Xa all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)),zero_zero(real))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),zero_zero(real)))))) # label(fact_2699_real__root__lt__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 301 (all B all Q_3 all R_3 all B_5 all Q_5 all R_4 (hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_3)),R_3) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_5),Q_5)),R_4) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_5),Q_5)),R_4))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_4),B_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_5),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_3),Q_5))))))))) # label(fact_584_zdiv__mono2__lemma) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 302 (all U all Ma all Na all J all I (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),I)) -> (hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),Ma) = Na <-> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I),U)),Ma) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J),U)),Na)))) # label(fact_867_nat__eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 303 (all X_a ti(fun(fun(X_a,bool),X_a),hilbert_Eps(X_a)) = hilbert_Eps(X_a)) # label(tsy_c_Hilbert__Choice_OEps_res) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 304 (all X all Y hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),X))),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),Y))))) # label(fact_4434_gcd__int__def) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 305 (all B all Bq all Br all A_1 all C_1 all Aq all Ar (hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,A_1),C_1),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Aq),Ar))) -> (hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,B),C_1),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Bq),Br))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C_1)) -> hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A_1),B)),C_1),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Aq),Bq)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ar),Br)),C_1))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ar),Br)),C_1)))))))) # label(fact_3122_divmod__nat__rel__add1__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 306 (all N hAPP(nat,nat,suc,zero_zero(nat)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(nat,nat,suc,zero_zero(nat))),N)) # label(fact_1404_power__Suc__0) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 307 (all X_b all X_a (real_normed_vector(X_b) & topological_space(X_a) -> (all G all F all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3)) -> (hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),G),A_3)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),plus_plus(X_b)),F)),G)),A_3))))))) # label(fact_4925_isCont__add) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 308 (all X_a (linord20386208strict(X_a) -> (all C_1 all D_3 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),D_3)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D_3)))))))))) # label(fact_481_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 309 (all X_a (linordered_idom(X_a) -> (all A_3 (zero_zero(X_a) = ti(X_a,A_3) <-> zero_zero(X_a) = hAPP(X_a,X_a,sgn_sgn(X_a),A_3))))) # label(fact_2261_sgn__0__0) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 310 (all X_a (linordered_field(X_a) -> (all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)))))))) # label(fact_1866_divide__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 311 (all Q_3 all B all R_3 all C_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_3),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Q_3),C_1))),R_3)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C_1))))))) # label(fact_1475_zmult2__lemma__aux4) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 312 (all X_a zero_frac(X_a) = hAPP(product_prod(int,int),X_a,abs_Frac(X_a),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)))) # label(fact_3818_Nitpick_Ozero__frac__def) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 313 (all Xa hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),sqrt),Xa))) # label(fact_4930_isCont__real__sqrt) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 314 (all X_a (linordered_field(X_a) -> (all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a)))),B)))))) # label(fact_1885_gt__half__sum) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 315 (all C (zero_zero(rat) = C <-> hBOOL(hAPP(fun(nat,rat),bool,vanishes,hAPP(rat,fun(nat,rat),combk(rat,nat),C))))) # label(fact_4877_vanishes__const) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 316 (all N (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N)) -> zero_zero(int) != hAPP(int,int,fact_fact(int),N))) # label(fact_3216_fact__nonzero__int) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 317 (all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> hAPP(real,real,hAPP(nat,fun(real,real),root,N),one_one(real)) = one_one(real))) # label(fact_2689_real__root__one) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 318 (all X_a all X_c all X_b all P all Q all R hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),P,R),hAPP(X_a,X_b,Q,R)) = hAPP(X_a,X_c,hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_a,fun(X_b,X_c)),fun(fun(X_a,X_b),fun(X_a,X_c)),combs(X_a,X_b,X_c),P),Q),R)) # label(help_COMBS_1_1_U) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 319 (all M all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,M)),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N))),N)) # label(fact_1525_mod__Suc__eq__Suc__mod) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 320 (all X_a (linordered_idom(X_a) -> (all N 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),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)))),zero_zero(X_a))))))) # label(fact_1512_odd__power__less__zero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 321 (all B all B_5 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),zero_zero(int)),B_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_5),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B_5))))))) # label(fact_1109_zdiv__mono2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 322 (all I_1 all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),I_1),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),I_1),N))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),I_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))))) # label(fact_882_power__dvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 323 (all N all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M)) -> M = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),N))) # label(fact_850_le__add__diff__inverse2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 324 (all X_a (semiring_1(X_a) -> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),one_one(X_a)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = one_one(X_a))) # label(fact_3_one__power2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 325 (all F (hBOOL(hAPP(fun(nat,real),bool,summable(real),F)) -> ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,F,N_1)))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(fun(nat,real),real,suminf(real),F)))))) # label(fact_4098_suminf__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 326 (all X_a (metric_space(X_a) -> (all X2_1 all Y all X1_1 all E_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Y),X1_1)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),E_1),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Y),X2_1)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),E_1),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X1_1),X2_1)),E_1))))))) # label(fact_3546_dist__triangle__half__r) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 327 (all X_a all B_1 all Xa all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa),A_2)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa),B_1))))) # label(fact_2205_set__rev__mp) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 328 (all X_a (linordered_idom(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_3)) <-> hAPP(X_a,X_a,sgn_sgn(X_a),A_3) = one_one(X_a))))) # label(fact_2304_sgn__1__pos) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 329 (all K_4 all L_4 (zero_zero(int) != hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),K_4),hAPP(int,int,uminus_uminus(int),L_4)) -> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),K_4),L_4) != zero_zero(int))) # label(fact_2081_zmod__zminus2__not__zero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 330 (all X_a (linord1117847801e_zero(X_a) -> (all Xa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Xa)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),one_one(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),Xa))))))) # label(fact_2630_one__le__inverse__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 331 (all X_a (real_normed_vector(X_a) -> (all K all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2)) -> hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),X_2),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K)))))))) # label(fact_3864_Bseq__ignore__initial__segment) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 332 (all X_a (field_char_0(X_a) -> (all B_2 all A_3 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),field_char_0_Rats(X_a))) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),field_char_0_Rats(X_a))) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_2)),field_char_0_Rats(X_a)))))))) # label(fact_3709_Rats__add) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 333 (all P_5 hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) # label(fact_1133_Euler_Oaux____2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 334 (all X_a all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2)) -> ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1)))) # label(fact_2190_equalityI) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 335 (all X_a all X_b all X_c all Q_1 all P_1 all Z (hBOOL(hAPP(X_a,bool,Q_1,hAPP(product_prod(X_b,X_c),X_a,hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),product_prod_case(X_b,X_c,X_a),P_1),Z))) -> -(all X_1 all Y_1 (ti(product_prod(X_b,X_c),Z) = hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),X_1),Y_1) -> -hBOOL(hAPP(X_a,bool,Q_1,hAPP(X_c,X_a,hAPP(X_b,fun(X_c,X_a),P_1,X_1),Y_1))))))) # label(fact_4795_splitE2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 336 (all X all Y all Xa_2 all Ya_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa_2),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Ya_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))))) # label(fact_2181_real__sqrt__sum__squares__mult__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 337 (all L_4 hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),pls),hAPP(int,int,bit1,L_4)) = hAPP(int,int,bit1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),min),L_4))) # label(fact_780_diff__bin__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 338 (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,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(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,Y) = ti(int,X)))))))) # label(fact_903_zcong__less__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 339 (all X hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),pi)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,cos,X))) # label(fact_2294_cos__periodic__pi) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 340 (all V ((-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V))) -> hAPP(int,int,number_number_of(int),V) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,number_number_of(nat),V))) & (hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V))) -> zero_zero(int) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,number_number_of(nat),V))))) # label(fact_1622_int__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 341 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,ln,X)),X)))) # label(fact_1744_ln__bound) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 342 (all A_3 all P_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4)) -> hBOOL(hAPP(fun(fun(int,bool),bool),bool,finite_finite_1(fun(int,bool)),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,A_3),P_4))))) # label(fact_2740_SetS__finite) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 343 (all X_a all X_b all X_c all X_d all X_e all Xa all P_1 ((all A_4 all B_4 all C_4 all D_2 all E hBOOL(hAPP(product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),bool,P_1,hAPP(product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e))),product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),hAPP(X_a,fun(product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e))),product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e))))),product_Pair(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),A_4),hAPP(product_prod(X_c,product_prod(X_d,X_e)),product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e))),hAPP(X_b,fun(product_prod(X_c,product_prod(X_d,X_e)),product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),product_Pair(X_b,product_prod(X_c,product_prod(X_d,X_e))),B_4),hAPP(product_prod(X_d,X_e),product_prod(X_c,product_prod(X_d,X_e)),hAPP(X_c,fun(product_prod(X_d,X_e),product_prod(X_c,product_prod(X_d,X_e))),product_Pair(X_c,product_prod(X_d,X_e)),C_4),hAPP(X_e,product_prod(X_d,X_e),hAPP(X_d,fun(X_e,product_prod(X_d,X_e)),product_Pair(X_d,X_e),D_2),E))))))) -> hBOOL(hAPP(product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),bool,P_1,Xa)))) # label(fact_3174_prod__induct5) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 344 (all X_a all F all A_2 ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2)) -> hBOOL(hAPP(int,bool,nat_is_nat,hAPP(X_a,int,F,X_1))))) -> hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),F),A_2) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),hAPP(fun(X_a,int),fun(X_a,nat),hAPP(fun(int,nat),fun(fun(X_a,int),fun(X_a,nat)),combb(int,nat,X_a),nat_1),F)),A_2)))) # label(fact_4887_transfer__int__nat__sum__prod2_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 345 (all X all Y hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),hAPP(real,real,inverse_inverse(real),Y)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),Y)) # label(fact_2605_divide__real__def) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 346 (all X_b all X_a (topological_space(X_b) & topological_space(X_a) -> (all K all A_3 hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(X_b,fun(X_a,X_b),combk(X_b,X_a),K)),A_3))))) # label(fact_4921_isCont__const) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 347 (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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Ma),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),K))))))) # label(fact_841_less__diff__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 348 (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),hAPP(int,int,bit1,pls)),Ya)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(int,X_a,number_number_of(X_a),Ya))))))) # label(fact_63_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 349 (all X_a (real_normed_field(X_a) -> (all A_1 all B (ti(X_a,B) != zero_zero(X_a) -> hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(X_a,real,norm_norm(X_a),A_1)),hAPP(X_a,real,norm_norm(X_a),B)) = hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)))))) # label(fact_2006_nonzero__norm__divide) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 350 (all X_a (ring_11004092258visors(X_a) -> (all A_3 (zero_zero(X_a) = ti(X_a,A_3) <-> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_3),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = zero_zero(X_a))))) # label(fact_5_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 351 (all Na all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma)) -> (ti(int,Na) = one_one(int) & ti(int,Ma) = one_one(int) <-> hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Ma),Na) = one_one(int)))) # label(fact_497_pos__zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 352 (all Xa (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)) -> -(all K_1 hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),K_1) != ti(int,Xa)))) # label(fact_1741_zEvenE) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 353 (all W all Z1 all Z2 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W),Z1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W),Z2)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z1),Z2))) # label(fact_407_zadd__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 354 (all X_a (ord(X_a) -> ord_max(X_a) = ti(fun(X_a,fun(X_a,X_a)),ord_max(X_a)))) # label(tsy_c_Orderings_Oord__class_Omax_res) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 355 (all N all M hAPP(code_code_numeral,product_prod(code_code_numeral,code_code_numeral),hAPP(code_code_numeral,fun(code_code_numeral,product_prod(code_code_numeral,code_code_numeral)),product_Pair(code_code_numeral,code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),N),M)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),N),M)) = hAPP(code_code_numeral,product_prod(code_code_numeral,code_code_numeral),hAPP(code_code_numeral,fun(code_code_numeral,product_prod(code_code_numeral,code_code_numeral)),code_d418564891umeral,N),M)) # label(fact_3106_div__mod__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 356 (all X_a (comm_semiring_1(X_a) -> (all M all A_1 all N all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),B)) -> (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M)),B))))))) # label(fact_748_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 357 (all X_a (linordered_idom(X_a) -> (all Xa all Na (-hBOOL(hAPP(nat,bool,even_odd_even(nat),Na)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),Na)),zero_zero(X_a))))))) # label(fact_2491_power__less__zero__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 358 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),ln),Xa),hAPP(real,real,inverse_inverse(real),Xa))))) # label(fact_2652_DERIV__ln) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 359 (all A_1 all B all X (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),X)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),X)))) # label(fact_3260_gcd__semilattice__nat_Ole__infI2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 360 (all X_a (number_ring(X_a) -> (all N hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,number_number_of(X_a),min)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)) = one_one(X_a)))) # label(fact_695_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 361 (all X all Y hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Y),hAPP(real,real,uminus_uminus(real),X))) = hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,uminus_uminus(real),Y)))) # label(fact_1994_abs__minus__add__cancel) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 362 (all N all A_1 all B (hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B) = one_one(int) -> one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),N)),B))) # label(fact_3475_zgcd__1__power__left__distrib) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 363 (all X_a (ring_1(X_a) -> (all A_3 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),ring_1_Ints(X_a))) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)),ring_1_Ints(X_a))))))) # label(fact_3832_Ints__minus) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 364 (all X_a (ordered_ab_group_add(X_a) -> (all A_3 all B_2 all C all D_1 (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),C),D_1) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_1))))))) # label(fact_737_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 365 (all X_a (number_ring(X_a) -> (all V all W hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),W)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),V),W))))) # label(fact_370_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 366 (all M all N all K_4 all L_4 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_4),L_4)) -> (hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),L_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_4),N) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))))) # label(fact_139_less__add__eq__less) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 367 (all K_4 all L_4 hAPP(product_prod(int,int),rat,frct,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,number_number_of(int),K_4)),hAPP(int,int,number_number_of(int),L_4))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),hAPP(int,rat,number_number_of(rat),K_4)),hAPP(int,rat,number_number_of(rat),L_4))) # label(fact_3729_Frct__code__post_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 368 (all K_4 hAPP(int,int,pred,hAPP(int,int,bit1,K_4)) = hAPP(int,int,bit0,K_4)) # label(fact_2926_pred__Bit1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 369 (all X_a (linorder(X_a) -> (all Y all X (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)))))) # label(fact_565_not__leE) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 370 (all X zero_zero(real) != hAPP(real,real,cos,hAPP(real,real,arctan,X))) # label(fact_2277_cos__arctan__not__zero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 371 (all P_5 hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(product_prod(int,int),int,product_fst(int,int),P_5)),hAPP(product_prod(int,int),int,product_snd(int,int),P_5)) = hAPP(product_prod(int,int),rat,frct,P_5)) # label(fact_3800_Frct__def) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 372 (all Xa all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma)) -> (hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,Xa),Ma) = one_one(int) -> hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),noXRRset,Ma),Xa)),Ma))))) # label(fact_3643_noX__is__RRset) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 373 (all X all A_1 hAPP(real,int,archimedean_ceiling(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),hAPP(int,real,real_1(int),A_1))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(real,int,archimedean_ceiling(real),X)),A_1)) # label(fact_3399_ceiling__subtract) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 374 (all X_a all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1)) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)))) # label(fact_2199_psubset__imp__subset) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 375 (all C_1 all N all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_1),B)) -> (hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),C_1),N) -> (exists R_1 exists S_1 (hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),R_1),N) = A_1 & B = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),S_1),N)))))) # label(fact_3450_coprime__pow) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 376 (all A_1 -(-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),one_one(nat)),A_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),one_one(nat))))) # label(fact_1256_gcd__lcm__complete__lattice__nat_Onot__less__bot) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 377 (all Xa hAPP(real,real,sin,Xa) = hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),sin_coeff)),hAPP(real,fun(nat,real),power_power(real),Xa)))) # label(fact_3996_sin__def) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 378 (all X_b all X_a (real_normed_vector(X_b) & topological_space(X_a) -> (all Ra all G all L all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),Ra)),G)),hAPP(X_b,X_b,hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),Ra),L)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4757_scaleR__right_OLIM) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 379 (all X_b all X_a (real_normed_field(X_a) -> (all G all A_2 all Ya hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),G),A_2)),Ya) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(X_a,fun(X_b,X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(X_a,fun(X_b,X_a)),combc(X_b,X_a,X_a),hAPP(fun(X_b,X_a),fun(X_b,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_b),inverse_divide(X_a)),G)),Ya)),A_2)))) # label(fact_4208_divide_Osetsum) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 380 (all X_a all X_b (ordere216010020id_add(X_b) -> (all F all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) -> ((all B_4 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_4),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),B_1),A_2))) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(X_a,X_b,F,B_4))))) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),B_1))))))))) # label(fact_4320_setsum__mono2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 381 (all Z_1 hAPP(complex,real,im,hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),Z_1),hAPP(complex,complex,cnj,Z_1))) = zero_zero(real)) # label(fact_3009_complex__In__mult__cnj__zero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 382 (exists K_2 all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,cnj,X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_1)),K_2)))) # label(fact_2993_cnj_Obounded) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 383 (all B_1 all A_2 (hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2)) -> (hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,B_1)) -> (hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2) = hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),B_1) <-> B_1 = A_2)))) # label(fact_3687_transfer__int__nat__set__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 384 (all X_a (linorder(X_a) -> (all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)))) # label(fact_4834_min__max_Oinf__left__idem) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 385 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,K)),hAPP(int,int,bit0,L))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)))) # label(fact_34_rel__simps_I14_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 386 (all Na all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Na),Ma)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),Na)),one_one(real))),hAPP(int,real,real_1(int),Ma))))) # label(fact_3416_int__less__real__le) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 387 (all A_3 all P_4 (hBOOL(hAPP(int,bool,zprime,P_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),zero_zero(int)),P_4)) -> hAPP(int,int,zfact,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))) = hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),hAPP(fun(fun(int,bool),bool),fun(int,bool),complete_Sup_Sup(fun(int,bool)),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,A_3),P_4))))))) # label(fact_3962_Union__SetS__setprod__prop2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 388 (all A_1 all B ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A_1) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),B)) & (-(hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int)))) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),B) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B)))))) # label(fact_1647_posDivAlg_Osimps) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 389 (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),Na),hAPP(nat,nat,suc,Ma))) <-> Na = Ma))) # label(fact_1387_le__less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 390 (all X_a (real_normed_vector(X_a) -> (all K all Sb all F (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) -> (hAPP(fun(nat,X_a),X_a,suminf(X_a),F) = ti(X_a,Sb) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Sb),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K))) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K)))))))) # label(fact_4196_suminf__minus__initial__segment) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 391 (all X_a (semiring_div(X_a) -> (all A_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),one_one(X_a)) = zero_zero(X_a)))) # label(fact_1353_mod__by__1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 392 (all X_a (ab_semigroup_add(X_a) -> ti(fun(X_a,fun(X_a,X_a)),plus_plus(X_a)) = plus_plus(X_a))) # label(tsy_c_Groups_Oplus__class_Oplus_2_res) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 393 (all X_a (number_ring(X_a) -> (all Xa all Ya (hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xa),hAPP(int,int,uminus_uminus(int),Ya))))) <-> hAPP(int,X_a,number_number_of(X_a),Xa) = hAPP(int,X_a,number_number_of(X_a),Ya))))) # label(fact_2162_eq__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 394 (all Y (Y != zero_zero(code_code_numeral) -> -(all Code_numeral_1 Y != hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,Code_numeral_1)))) # label(fact_2996_code__numeral_Oexhaust) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 395 (all X_a (real_normed_vector(X_a) -> (all B zero_zero(X_a) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),zero_zero(real)),B)))) # label(fact_4738_scaleR_Ozero__left) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 396 (all X_a (linordered_idom(X_a) -> (all A_1 hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(X_a,X_a,abs_abs(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) # label(fact_1051_abs__power2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 397 (all X_a (ordere223160158up_add(X_a) -> (all C_1 all D_3 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),D_3)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D_3)))))))) # label(fact_234_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 398 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) -> (exists K_1 N = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K_1))))) # label(fact_1524_less__imp__Suc__add) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 399 (all X_b all X_a (topological_space(X_a) & real_normed_vector(X_b) -> (all C all F all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),C)),F)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4760_scaleR_OLIM__right__zero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 400 (all X_a (linordered_idom(X_a) -> (all Xa all Ya (ti(X_a,Xa) = zero_zero(X_a) & zero_zero(X_a) = ti(X_a,Ya) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Ya),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),zero_zero(X_a))))))) # label(fact_333_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 401 (all X_a (number_ring(X_a) -> (all V all W hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),hAPP(int,int,uminus_uminus(int),W))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),W))))) # label(fact_2082_diff__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 402 (all Y_2 all Ya all X_2 all Xa (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),X_2),Xa),sequentially)) -> (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),Y_2),Ya),sequentially)) -> ((exists N_2 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,X_2,N_1)),hAPP(nat,real,Y_2,N_1))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya)))))) # label(fact_4519_LIMSEQ__le) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 403 (all X hAPP(real,real,cos,X) = hAPP(real,real,cos,hAPP(real,real,uminus_uminus(real),X))) # label(fact_2255_cos__minus) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 404 (all X_a all B_1 all C all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),A_2)) -> (-hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),B_1)) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)))))) # label(fact_2233_DiffI) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 405 (all Xa all A_2 (hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),Xa),A_2)) <-> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(nat,int,semiring_1_of_nat(int),Xa)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2))))) # label(fact_3673_transfer__nat__int__set__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 406 (all I_1 all J_2 all K_4 hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I_1),J_2)),K_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),K_4))) # label(fact_812_diff__diff__left) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 407 (all P_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4)) -> hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),sRStar,P_4))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int)))) # label(fact_1661_SRStar__card) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 408 (all X_a all X_c all X_b all P all Q all R hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),P,R),Q) = hAPP(X_a,X_c,hAPP(X_b,fun(X_a,X_c),hAPP(fun(X_a,fun(X_b,X_c)),fun(X_b,fun(X_a,X_c)),combc(X_a,X_b,X_c),P),Q),R)) # label(help_COMBC_1_1_U) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 409 (all X_a (archim1727834104eiling(X_a) -> (all Z_1 all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)),X)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)),one_one(X_a)))) -> ti(int,Z_1) = hAPP(X_a,int,archim2085082626_floor(X_a),X)))))) # label(fact_3630_floor__unique) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 410 (all A_3 all Ma all Xa (zero_zero(int) != ti(int,Xa) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),Ma)) -> hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,A_3),Ma))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Xa),hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,A_3),Ma)))) = hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,int),fun(fun(int,bool),fun(int,bool)),image(int,int),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),times_times(int)),Xa)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,A_3),Ma)))))) # label(fact_3959_Bnor__prod__power) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 411 (all X_a all C all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1))) -> -hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),B_1)))) # label(fact_2241_DiffD2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 412 (all X_a all X_b (comm_monoid_mult(X_b) -> (all G all A_2 (-hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),A_2) = one_one(X_b))))) # label(fact_3979_setprod_Oinfinite) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 413 (all M all N (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),M),N)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,fact_fact(int),M)),hAPP(int,int,fact_fact(int),N))))) # label(fact_3206_fact__mono__int) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 414 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),A_1)) -> zero_zero(int) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)))) # label(fact_1103_div__neg__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 415 (all X_a (linorder(X_a) -> (all X all Y all Z_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Z_1))))) # label(fact_4976_min__max_Osup__inf__distrib1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 416 (all M all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,M)),N)) # label(fact_1395_mult__Suc) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 417 (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),hAPP(int,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_235_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 418 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),X))),hAPP(real,real,uminus_uminus(real),X)))))) # label(fact_2040_ln__one__minus__pos__upper__bound) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 419 (all X_a all X_b all X_c (topological_space(X_b) & topological_space(X_a) & topological_space(X_c) -> (all F all A_3 all G all L (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),hAPP(X_a,X_b,G,L)),hAPP(X_a,filter(X_a),at(X_a),L))) -> (hBOOL(hAPP(filter(X_c),bool,hAPP(X_a,fun(filter(X_c),bool),hAPP(fun(X_c,X_a),fun(X_a,fun(filter(X_c),bool)),tendsto(X_c,X_a),F),L),hAPP(X_c,filter(X_c),at(X_c),A_3))) -> hBOOL(hAPP(filter(X_c),bool,hAPP(X_b,fun(filter(X_c),bool),hAPP(fun(X_c,X_b),fun(X_b,fun(filter(X_c),bool)),tendsto(X_c,X_b),hAPP(fun(X_c,X_a),fun(X_c,X_b),hAPP(fun(X_a,X_b),fun(fun(X_c,X_a),fun(X_c,X_b)),combb(X_a,X_b,X_c),G),F)),hAPP(X_a,X_b,G,L)),hAPP(X_c,filter(X_c),at(X_c),A_3)))))))) # label(fact_4583_LIM__compose) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 420 (all A_1 all C_1 hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A_1),C_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,suc,zero_zero(nat))),C_1))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A_1),C_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,zero_zero(nat))),C_1))),C_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,suc,A_1)),C_1)) # label(fact_1555_div__Suc) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 421 (all Xa (zero_zero(int) != hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Xa),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) <-> one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Xa),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) # label(fact_1498_neq__one__mod__two) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 422 (all X_b all X_a (real_vector(X_a) -> (all G all A_2 all Xa hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(X_a,fun(X_b,X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(X_a,fun(X_b,X_a)),combc(X_b,X_a,X_a),hAPP(fun(X_b,real),fun(X_b,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(X_b,real),fun(X_b,fun(X_a,X_a))),combb(real,fun(X_a,X_a),X_b),scaleR_scaleR(X_a)),G)),Xa)),A_2) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(fun(X_b,bool),real,hAPP(fun(X_b,real),fun(fun(X_b,bool),real),big_co1399186613setsum(X_b,real),G),A_2)),Xa)))) # label(fact_4755_scaleR__left_Osetsum) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 423 (all Na all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),Ma)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,Na)),hAPP(nat,nat,suc,Ma))))) # label(fact_1342_Suc__le__mono) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 424 (all Na (hBOOL(hAPP(nat,bool,even_odd_even(nat),Na)) <-> hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Na),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) # label(fact_2523_even__even__mod__4__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 425 (all X_a (linord20386208strict(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1)))))))) # label(fact_452_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 426 (all X all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(real,nat,natceiling,X)),hAPP(real,nat,natceiling,Y))))) # label(fact_1669_natceiling__mono) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 427 (all M all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),hAPP(nat,nat,suc,N))) # label(fact_1396_mult__Suc__right) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 428 (all Na all Xa all Diff all F (hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F -> ((all M_2 all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_2)),X_1),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),X_1)))) -> (exists T_4 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),T_4)),hAPP(real,real,abs_abs(real),Xa))) & hAPP(real,real,F,Xa) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,Na),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),Na)))))))) # label(fact_4312_Maclaurin__all__le) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 429 (all X_a (real_vector(X_a) -> (all A_1 all X hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),X)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X))))) # label(fact_4734_scaleR__minus__right) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 430 (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,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),M)) -> ti(int,X) = zero_zero(int))))) # label(fact_914_Int2_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 431 (all X_a all X_b all A_3 all B_2 all A_8 all B_7 all Ra all Sb (hBOOL(hAPP(fun(product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b)),bool),bool,hAPP(product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b)),fun(fun(product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b)),bool),bool),member(product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b))),hAPP(product_prod(X_a,X_b),product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b)),hAPP(product_prod(X_a,X_b),fun(product_prod(X_a,X_b),product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b))),product_Pair(product_prod(X_a,X_b),product_prod(X_a,X_b)),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_3),B_2)),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_8),B_7))),hAPP(fun(product_prod(X_b,X_b),bool),fun(product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b)),bool),hAPP(fun(product_prod(X_a,X_a),bool),fun(fun(product_prod(X_b,X_b),bool),fun(product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b)),bool)),lex_prod(X_a,X_b),Ra),Sb))) <-> hBOOL(hAPP(fun(product_prod(X_a,X_a),bool),bool,hAPP(product_prod(X_a,X_a),fun(fun(product_prod(X_a,X_a),bool),bool),member(product_prod(X_a,X_a)),hAPP(X_a,product_prod(X_a,X_a),hAPP(X_a,fun(X_a,product_prod(X_a,X_a)),product_Pair(X_a,X_a),A_3),A_8)),Ra)) | hBOOL(hAPP(fun(product_prod(X_b,X_b),bool),bool,hAPP(product_prod(X_b,X_b),fun(fun(product_prod(X_b,X_b),bool),bool),member(product_prod(X_b,X_b)),hAPP(X_b,product_prod(X_b,X_b),hAPP(X_b,fun(X_b,product_prod(X_b,X_b)),product_Pair(X_b,X_b),B_2),B_7)),Sb)) & ti(X_a,A_8) = ti(X_a,A_3))) # label(fact_3512_in__lex__prod) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 432 (all X_b all X_a (metric_space(X_b) & metric_space(X_a) -> (all F all L_2 all A_3 ((all R_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1)) -> (exists S_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),S_1)) & (all X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X_1),A_3)),S_1)) & ti(X_a,X_1) != ti(X_a,A_3) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,hAPP(X_b,fun(X_b,real),dist_dist(X_b),hAPP(X_a,X_b,F,X_1)),L_2)),R_1)))))))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4789_metric__LIM__I) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 433 (all X_a (linorder(X_a) -> (all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ya),Xa)) <-> -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya)))))) # label(fact_571_linorder__not__le) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 434 (all X_a (order(X_a) -> (all Ya all Xa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ya),Xa)) -> (ti(X_a,Xa) = ti(X_a,Ya) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya))))))) # label(fact_509_order__antisym__conv) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 435 (all Wa all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Wa),Z)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z),one_one(int)))))) # label(fact_289_zle__add1__eq__le) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 436 (all X all Y hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,cos,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,X)),hAPP(real,real,sin,Y)))) # label(fact_2271_cos__add) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 437 (all X_a all X_b (linordered_semidom(X_b) -> (all F all A_2 ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),zero_zero(X_b)),hAPP(X_a,X_b,F,X_1))))) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),zero_zero(X_b)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2))))))) # label(fact_4176_setprod__pos) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 438 (all Ma all Na (zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),Na) <-> Ma = zero_zero(nat) & Na = zero_zero(nat))) # label(fact_157_add__is__0) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 439 (all Z 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)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Wa),Z)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(int,nat,nat_1,Wa)),hAPP(int,nat,nat_1,Z)))))) # label(fact_1039_nat__le__eq__zle) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 440 (all X_a (comm_semiring_1(X_a) -> (all X all Q_3 hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,suc,Q_3)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),Q_3))))) # label(fact_1367_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 441 (all A_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),one_one(nat))) -> one_one(nat) = A_1)) # label(fact_1260_gcd__lcm__complete__lattice__nat_Ole__bot) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 442 (all Z_1 hBOOL(hAPP(int,bool,nat_is_nat,hAPP(nat,int,semiring_1_of_nat(int),Z_1)))) # label(fact_4875_Nat__Transfer_Otransfer__int__nat__function__closures_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 443 (all Na hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),hAPP(bool,fun(real,fun(real,real)),if(real),hAPP(nat,bool,even_odd_even(nat),Na)),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,zero_zero(nat)))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),zero_zero(real)),Na)) = zero_zero(real)) # label(fact_2526_lemma__STAR__sin) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 444 (all A_3 all B_2 all K all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Ma)) -> (hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K),Ma) = one_one(int) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),Ma)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_3),K)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_2),K)),Ma)))))) # label(fact_3496_zcong__cancel) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 445 (all X hAPP(int,real,number_number_of(real),X) = hAPP(int,real,ring_1_of_int(real),X)) # label(fact_3592_number__of__real__def) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 446 (all X_a all P ti(X_a,P) = hAPP(X_a,X_a,combi(X_a),P)) # label(help_COMBI_1_1_U) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 447 (all B_2 all A_3 all P_4 (hBOOL(hAPP(int,bool,zprime,P_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int)))) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,A_3),P_4))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_2),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int)))))))) # label(fact_1207_wset__less) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 448 (all Na hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),hAPP(nat,real,real_1(nat),Na)),ring_1_Ints(real)))) # label(fact_3826_Ints__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 449 (all X_a (number_ring(X_a) -> (all V all W hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),V),W)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),W))))) # label(fact_716_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 450 (all X_a (comm_semiring_1(X_a) -> (all A_1 all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),A_1)))) # label(fact_301_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 451 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),min)),Y)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,arcsin,Y))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,arcsin,Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) # label(fact_2228_arcsin__lt__bounded) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 452 (all M hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,fact_fact(int),M)))) # label(fact_3215_fact__ge__zero__int) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 453 (all X_a all X_b all X_c (topological_space(X_c) & topological_space(X_a) & topological_space(X_b) -> (all F all A_3 all G all L (hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),G),L)) -> (hBOOL(hAPP(filter(X_c),bool,hAPP(X_a,fun(filter(X_c),bool),hAPP(fun(X_c,X_a),fun(X_a,fun(filter(X_c),bool)),tendsto(X_c,X_a),F),L),hAPP(X_c,filter(X_c),at(X_c),A_3))) -> hBOOL(hAPP(filter(X_c),bool,hAPP(X_b,fun(filter(X_c),bool),hAPP(fun(X_c,X_b),fun(X_b,fun(filter(X_c),bool)),tendsto(X_c,X_b),hAPP(fun(X_c,X_a),fun(X_c,X_b),hAPP(fun(X_a,X_b),fun(fun(X_c,X_a),fun(X_c,X_b)),combb(X_a,X_b,X_c),G),F)),hAPP(X_a,X_b,G,L)),hAPP(X_c,filter(X_c),at(X_c),A_3)))))))) # label(fact_4901_isCont__LIM__compose) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 454 (all X all A_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)))) # label(fact_3372_powr__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 455 (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_191_of__nat__0) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 456 (all M hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),one_one(nat)) = one_one(nat)) # label(fact_3277_gcd__1__nat) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 457 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),Y)),one_one(real))) -> hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(real,real,sin,hAPP(real,real,arccos,Y)))) # label(fact_2362_sin__arccos__abs) # label(axiom) # label(non_clause). [assumption]. 7.53/7.60 458 (all Ma all Na (hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),Ma),Na) = zero_zero(int) <-> ti(int,Ma) = zero_zero(int) & ti(int,Na) = zero_zero(int))) # label(fact_4410_gcd__zero__int) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 459 (all A_1 all M all B hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),M)),M))) # label(fact_898_zcong__zmult__self) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 460 (all X_a all F all G all Ma all S ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),S)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(X_a,int,F,X_1)),hAPP(X_a,int,G,X_1)),Ma)))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),F),S)),hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),G),S)),Ma)))) # label(fact_4360_setsum__same__function__zcong) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 461 (all V (hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V))) -> hAPP(int,nat,number_number_of(nat),V) = zero_zero(nat))) # label(fact_1616_neg__imp__number__of__eq__0) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 462 (all X_a (real_normed_field(X_a) -> (all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),zero_zero(X_a)),Y) = zero_zero(X_a)))) # label(fact_1750_divide_Ozero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 463 (all X all Y hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,real_1(int),X)),hAPP(int,real,real_1(int),Y)) = hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),Y))) # label(fact_3360_real__of__int__diff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 464 (all Na hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min))),suc)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Na))) = zero_zero(real)) # label(fact_4264_sumr__minus__one__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 465 (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),X),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,ln,X)),zero_zero(real)))))) # label(fact_1720_ln__less__zero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 466 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),hAPP(real,real,exp(real),Xa))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)))) # label(fact_1786_one__less__exp__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 467 (all X_a (comm_ring_1(X_a) -> (all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Xa),hAPP(X_a,X_a,uminus_uminus(X_a),Ya))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Xa),Ya)))))) # label(fact_1920_dvd__minus__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 468 (all X_b all X_a (ab_group_add(X_a) -> (all F all G all A_2 hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,X_a)),combs(X_b,X_a,X_a),hAPP(fun(X_b,X_a),fun(X_b,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_b),minus_minus(X_a)),F)),G)),A_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2)),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),G),A_2))))) # label(fact_4247_setsum__subtractf) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 469 (all C_1 all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),C_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C_1))))) # label(fact_827_divides__mul__r) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 470 (all M all N hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M),N) = hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N))) # label(fact_3119_divmod__nat__div__mod) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 471 (all X (hAPP(real,real,sin,X) = zero_zero(real) -> one_one(real) = hAPP(real,real,abs_abs(real),hAPP(real,real,cos,X)))) # label(fact_2268_sin__zero__abs__cos__one) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 472 (all X_a (real_normed_algebra(X_a) -> (all Y_2 all B_2 all X_2 all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially)) -> (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),Y_2),B_2),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),X_2)),Y_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_2)),sequentially))))))) # label(fact_4484_LIMSEQ__mult) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 473 (all M all N hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M),N),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N))))) # label(fact_3128_divmod__nat__rel) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 474 (all A_3 all P_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4)) -> hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(fun(fun(int,bool),bool),fun(int,bool),complete_Sup_Sup(fun(int,bool)),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,A_3),P_4)))))) # label(fact_2725_Union__SetS__finite) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 475 (all X_a (power(X_a) -> (all M all N ((zero_zero(nat) != N -> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),M),N) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),M),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))))) & (zero_zero(nat) = N -> one_one(X_a) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),M),N)))))) # label(fact_924_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 476 (all Na (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(int,real,real_1(int),Na))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Na)))) # label(fact_3410_real__of__int__ge__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 477 (all I all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),I)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),I),K)))))) # label(fact_1111_pos__imp__zdiv__pos__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 478 (all X_a all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1)) -> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))))) # label(fact_2742_finite__subset) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 479 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),K)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),hAPP(int,int,bit0,K))))) # label(fact_678_rel__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 480 (all A_3 all B_2 all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),powr,Xa),A_3)),hAPP(real,real,hAPP(real,fun(real,real),powr,Xa),B_2)))))) # label(fact_3403_powr__le__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 481 (all X_a (ord(X_a) -> ord_atMost(X_a) = ti(fun(X_a,fun(X_a,bool)),ord_atMost(X_a)))) # label(tsy_c_SetInterval_Oord__class_OatMost_res) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 482 (all X_a ti(fun(fun(product_prod(X_a,X_a),bool),bool),refl(X_a)) = refl(X_a)) # label(tsy_c_Nitpick_Orefl_H_res) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 483 (all P_5 all A_1 hAPP(int,int,hAPP(int,fun(int,int),inv,P_5),A_1) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),P_5)) # label(fact_1515_WilsonRuss_Oinv__def) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 484 (all Ma all Z (ti(int,Z) = hAPP(nat,int,semiring_1_of_nat(int),Ma) <-> Ma = hAPP(int,nat,nat_1,Z) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)))) # label(fact_1023_int__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 485 (all I_1 all K_4 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,I_1)),K_4)) -> -(all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_1)) -> K_4 != hAPP(nat,nat,suc,J_1))))) # label(fact_2752_Suc__lessE) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 486 (all X_a (monoid_mult(X_a) -> (all A_1 all N hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) # label(fact_1505_power__odd__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 487 (all N all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N),M)) -> ti(int,M) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),N),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),M),N)))) # label(fact_1083_zdvd__mult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 488 (all M all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,suc,M)))),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),M)),N)) # label(fact_1493_Suc__div__eq__add3__div) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 489 (all X_a (division_ring(X_a) -> (all A_1 all B all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C_1))))) # label(fact_1827_times__divide__eq__right) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 490 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),Xa)),one_one(real))) -> hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),inverse_divide(real),one_one(real))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat)))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),Xa)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat)))))) = hAPP(real,real,arctan,Xa))) # label(fact_3993_arctan__series) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 491 (all Z_1 hAPP(real,real,hAPP(real,fun(real,real),times_times(real),one_one(real)),Z_1) = Z_1) # label(fact_623_real__mult__1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 492 (all X_a (linorder(X_a) -> (all Ma all I all Na (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),I),Na)) -> hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),Na),Ma) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),I),Ma)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),I),Na)))))) # label(fact_4220_ivl__diff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 493 (all P_1 ((all X1 hBOOL(hAPP(nat,bool,P_1,X1))) <-> (all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) -> hBOOL(hAPP(nat,bool,P_1,hAPP(int,nat,nat_1,X_1))))))) # label(fact_1002_all__nat) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 494 (all Na all Xa exists T_4 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),T_4)),hAPP(real,real,abs_abs(real),Xa))) & hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(real,fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(real,fun(nat,fun(real,real))),combc(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),zero_zero(real))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),div_div(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),minus_minus(nat)),hAPP(nat,nat,suc,zero_zero(nat))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat)))))),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),T_4),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,real_1(nat),Na))),pi)))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),Na))) = hAPP(real,real,sin,Xa))) # label(fact_4268_Maclaurin__sin__expansion2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 495 (all M all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N)),N)) # label(fact_1548_div__mod__equality_H) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 496 (all X_a (topological_space(X_a) -> (all F all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,real),fun(X_a,real),hAPP(fun(real,real),fun(fun(X_a,real),fun(X_a,real)),combb(real,real,X_a),abs_abs(real)),F)),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3))) <-> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4619_LIM__rabs__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 497 (all A_2 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Ma)) -> (hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),Ma)) -> hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,int),fun(fun(int,bool),fun(int,bool)),image(int,int),hAPP(int,fun(int,int),hAPP(fun(int,bool),fun(int,fun(int,int)),rRset2norRR,A_2),Ma)),A_2) = hAPP(int,fun(int,bool),norRRset,Ma)))) # label(fact_3653_RRset2norRR__eq__norR) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 498 (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_165_zero__less__int__conv) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 499 (all K_4 all A_1 all B all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),K_4)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),K_4)),M)))) # label(fact_895_zcong__scalar) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 500 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,K)),hAPP(int,int,bit0,L))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)))) # label(fact_278_rel__simps_I33_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 501 (all A_3 all B_2 (hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),negDivAlg_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_3),B_2))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_3),B_2))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),zero_zero(int))) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,number_number_of(int),min)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_3),B_2)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_3),B_2)) & (-(hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_3),B_2))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),zero_zero(int)))) -> hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,B_2),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_3),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B_2))) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_3),B_2)))) # label(fact_1657_negDivAlg_Opsimps) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 502 (all M all K_4 all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M)),N))) # label(fact_3288_gcd__add__mult__nat) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 503 (all X_a (metric_space(X_a) -> (all X_2 all L_2 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),L_2),sequentially)) <-> (all R_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1)) -> (exists No all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),No),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),hAPP(nat,X_a,X_2,N_1)),L_2)),R_1)))))))))) # label(fact_4545_LIMSEQ__def) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 504 (all X_a (real_normed_algebra(X_a) -> (all X exists K_2 ((all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_1)),K_2)))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2)))))) # label(fact_2051_mult__right_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 505 (all X_a (real_normed_algebra(X_a) -> (all A_1 all A_6 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_6),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),A_6)),B)))) # label(fact_1248_mult_Odiff__left) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 506 (all P_1 all Na all K (hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Na),K))) <-> (zero_zero(int) = ti(int,K) -> hBOOL(hAPP(int,bool,P_1,zero_zero(int)))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K)) -> (all I_2 all J_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J_1)) & ti(int,Na) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),I_2)),J_1) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),J_1),K)) -> hBOOL(hAPP(int,bool,P_1,I_2))))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),zero_zero(int))) -> (all I_2 all J_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),J_1),zero_zero(int))) & hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),I_2)),J_1) = ti(int,Na) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),J_1)) -> hBOOL(hAPP(int,bool,P_1,I_2))))))) # label(fact_1126_split__zdiv) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 507 (all X_a (archim1727834104eiling(X_a) -> (all X hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),one_one(int)) = hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),one_one(X_a)))))) # label(fact_2915_floor__diff__one) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 508 (all X_a (real_normed_vector(X_a) -> (all B_2 all F all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),A_3),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),F)),B_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2)),sequentially)))))) # label(fact_4503_LIMSEQ__diff__const) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 509 (all N hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N),one_one(int))),N) = one_one(int)) # label(fact_4394_coprime__plus__one__int) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 510 (all Ma all Na ((hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) | zero_zero(nat) = Na -> hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),Ma) = hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,Ma),Na)) & (-(hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) | Na = zero_zero(nat)) -> hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,Ma),Na) = hAPP(product_prod(nat,nat),product_prod(nat,nat),hAPP(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_prod_case(nat,nat,product_prod(nat,nat)),hAPP(fun(nat,nat),fun(nat,fun(nat,product_prod(nat,nat))),hAPP(fun(nat,fun(nat,product_prod(nat,nat))),fun(fun(nat,nat),fun(nat,fun(nat,product_prod(nat,nat)))),combb(nat,fun(nat,product_prod(nat,nat)),nat),product_Pair(nat,nat)),suc)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Ma),Na)),Na))))) # label(fact_4079_divmod__nat__if) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 511 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Ma)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),Na))))) # label(fact_147_add__gr__0) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 512 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real))) -> Y = hAPP(real,real,sin,hAPP(real,real,arcsin,Y))))) # label(fact_2320_sin__arcsin) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 513 (all X 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)),X)) -> X = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)),N)))) # label(fact_2715_real__root__pow__pos) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 514 (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),Xa),Ya)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)))) # label(fact_960_dvd_Ole__less) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 515 (all N all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M)),N) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)))) # label(fact_1400_Suc__diff__le) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 516 (all Xa all F all A_3 all B_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),B_2)) -> ((all X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),X_1)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),B_2)) -> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),F),X_1)))) -> ((all X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),B_2)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),X_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),zero_zero(real))))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),B_2)) -> hAPP(real,real,F,Xa) = hAPP(real,real,F,A_3))))))) # label(fact_4952_DERIV__isconst2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 517 (all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),M)) -> -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,one_one(int)),hAPP(int,int,number_number_of(int),min)),M)))) # label(fact_793_one__not__neg__one__mod__m) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 518 (all X_a (number_ring(X_a) -> (all V all W hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),W)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),W))))) # label(fact_91_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 519 (all X_a (metric_space(X_a) -> (all X all Y all Z_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Z_1)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Y),Z_1))))))) # label(fact_3531_dist__triangle2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 520 (all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,real_1(nat),hAPP(real,nat,natfloor,X))),one_one(real))))) # label(fact_2412_real__natfloor__add__one__gt) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 521 (all X_a (number(X_a) & field_inverse_zero(X_a) -> (all A_3 all B_2 all Wa ((hAPP(int,X_a,number_number_of(X_a),Wa) = zero_zero(X_a) -> ti(X_a,A_3) = zero_zero(X_a)) & (hAPP(int,X_a,number_number_of(X_a),Wa) != zero_zero(X_a) -> ti(X_a,B_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),hAPP(int,X_a,number_number_of(X_a),Wa))) <-> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),hAPP(int,X_a,number_number_of(X_a),Wa)) = ti(X_a,A_3))))) # label(fact_1776_eq__divide__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 522 (all X_a (semiring_0(X_a) & power(X_a) -> (all N hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),zero_zero(X_a)),hAPP(nat,nat,suc,N)) = zero_zero(X_a)))) # label(fact_1364_power__0__Suc) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 523 (all D_1 all A_3 all B_2 (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_3),B_2)) <-> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_1),B_2) = one_one(int) & hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_1),A_3) = one_one(int))) # label(fact_4412_coprime__mul__eq__int) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 524 (all N (-hBOOL(hAPP(nat,bool,even_odd_even(nat),N)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)))) # label(fact_2477_odd__pos) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 525 (all X_a (field(X_a) -> (all Na all Xa (ti(X_a,Xa) != one_one(X_a) -> hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),Na)),one_one(X_a))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Xa),one_one(X_a))))))) # label(fact_4197_geometric__sum) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 526 (all X_a (ab_group_add(X_a) -> (all F all P_4 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_eq(nat),Na),P_4)) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Ma),P_4))),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Ma),Na))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Na),P_4))))))) # label(fact_4229_setsum__diff__nat__ivl) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 527 (all X_a all F1 all F2 all Nat hAPP(nat,X_a,F2,Nat) = hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(X_a,fun(fun(nat,X_a),fun(nat,X_a)),nat_case(X_a),F1),F2),hAPP(nat,nat,suc,Nat))) # label(fact_4138_nat__case__Suc) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 528 (all P_5 all Q_3 hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,P_5),Q_3) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),P_5),Q_3)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),P_5),Q_3))) # label(fact_3105_divmod__int__mod__div) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 529 (all X_a all X_c all X_b all P all Q all R hAPP(X_a,X_c,hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c)),combb(X_b,X_c,X_a),P),Q),R) = hAPP(X_b,X_c,P,hAPP(X_a,X_b,Q,R))) # label(help_COMBB_1_1_U) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 530 (all I_1 all J_2 all K_4 hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I_1),J_2)),K_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I_1),K_4)),J_2)) # label(fact_801_diff__commute) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 531 (all X_a all X_b (comm_monoid_add(X_b) -> (all G all H all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2)) -> hAPP(X_a,X_b,H,X_1) = hAPP(X_a,X_b,G,X_1))) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),H),A_2) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),A_2)))))) # label(fact_4799_setsum_Ocong) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 532 (all X_a (group_add(X_a) -> (all A_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)) = zero_zero(X_a)))) # label(fact_1957_right__minus) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 533 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real))) -> zero_zero(nat) = hAPP(real,nat,natfloor,X))) # label(fact_1672_natfloor__neg) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 534 (all K_4 hAPP(int,int,uminus_uminus(int),K_4) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),min),K_4)) # label(fact_2079_mult__Min) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 535 (all N hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),N)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) # label(fact_2437_fact__lemma) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 536 (all P_5 all Q_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Q_3)) -> (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),P_5),Q_3) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_5),Q_3) = hAPP(product_prod(int,int),product_prod(int,int),normalize,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_5),Q_3))))) # label(fact_4435_normalize__stable) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 537 (all X_a (archim1727834104eiling(X_a) -> (all V hAPP(int,int,number_number_of(int),V) = hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(int,X_a,number_number_of(X_a),V))))) # label(fact_2898_floor__number__of) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 538 (all Y all X all U_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),U_1)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),U_1),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),U_1),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),U_1)))))))) # label(fact_2116_lemma__sqrt__hcomplex__capprox) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 539 (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),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),M)) -> zero_zero(int) = ti(int,A_1))))) # label(fact_915_zcong__zless__0) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 540 (all X_a (real_normed_algebra(X_a) -> (all Xa all A_3 hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),A_3)),hAPP(X_a,filter(X_a),at(X_a),A_3)))))) # label(fact_4587_mult__right_Ocont) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 541 (all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),plus_plus(real)),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),plus_plus(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),sin),uminus_uminus(real)))),sin))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),minus_minus(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),cos),uminus_uminus(real)))),cos))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X_1),zero_zero(real)))) # label(fact_3841_lemma__DERIV__sin__cos__minus) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 542 (all X_a (ordere216010020id_add(X_a) -> (all B all C_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),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)))))))) # label(fact_228_add__increasing) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 543 (all A_3 all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),A_3)),Xa)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(real,int,archimedean_ceiling(real),Xa))))) # label(fact_3391_less__ceiling__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 544 (all I_1 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),N)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),I_1)) = I_1)) # label(fact_818_diff__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 545 (all Va all Na ((-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),Na))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(int,nat,number_number_of(nat),Va)),hAPP(nat,nat,suc,Na))))) # label(fact_3925_le__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 546 (all X all Y hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(complex,complex,cnj,X)),hAPP(complex,complex,cnj,Y)) = hAPP(complex,complex,cnj,hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),X),Y))) # label(fact_2964_cnj_Odiff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 547 (all G all F all Xa all D_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_1)) -> ((all Z_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),Xa))),D_1)) -> hAPP(real,real,G,hAPP(real,real,F,Z_2)) = Z_2)) -> ((all Z_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),Xa))),D_1)) -> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),F),Z_2)))) -> (exists Z_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,Z_2)),hAPP(real,real,F,Xa))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),Xa))),D_1)))))))) # label(fact_4956_lemma__isCont__inj2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 548 (all X_a (ord(X_a) -> ord_lessThan(X_a) = ti(fun(X_a,fun(X_a,bool)),ord_lessThan(X_a)))) # label(tsy_c_SetInterval_Oord__class_OlessThan_res) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 549 (all Diff all Na all Xa (zero_zero(real) = Xa -> (zero_zero(nat) != Na -> hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na)) = hAPP(real,real,hAPP(nat,fun(real,real),Diff,zero_zero(nat)),zero_zero(real))))) # label(fact_4257_Maclaurin__zero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 550 (all X_a all X_b (real_normed_algebra(X_b) -> (all Xa all G all A_3 all A_2 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),A_3),A_2)) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),hAPP(X_b,fun(X_b,X_b),times_times(X_b),Xa)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Xa),A_3)),A_2)))))) # label(fact_4479_mult__right_Otendsto) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 551 (all X_a (archim1727834104eiling(X_a) -> (all Z all Xa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)),one_one(X_a))),Xa)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),hAPP(X_a,int,archimedean_ceiling(X_a),Xa))))))) # label(fact_3637_le__ceiling__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 552 (all M all K_4 (M != hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),K_4) -> (M != hAPP(nat,nat,suc,zero_zero(nat)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),M))))) # label(fact_1444_one__less__m) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 553 (all X_a (idom(X_a) -> (all A_3 all C all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),C))) <-> ti(X_a,C) = zero_zero(X_a) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_3),B_2)))))) # label(fact_746_dvd__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 554 (all Xa hAPP(fun(real,bool),real,complete_Sup_Sup(real),hAPP(real,fun(real,bool),ord_atMost(real),Xa)) = Xa) # label(fact_4971_Sup__atMost) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 555 (all A_1 all B hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)) # label(fact_2067_zdiv__zminus__zminus) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 556 (all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1)) -> hAPP(nat,nat,suc,hAPP(int,nat,nat_1,Z_1)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),Z_1)))) # label(fact_1491_Suc__nat__eq__nat__zadd1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 557 (all Z_1 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(real,nat,natfloor,Z_1)),one_one(nat))),N)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Z_1),hAPP(nat,real,real_1(nat),N))))) # label(fact_2426_ge__natfloor__plus__one__imp__gt) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 558 (all X_a (real_normed_vector(X_a) -> (all X all Y all C_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C_1),zero_zero(real))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),X)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C_1),hAPP(X_a,real,norm_norm(X_a),Y)))) -> zero_zero(X_a) = ti(X_a,X)))))) # label(fact_2028_norm__ratiotest__lemma) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 559 (all D_3 all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_3),B)))) # label(fact_2878_coprime__lmul2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 560 (all X_a (monoid_mult(X_a) -> (all Z_1 all W hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,W))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1),hAPP(int,nat,number_number_of(nat),W))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1),hAPP(int,nat,number_number_of(nat),W)))))) # label(fact_3881_power__number__of__even) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 561 (all X_a (number_ring(X_a) -> zero_zero(X_a) = hAPP(int,X_a,number_number_of(X_a),pls))) # label(fact_86_semiring__norm_I112_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 562 (all X all N hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(complex,complex,cnj,X)),N) = hAPP(complex,complex,cnj,hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),X),N))) # label(fact_2956_complex__cnj__power) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 563 (all X_a all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)),hAPP(fun(X_a,bool),nat,finite_card(X_a),B_1)))))) # label(fact_2749_psubset__card__mono) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 564 (all Wa all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Wa),one_one(int))),Z)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z)))) # label(fact_288_add1__zle__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 565 (all A_1 all B all C_1 hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B)),C_1) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),B),C_1))) # label(fact_4407_gcd__assoc__int) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 566 (all X_a (linordered_idom(X_a) -> (all Xa all Wa (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),hAPP(int,nat,number_number_of(nat),Wa)))) <-> -hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,number_number_of(nat),Wa))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Xa)) | hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,number_number_of(nat),Wa))))))) # label(fact_2510_zero__le__power__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 567 (all Ya all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Ya)) -> (Ya = Xa <-> hAPP(real,real,ln,Ya) = hAPP(real,real,ln,Xa))))) # label(fact_1706_ln__inj__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 568 (all X_a (t2_space(X_a) & comm_monoid_add(X_a) -> (all F all Na ((all M_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),M_2)) -> hAPP(nat,X_a,F,M_2) = zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na)))))))) # label(fact_4318_series__zero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 569 (all X_a (order(X_a) -> (all A_3 all B_2 all C all D_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),A_3),B_2)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),C),D_1))) <-> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),D_1)) & (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),D_1)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),A_3))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),A_3)) | -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_1)))))) # label(fact_4552_atLeastatMost__psubset__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 570 (all X_a (linord581940658strict(X_a) -> (all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),Xa)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ya),Ya))),zero_zero(X_a))) <-> ti(X_a,Ya) = zero_zero(X_a) & ti(X_a,Xa) = zero_zero(X_a))))) # label(fact_487_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 571 (all X_c all X_b all X_a all Q_1 all F all P_1 all P_4 (hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,hAPP(product_prod(X_b,X_a),X_a,product_snd(X_b,X_a),P_4)),hAPP(product_prod(X_b,X_a),X_b,product_fst(X_b,X_a),P_4))) -> ((all X_1 all Y_1 (hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,Y_1),X_1)) -> hBOOL(hAPP(X_c,bool,Q_1,hAPP(X_a,X_c,hAPP(X_b,fun(X_a,X_c),F,X_1),Y_1))))) -> hBOOL(hAPP(X_c,bool,Q_1,hAPP(product_prod(X_b,X_a),X_c,hAPP(fun(X_b,fun(X_a,X_c)),fun(product_prod(X_b,X_a),X_c),product_prod_case(X_b,X_a,X_c),F),P_4)))))) # label(fact_4153_exE__realizer) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 572 (all X_a (field_char_0(X_a) -> (all B_2 all A_3 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),field_char_0_Rats(X_a))) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),field_char_0_Rats(X_a))) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2)),field_char_0_Rats(X_a)))))))) # label(fact_3712_Rats__diff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 573 (all X_a (archim1727834104eiling(X_a) -> (all Xa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),zero_zero(X_a))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archimedean_ceiling(X_a),Xa)),zero_zero(int))))))) # label(fact_3323_ceiling__le__zero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 574 (all X_a (field_char_0(X_a) -> ti(fun(X_a,bool),field_char_0_Rats(X_a)) = field_char_0_Rats(X_a))) # label(tsy_c_Rat_Ofield__char__0__class_ORats_res) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 575 (all X_a (real_normed_field(X_a) -> (all Ya all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2)) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),X_2)),Ya) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),X_2)),Ya)))))) # label(fact_4011_divide_Osuminf) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 576 (all X_a all X_b (metric_space(X_b) & real_n2089651433ebra_1(X_a) -> (all K all L_2 all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),combk(X_b,X_a),K)),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> ti(X_b,K) = ti(X_b,L_2))))) # label(fact_4584_LIM__const__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 577 (all X all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5)) -> (hBOOL(hAPP(int,bool,zprime,P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),X)),X)),one_one(int)),P_5)))))) # label(fact_1195_MultInv__prop2a) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 578 (all Xa hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less_eq(nat)),Xa))))) # label(fact_3870_bdd__nat__set__le__finite) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 579 (all Xa all P_4 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),sr,P_4))) -> hAPP(int,int,hAPP(int,fun(int,int),standardRes,P_4),Xa) = ti(int,Xa))) # label(fact_1273_StandardRes__SR__prop) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 580 (all K_4 hAPP(int,int,pred,K_4) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),min),K_4)) # label(fact_2933_add__Min) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 581 (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),hAPP(int,X_a,number_number_of(X_a),Xa)),hAPP(int,X_a,number_number_of(X_a),Ya))))))) # label(fact_18_less__number__of) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 582 (all X_a all X_b (real_normed_vector(X_b) -> (all X_2 ((exists K_2 ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_a,X_b,X_2,N_1))),K_2))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2)))) <-> (exists N_2 all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_a,X_b,X_2,N_1))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_2))))))))) # label(fact_2417_lemma__NBseq__def) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 583 (all X_a (archim1727834104eiling(X_a) -> (all Xa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),min)),Xa)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(X_a,int,archimedean_ceiling(X_a),Xa))))))) # label(fact_3335_zero__le__ceiling) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 584 (all X_a (linord1117847801e_zero(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)) = hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B))))) # label(fact_1832_abs__divide) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 585 (all A_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A_1)) -> zero_zero(nat) = A_1)) # label(fact_1251_gcd__lcm__complete__lattice__nat_Otop__le) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 586 (all X_a all X_b (ord(X_b) -> (all Xa all F all G (hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,Xa)),hAPP(X_a,X_b,G,Xa))))))) # label(fact_510_le__funD) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 587 (all X_b all X_a all B_1 all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_1)) -> hBOOL(hAPP(fun(sum_sum(X_a,X_b),bool),bool,finite_finite_1(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_2),B_1)))))) # label(fact_3208_finite__Plus) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 588 (all X_a (linordered_field(X_a) -> (all X all Z_1 all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(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),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),Y))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),Z_1))))))) # label(fact_1893_mult__imp__div__pos__le) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 589 (all X_a (comm_semiring_1(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),zero_zero(X_a))))) # label(fact_319_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 590 (all N N = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),one_one(nat)),N)) # label(fact_622_nat__mult__1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 591 (all X all M hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),X)),M) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),M))),M)) # label(fact_2069_zminus__zmod) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 592 (all X_a (monoid_mult(X_a) -> (all A_1 all N hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,N)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),A_1)))) # label(fact_1370_power__Suc2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 593 (all X_a (metric_space(X_a) -> (all X_2 all L_2 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),L_2),sequentially)) -> hAPP(fun(nat,X_a),X_a,lim(X_a),X_2) = ti(X_a,L_2))))) # label(fact_4455_limI) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 594 (all N all M all K_4 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_4)) -> (hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N) -> N = M))) # label(fact_661_mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 595 (all X_b all X_a (field(X_a) -> (all F all A_2 all Ra hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(X_a,fun(X_b,X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(X_a,fun(X_b,X_a)),combc(X_b,X_a,X_a),hAPP(fun(X_b,X_a),fun(X_b,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_b),inverse_divide(X_a)),F)),Ra)),A_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2)),Ra)))) # label(fact_4244_setsum__divide__distrib) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 596 (all X_a (preorder(X_a) -> (all X all Y (Y = X -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)))))) # label(fact_511_order__eq__refl) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 597 (all B all D_3 all A_1 (one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_3),A_1) -> (one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_3),B) -> one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B))))) # label(fact_3294_coprime__mult__nat) # label(axiom) # label(non_clause). [assumption]. 7.53/7.61 598 (all X_a (linord1117847801e_zero(X_a) -> (all A_3 all C all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C))) <-> (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_eq(X_a),B_2),A_3))) & (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2))))))) # label(fact_1859_divide__le__cancel) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 599 (all X_a (real_normed_vector(X_a) -> (all X all Y all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),X),Y)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Y),B))),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),A_1)),B))),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Y),B)))))) # label(fact_4691_scaleR_Oprod__diff__prod) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 600 (all X_a (number_ring(X_a) -> (all X ((-hBOOL(hAPP(nat,bool,even_odd_even(nat),X)) -> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,number_number_of(X_a),min)),X) = hAPP(int,X_a,number_number_of(X_a),min)) & (hBOOL(hAPP(nat,bool,even_odd_even(nat),X)) -> one_one(X_a) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,number_number_of(X_a),min)),X)))))) # label(fact_2512_neg__one__even__odd__power) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 601 (all X_b all X_a (real_n1866405975lgebra(X_b) & topological_space(X_a) -> (all F all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3)) -> (zero_zero(X_b) != hAPP(X_a,X_b,F,A_3) -> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),inverse_inverse(X_b)),F)),A_3))))))) # label(fact_4910_isCont__inverse) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 602 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,Ma)),Na)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)))) # label(fact_1390_Suc__le__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 603 (all X_a (plus(X_a) & linorder(X_a) & dvd(X_a) -> (all D_1 all Sb exists Z_2 all X_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Z_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),Sb))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),Sb)))))))) # label(fact_1214_minf_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 604 (all X_c all X_a all X_b all F all A_3 all B_2 all A_2 (hBOOL(hAPP(fun(product_prod(X_a,X_b),bool),bool,hAPP(product_prod(X_a,X_b),fun(fun(product_prod(X_a,X_b),bool),bool),member(product_prod(X_a,X_b)),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_3),B_2)),A_2)) -> hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),F,A_3),B_2)),hAPP(fun(product_prod(X_a,X_b),bool),fun(X_c,bool),hAPP(fun(product_prod(X_a,X_b),X_c),fun(fun(product_prod(X_a,X_b),bool),fun(X_c,bool)),image(product_prod(X_a,X_b),X_c),hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),F)),A_2))))) # label(fact_4008_pair__imageI) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 605 (all X_a all X_b (real_normed_field(X_b) -> (all F all Xa all F_2 all S (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S)) -> ((all N_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),N_1),S)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,X_b),fun(X_b,fun(X_b,bool)),deriv(X_b),hAPP(X_a,fun(X_b,X_b),hAPP(fun(X_b,fun(X_a,X_b)),fun(X_a,fun(X_b,X_b)),combc(X_b,X_a,X_b),F),N_1)),Xa),hAPP(X_a,X_b,hAPP(X_b,fun(X_a,X_b),F_2,Xa),N_1))))) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,X_b),fun(X_b,fun(X_b,bool)),deriv(X_b),hAPP(fun(X_a,bool),fun(X_b,X_b),hAPP(fun(X_b,fun(fun(X_a,bool),X_b)),fun(fun(X_a,bool),fun(X_b,X_b)),combc(X_b,fun(X_a,bool),X_b),hAPP(fun(X_b,fun(X_a,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),hAPP(fun(fun(X_a,X_b),fun(fun(X_a,bool),X_b)),fun(fun(X_b,fun(X_a,X_b)),fun(X_b,fun(fun(X_a,bool),X_b))),combb(fun(X_a,X_b),fun(fun(X_a,bool),X_b),X_b),big_co1399186613setsum(X_a,X_b)),F)),S)),Xa),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),hAPP(X_b,fun(X_a,X_b),F_2,Xa)),S)))))))) # label(fact_4331_DERIV__setsum) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 606 (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),ord_less(nat),hAPP(nat,nat,fact_fact(nat),M)),hAPP(nat,nat,fact_fact(nat),N)))))) # label(fact_2545_fact__less__mono__nat) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 607 (all X_a (real_normed_field(X_a) -> (all Y exists K_2 ((all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X_1),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_1)),K_2)))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2)))))) # label(fact_2049_divide_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 608 (all A_3 all X_2 all Xa (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),X_2),Xa),sequentially)) -> ((exists N_2 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),hAPP(nat,real,X_2,N_1))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),Xa))))) # label(fact_4517_LIMSEQ__le__const) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 609 (all X all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),Y)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,tan,Y)),hAPP(real,real,tan,X))))))) # label(fact_2448_tan__monotone) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 610 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) -> (M != N -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))))) # label(fact_403_le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 611 (all M (zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) | hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) | hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) | one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) # label(fact_1562_mod__exhaust__less__4) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 612 (all X hAPP(real,nat,natfloor,X) = hAPP(int,nat,nat_1,hAPP(real,int,archim2085082626_floor(real),X))) # label(fact_2886_natfloor__def) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 613 (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)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) -> ti(X_a,Y) = ti(X_a,X)))))) # label(fact_504_order__antisym) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 614 (all R_3 all A_1 all B hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),R_3),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),A_1)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),B))) # label(fact_4724_complex__scaleR) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 615 (all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),hAPP(real,real,sin,X)))) # label(fact_2286_sin__ge__minus__one) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 616 (all X_a (real_normed_vector(X_a) -> (all Y_2 all B_2 all X_2 all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),X_2),A_3),sequentially)) -> (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),Y_2),B_2),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),X_2)),Y_2)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_3),B_2)),sequentially))))))) # label(fact_4765_scaleR_OLIMSEQ) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 617 (all X_a (real_normed_vector(X_a) -> (all A_1 hAPP(real,real,abs_abs(real),hAPP(X_a,real,norm_norm(X_a),A_1)) = hAPP(X_a,real,norm_norm(X_a),A_1)))) # label(fact_1946_abs__norm__cancel) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 618 (all K_4 all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),N)))) # label(fact_3268_dvd__gcd__D2__nat) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 619 (all X_a (semiring_div(X_a) -> (all C_1 all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B))))) # label(fact_1286_mod__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 620 (all B all A_1 all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_1),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),B)) -> (exists D_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_2)) & (all Y_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y_1))),D_2)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y_1),B)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_1),Y_1))))))))) # label(fact_1646_lemma__interval__lt) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 621 (all X_a (ordere142940540dd_abs(X_a) -> (all A_3 (zero_zero(X_a) = ti(X_a,A_3) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_3)),zero_zero(X_a))))))) # label(fact_987_abs__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 622 (all X_a (comm_monoid_add(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),zero_zero(X_a))))) # label(fact_318_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 623 (all K_3 all K all F ((all N_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),K),hAPP(nat,real,F,N_1))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,F,N_1)),K_3)))) -> hBOOL(hAPP(fun(nat,real),bool,bseq(real),F)))) # label(fact_4339_BseqI2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 624 (all X_a (topological_space(X_a) & comm_monoid_add(X_a) -> (all F hAPP(fun(nat,X_a),X_a,suminf(X_a),F) = hAPP(fun(X_a,bool),X_a,the(X_a),hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F))))) # label(fact_4162_suminf__def) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 625 (all D_3 all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_3),A_1)))) # label(fact_2879_coprime__rmul2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 626 (all N all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M)) -> (M = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N) -> N = hAPP(nat,nat,suc,zero_zero(nat))))) # label(fact_1441_mn__eq__m__one) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 627 (all X_a (semiring_div(X_a) -> (all A_1 all B all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C_1))),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C_1)))) # label(fact_1295_mod__add__right__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 628 (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),hAPP(int,X_a,number_number_of(X_a),Wa)),hAPP(int,X_a,number_number_of(X_a),Va))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,number_number_of(X_a),Va)),hAPP(int,X_a,number_number_of(X_a),Wa))))))) # label(fact_250_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 629 (all Xa all Na (hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Xa),Na))) <-> hBOOL(hAPP(int,bool,even_odd_even(int),Xa)) & Na != zero_zero(nat))) # label(fact_1609_even__power) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 630 (all Z1 all Z2 all Z3 hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z1),Z2)),Z3) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z2),Z3))) # label(fact_368_zmult__assoc) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 631 (all V all W hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),hAPP(int,int,uminus_uminus(int),W))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,int,number_number_of(int),W))) # label(fact_2087_minus__numeral__code_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 632 (all X_a (comple44059174attice(X_a) -> (all Z all A_2 ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Z)))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(X_a,bool),X_a,complete_Sup_Sup(X_a),A_2)),Z)))))) # label(fact_2726_complete__lattice__class_OSup__least) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 633 (all X_a all G all B_2 all F all A_3 all Net (hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),A_3),Net)) -> (hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),G),B_2),Net)) -> hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),hAPP(fun(X_a,real),fun(X_a,complex),hAPP(fun(X_a,fun(real,complex)),fun(fun(X_a,real),fun(X_a,complex)),combs(X_a,real,complex),hAPP(fun(X_a,real),fun(X_a,fun(real,complex)),hAPP(fun(real,fun(real,complex)),fun(fun(X_a,real),fun(X_a,fun(real,complex))),combb(real,fun(real,complex),X_a),complex_1),F)),G)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_3),B_2)),Net))))) # label(fact_4530_tendsto__Complex) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 634 (all Ya all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Ya),Xa)) -> Xa = hAPP(fun(real,bool),real,complete_Sup_Sup(real),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastAtMost(real),Ya),Xa)))) # label(fact_4549_Sup__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 635 (all B all A_1 all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_1),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),B)) -> (exists D_2 ((all Y_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y_1))),D_2)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y_1),B)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_1),Y_1)))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_2))))))) # label(fact_1584_lemma__interval) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 636 (all Na hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(fun(nat,real),fun(nat,fun(real,real))),combs(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),div_div(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),zero_zero(real)))),hAPP(real,fun(nat,real),power_power(real),zero_zero(real)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,Na))) = one_one(real)) # label(fact_4267_sumr__cos__zero__one) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 637 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) -> N != zero_zero(nat))) # label(fact_143_gr__implies__not0) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 638 (all X_a (linordered_idom(X_a) -> (all Xa all A_3 all Ra (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),Ra)),Xa)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),Ra))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Xa),A_3))),Ra)))))) # label(fact_1138_abs__diff__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 639 (all X_a (linordered_idom(X_a) -> (all A_1 hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) # label(fact_1050_power2__abs) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 640 (all Z_1 all W hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,uminus_uminus(int),Z_1)),W) = hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z_1),W))) # label(fact_2061_zmult__zminus) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 641 (all K_4 hAPP(int,real,number_number_of(real),K_4) = hAPP(rat,real,ratreal,hAPP(int,rat,number_number_of(rat),K_4))) # label(fact_3785_Ratreal__number__collapse_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 642 (all X_a (linord219039673up_add(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),A_3))))))) # label(fact_232_zero__le__double__add__iff__zero__le__single__add) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 643 (all X all Y all M hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),Y)),M) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Y),M))),M)) # label(fact_1311_zdiff__zmod__right) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 644 (all Xa all Ya all Z (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Ya),Z))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Z)))) # label(fact_3257_gcd__semilattice__nat_Ole__inf__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 645 (all N zero_zero(real) = hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),pi))) # label(fact_2408_sin__npi) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 646 (all P_1 all A_3 all B_2 ((all D_2 (A_3 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B_2),D_2) -> hBOOL(hAPP(nat,bool,P_1,D_2)))) & (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_3),B_2)) -> hBOOL(hAPP(nat,bool,P_1,zero_zero(nat)))) <-> hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A_3),B_2))))) # label(fact_860_nat__diff__split) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 647 (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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),M))))) # label(fact_832_diff__less) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 648 (all B all A_1 (A_1 != zero_zero(nat) | zero_zero(nat) != B -> one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B))))) # label(fact_3316_div__gcd__coprime__nat) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 649 (all D_3 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_3),hAPP(int,int,abs_abs(int),D_3)))) # label(fact_1226_zdvd__self__abs1) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 650 (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),hAPP(int,int,bit1,K)),min)))) # label(fact_679_rel__simps_I30_J) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 651 (all M all N all K_4 hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)),K_4)) # label(fact_1529_mod__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 652 (all X all Y 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_eq(real),X),Y)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),Y)))))) # label(fact_2682_real__root__le__mono) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 653 (all I_1 all J_2 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I_1),J_2)),J_2))) # label(fact_3453_zgcd__zdvd2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 654 (all X_a all Xs all Ys (hAPP(list(X_a),nat,size_size(list(X_a)),Xs) != hAPP(list(X_a),nat,size_size(list(X_a)),Ys) -> Ys != Xs)) # label(fact_3581_neq__if__length__neq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 655 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,arcsin,Y)),pi)) & Y = hAPP(real,real,sin,hAPP(real,real,arcsin,Y)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,arcsin,Y)))))) # label(fact_2346_arcsin__pi) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 656 (all X_a (real_normed_algebra(X_a) -> (all B zero_zero(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),zero_zero(X_a)),B)))) # label(fact_1239_mult_Ozero__left) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 657 (all K_4 hAPP(int,int,bit0,K_4) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),K_4)) # label(fact_81_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 658 (all X_a (ring_div(X_a) -> (all A_1 all B all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C_1)),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),C_1)))) # label(fact_1300_mod__diff__left__eq) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 659 (all N (zero_zero(nat) != N -> hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),N)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),N))) # label(fact_1507_lemma__Suc__Suc__4n__diff__2) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 660 (all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),hAPP(nat,nat,suc,zero_zero(nat)))) <-> zero_zero(nat) = Na)) # label(fact_1375_less__Suc0) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 661 (all P_4 hAPP(product_prod(int,int),product_prod(int,int),hAPP(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_prod_case(int,int,product_prod(int,int)),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int)),uminus_uminus(int))),hAPP(rat,product_prod(int,int),quotient_of,P_4)) = hAPP(rat,product_prod(int,int),quotient_of,hAPP(rat,rat,uminus_uminus(rat),P_4))) # label(fact_4055_rat__uminus__code) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 662 (all X_a (real_algebra_1(X_a) & real_normed_vector(X_a) -> (all A_3 hBOOL(hAPP(real,bool,hAPP(fun(real,X_a),fun(real,bool),isCont(real,X_a),of_real(X_a)),A_3))))) # label(fact_4916_of__real_OisCont) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 663 (all D_3 all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_3),A_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_3),B))))) # label(fact_2895_coprime__divprod) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 664 (all X_a (number_ring(X_a) -> (all A_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),pls)),A_1) = ti(X_a,A_1)))) # label(fact_87_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 665 (all P_5 ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(product_prod(int,int),int,product_snd(int,int),P_5))) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(product_prod(int,int),int,product_fst(int,int),P_5)),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(product_prod(int,int),int,product_fst(int,int),P_5)),hAPP(product_prod(int,int),int,product_snd(int,int),P_5)))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(product_prod(int,int),int,product_snd(int,int),P_5)),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(product_prod(int,int),int,product_fst(int,int),P_5)),hAPP(product_prod(int,int),int,product_snd(int,int),P_5)))) = hAPP(product_prod(int,int),product_prod(int,int),normalize,P_5)) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(product_prod(int,int),int,product_snd(int,int),P_5))) -> (zero_zero(int) = hAPP(product_prod(int,int),int,product_snd(int,int),P_5) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) = hAPP(product_prod(int,int),product_prod(int,int),normalize,P_5)) & (zero_zero(int) != hAPP(product_prod(int,int),int,product_snd(int,int),P_5) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(product_prod(int,int),int,product_fst(int,int),P_5)),hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(product_prod(int,int),int,product_fst(int,int),P_5)),hAPP(product_prod(int,int),int,product_snd(int,int),P_5))))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(product_prod(int,int),int,product_snd(int,int),P_5)),hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(product_prod(int,int),int,product_fst(int,int),P_5)),hAPP(product_prod(int,int),int,product_snd(int,int),P_5))))) = hAPP(product_prod(int,int),product_prod(int,int),normalize,P_5))))) # label(fact_4371_normalize__def) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 666 (all M zero_zero(nat) != hAPP(nat,nat,suc,M)) # label(fact_1317_Suc__neq__Zero) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 667 (all X_a (real_normed_algebra(X_a) -> (exists K_2 ((all A_4 all B_4 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_4),B_4))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),A_4)),hAPP(X_a,real,norm_norm(X_a),B_4))),K_2)))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2)))))) # label(fact_2050_mult_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 668 (all X_a all X_b (comm_monoid_mult(X_b) -> (all G all H all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),G)),H)),A_2) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),H),A_2)))))) # label(fact_3981_setprod_Odistrib) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 669 (all X_a (sgn_if(X_a) -> zero_zero(X_a) = hAPP(X_a,X_a,sgn_sgn(X_a),zero_zero(X_a)))) # label(fact_2260_sgn0) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 670 (all R_3 hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,of_real(complex),R_3)),ii) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,zero_zero(real)),R_3)) # label(fact_2822_complex__of__real__i) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 671 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) -> (Xa = one_one(real) <-> hAPP(real,real,ln,Xa) = zero_zero(real)))) # label(fact_1722_ln__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.53/7.62 672 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),Xa)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,ln,Xa)))))) # label(fact_1723_ln__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 673 (all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,re,X)),hAPP(complex,real,norm_norm(complex),X)))) # label(fact_3023_complex__Re__le__cmod) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 674 (all X_c all X_b all X_a (real_normed_vector(X_a) & topological_space(X_c) & topological_space(X_b) -> (all G all L all F all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3)) -> (hBOOL(hAPP(filter(X_b),bool,hAPP(X_c,fun(filter(X_b),bool),hAPP(fun(X_b,X_c),fun(X_c,fun(filter(X_b),bool)),tendsto(X_b,X_c),G),L),hAPP(X_b,filter(X_b),at(X_b),hAPP(X_a,X_b,F,A_3)))) -> ((exists D_2 ((all X_1 (ti(X_a,X_1) != ti(X_a,A_3) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_1),A_3))),D_2)) -> hAPP(X_a,X_b,F,A_3) != hAPP(X_a,X_b,F,X_1))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_2)))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_c,fun(filter(X_a),bool),hAPP(fun(X_a,X_c),fun(X_c,fun(filter(X_a),bool)),tendsto(X_a,X_c),hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c)),combb(X_b,X_c,X_a),G),F)),L),hAPP(X_a,filter(X_a),at(X_a),A_3))))))))) # label(fact_4900_isCont__LIM__compose2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 675 (all N ti(int,N) = hAPP(real,int,archimedean_ceiling(real),hAPP(int,real,real_1(int),N))) # label(fact_3366_ceiling__real__of__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 676 (all X_a (archim1727834104eiling(X_a) -> (all X all Y (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archimedean_ceiling(X_a),X)),hAPP(X_a,int,archimedean_ceiling(X_a),Y))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)))))) # label(fact_3305_ceiling__less__cancel) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 677 (all X hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,X)) = hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) # label(fact_2306_minus__sin__cos__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 678 (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_eq(nat),N),M)) -> hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M),N) = hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),N))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),N))))) # label(fact_3114_divmod__nat__step) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 679 (all Xa all P_4 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),sRStar,P_4))) -> -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Xa),zero_zero(int)),P_4)))) # label(fact_1551_StandardRes__SRStar__prop1a) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 680 (all X_a (ring(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B))))) # label(fact_1906_minus__mult__commute) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 681 (all X_a (field_inverse_zero(X_a) -> (all Xa (hAPP(X_a,X_a,inverse_inverse(X_a),Xa) = one_one(X_a) <-> ti(X_a,Xa) = one_one(X_a))))) # label(fact_2566_inverse__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 682 (all X_a all F all S (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S)) -> ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),S)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(X_a,nat,F,X_1))))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co302546470etprod(X_a,nat),F),S)))))) # label(fact_3990_setprod__pos__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 683 (all X_a all G all A_3 all A_2 (hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),A_3),A_2)) -> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,complex),fun(X_a,real),hAPP(fun(complex,real),fun(fun(X_a,complex),fun(X_a,real)),combb(complex,real,X_a),im),G)),hAPP(complex,real,im,A_3)),A_2)))) # label(fact_4533_Im_Otendsto) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 684 (all X_a size_size(list(X_a)) = ti(fun(list(X_a),nat),size_size(list(X_a)))) # label(tsy_c_Nat_Osize__class_Osize_0_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 685 (all L all U hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),L),U)))) # label(fact_4558_finite__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 686 (all X_a (linord219039673up_add(X_a) -> (all A_3 (zero_zero(X_a) = ti(X_a,A_3) <-> zero_zero(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),A_3))))) # label(fact_65_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 687 (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)) -> hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),nat_tsub,X),Y)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,nat_1,X)),hAPP(int,nat,nat_1,Y))))) # label(fact_1266_Nat__Transfer_Otransfer__nat__int__functions_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 688 (all X_a (real_normed_vector(X_a) -> (all Xa (zero_zero(X_a) != ti(X_a,Xa) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(X_a,real,norm_norm(X_a),Xa))))))) # label(fact_1998_zero__less__norm__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 689 (all Xa (-hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)) <-> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd)))) # label(fact_1703_odd__iff__not__even) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 690 (all M hAPP(nat,nat,suc,M) != zero_zero(nat)) # label(fact_1314_Suc__not__Zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 691 (all X_a ti(fun(fun(real,fun(real,X_a)),fun(complex,X_a)),complex_case(X_a)) = complex_case(X_a)) # label(tsy_c_Complex_Ocomplex_Ocomplex__case_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 692 (all X_a (field_inverse_zero(X_a) & number(X_a) -> (all Wa all B_2 all C ((ti(X_a,C) != zero_zero(X_a) -> ti(X_a,B_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),C)) & (ti(X_a,C) = zero_zero(X_a) -> zero_zero(X_a) = hAPP(int,X_a,number_number_of(X_a),Wa)) <-> hAPP(int,X_a,number_number_of(X_a),Wa) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C))))) # label(fact_1777_eq__divide__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 693 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),K)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),hAPP(int,int,bit0,K))))) # label(fact_55_rel__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 694 (all Na all Ma (hBOOL(hAPP(int,bool,nat_is_nat,Ma)) -> (hBOOL(hAPP(int,bool,nat_is_nat,Na)) -> hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(int,nat,nat_1,Ma)),hAPP(int,nat,nat_1,Na))) = hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),Ma),Na)))) # label(fact_4864_SetInterval_Otransfer__int__nat__set__functions) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 695 (all B all D_3 all A_1 all C_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),D_3)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),C_1),D_3)))))) # label(fact_3266_gcd__semilattice__nat_Oinf__mono) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 696 (all K all F (hBOOL(hAPP(fun(nat,real),bool,summable(real),F)) -> ((all D_2 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))),D_2)))),hAPP(nat,real,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))),D_2)),one_one(nat)))))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K))),hAPP(fun(nat,real),real,suminf(real),F)))))) # label(fact_4298_sumr__pos__lt__pair) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 697 (all X_b all X_a (topological_space(X_a) & real_normed_field(X_b) -> (all Ya all G all L all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),inverse_divide(X_b)),G)),Ya)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),inverse_divide(X_b),L),Ya)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4579_divide_OLIM) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 698 (all N hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))))) # label(fact_2636_inv__real__of__nat__fact__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 699 (all X all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),pi)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,cos,X)),hAPP(real,real,cos,Y))))))) # label(fact_2287_cos__monotone__0__pi_H) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 700 (all X_a all F1 all Real1 all Real2 hAPP(complex,X_a,hAPP(fun(real,fun(real,X_a)),fun(complex,X_a),complex_rec(X_a),F1),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Real1),Real2)) = hAPP(real,X_a,hAPP(real,fun(real,X_a),F1,Real1),Real2)) # label(fact_2776_complex_Orecs) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 701 (all K_4 all L_4 ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L_4)) -> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),K_4),L_4) = hAPP(int,int,hAPP(int,fun(int,int),z3div,K_4),L_4)) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L_4)) -> hAPP(int,int,hAPP(int,fun(int,int),z3div,K_4),L_4) = hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),K_4),hAPP(int,int,uminus_uminus(int),L_4)))))) # label(fact_2188_z3div__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 702 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,K)),hAPP(int,int,bit1,L))))) # label(fact_280_rel__simps_I15_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 703 (all X_a all R_2 all F all Xa all Ya (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(X_a,nat,F,Xa)),hAPP(X_a,nat,F,Ya))) -> hBOOL(hAPP(fun(product_prod(X_a,X_a),bool),bool,hAPP(product_prod(X_a,X_a),fun(fun(product_prod(X_a,X_a),bool),bool),member(product_prod(X_a,X_a)),hAPP(X_a,product_prod(X_a,X_a),hAPP(X_a,fun(X_a,product_prod(X_a,X_a)),product_Pair(X_a,X_a),Xa),Ya)),hAPP(fun(product_prod(X_a,X_a),bool),fun(product_prod(X_a,X_a),bool),hAPP(fun(X_a,nat),fun(fun(product_prod(X_a,X_a),bool),fun(product_prod(X_a,X_a),bool)),mlex_prod(X_a),F),R_2))))) # label(fact_3837_mlex__less) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 704 (all N all P_5 (hBOOL(hAPP(nat,bool,prime,P_5)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,P_5),N)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),N)))) # label(fact_3087_prime__coprime__strong) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 705 (all X_a (topological_space(X_a) -> (all G all L all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),hAPP(fun(X_a,complex),fun(X_a,complex),hAPP(fun(complex,complex),fun(fun(X_a,complex),fun(X_a,complex)),combb(complex,complex,X_a),cnj),G)),hAPP(complex,complex,cnj,L)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4574_cnj_OLIM) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 706 (all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> hAPP(int,nat,phi,P_5) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))))) # label(fact_1216_phi__prime) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 707 (all X_a (linord1117847801e_zero(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C_1)))))))) # label(fact_1835_divide__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 708 (all X_b all X_a (real_vector(X_a) -> (all A_3 all G all A_2 hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_3)),G)),A_2) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_3),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),G),A_2))))) # label(fact_4756_scaleR__right_Osetsum) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 709 (all X_a (real_normed_algebra(X_a) -> (all R_3 all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),A_1)),B) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))))) # label(fact_4722_mult_OscaleR__left) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 710 (all U all Ma all Na all I all J (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I),U)),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J),U)),Na))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),I)),U)),Na)))))) # label(fact_866_nat__le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 711 (all X_a (ordered_ab_group_add(X_a) -> (all B_2 all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),B_2)),hAPP(X_a,X_a,uminus_uminus(X_a),A_3))))))) # label(fact_1932_neg__less__iff__less) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 712 (all N hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)))) # label(fact_550_two__realpow__ge__one) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 713 (all Y all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Y),X)) -> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,abs_abs(int),X)),hAPP(int,int,abs_abs(int),Y)) = hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),Y)))) # label(fact_1084_abs__div) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 714 (all V hAPP(complex,real,re,hAPP(int,complex,number_number_of(complex),V)) = hAPP(int,real,number_number_of(real),V)) # label(fact_3030_complex__Re__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 715 (all X_a (order(X_a) -> ti(fun(X_a,fun(X_a,fun(X_a,bool))),ord_atLeastAtMost(X_a)) = ord_atLeastAtMost(X_a))) # label(tsy_c_SetInterval_Oord__class_OatLeastAtMost_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 716 (all X all Y hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y)))) # label(fact_4425_gcd__ge__0__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 717 (all X_a (preorder(X_a) -> (all X all 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_eq(X_a),X),Y)))))) # label(fact_560_order__less__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 718 (all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1)) -> (exists P_6 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_6),A_1)) & hBOOL(hAPP(int,bool,zprime,P_6)))))) # label(fact_928_zprime__factor__exists) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 719 (all X_a all F all C all H ((all N_1 hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),C),H),N_1) = hAPP(nat,X_a,F,N_1)) -> hAPP(nat,X_a,F,zero_zero(nat)) = ti(X_a,C))) # label(fact_4359_def__nat__rec__0) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 720 (all L all U hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),L),U)))) # label(fact_4191_finite__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 721 (all X_a (ring_11004092258visors(X_a) -> (all Xa (hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) = ti(X_a,Xa) | ti(X_a,Xa) = one_one(X_a) <-> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),Xa) = one_one(X_a))))) # label(fact_1965_square__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 722 (all N all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),N)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N)))) -> N = hAPP(int,nat,nat_1,hAPP(real,int,archim2085082626_floor(real),X))))) # label(fact_2918_floor__eq4) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 723 (all Ya all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),plus_plus(real)),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),minus_minus(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),sin),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),plus_plus(real)),Ya)))),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),plus_plus(real)),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),sin)),hAPP(real,real,cos,Ya)))),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),cos)),hAPP(real,real,sin,Ya)))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),minus_minus(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),cos),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),plus_plus(real)),Ya)))),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),minus_minus(real)),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),cos)),hAPP(real,real,cos,Ya)))),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),sin)),hAPP(real,real,sin,Ya)))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X_1),zero_zero(real)))) # label(fact_3944_lemma__DERIV__sin__cos__add) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 724 (all X_a (comple44059174attice(X_a) -> ti(fun(fun(X_a,bool),X_a),complete_Sup_Sup(X_a)) = complete_Sup_Sup(X_a))) # label(tsy_c_Complete__Lattice_OSup__class_OSup_1_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 725 (all N all A_1 all B all P_5 (hBOOL(hAPP(nat,bool,prime,P_5)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_1),B)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),N)),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),N)),A_1)))))) # label(fact_3102_prime__divprod__pow) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 726 (all X all M hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),M)),pi)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,M))),pi)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) # label(fact_2454_sin__expansion__lemma) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 727 (all A_1 all B hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,uminus_uminus(int),B)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,uminus_uminus(int),A_1)),B)) # label(fact_2068_zdiv__zminus2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 728 (all D_3 all A_1 all B (hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_3),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)) = one_one(int) -> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_3),A_1) = one_one(int))) # label(fact_4415_coprime__lmult__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 729 (all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),N) = N) # label(fact_4990_max__0L) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 730 (all N (hBOOL(hAPP(nat,bool,even_odd_even(nat),N)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) # label(fact_2525_lemma__even__div2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 731 (all X_a (linorder(X_a) -> (all Ma all Na (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ma),Na)) <-> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),ord_lessThan(X_a),Ma)),hAPP(X_a,fun(X_a,bool),ord_lessThan(X_a),Na))))))) # label(fact_4347_lessThan__strict__subset__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 732 (all X_a ti(fun(fun(X_a,nat),fun(fun(product_prod(X_a,X_a),bool),fun(product_prod(X_a,X_a),bool))),mlex_prod(X_a)) = mlex_prod(X_a)) # label(tsy_c_Wellfounded_Omlex__prod_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 733 (all Xa all Ra (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ra)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),Ra)),Xa)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),Xa)),Ra)))) # label(fact_1991_abs__le__interval__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 734 (all X_a (archim1727834104eiling(X_a) -> (all Va all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),Va)),hAPP(X_a,int,archim2085082626_floor(X_a),Xa))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,number_number_of(X_a),Va)),Xa)))))) # label(fact_2912_number__of__le__floor) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 735 (all K_4 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,suc,zero_zero(nat))),K_4))) # label(fact_1272_dvd__1__left) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 736 (all I_1 all K_4 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),J_2)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),J_2)),K_4))) # label(fact_848_add__diff__assoc) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 737 (all X all K_4 all N all P_5 (hBOOL(hAPP(nat,bool,prime,P_5)) -> (zero_zero(nat) != N -> (hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),K_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),N) -> (exists I_2 hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),I_2) = X))))) # label(fact_3112_prime__power__exp) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 738 (all K_4 all I_1 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J_2),K_4)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,I_1)),K_4))))) # label(fact_1327_less__trans__Suc) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 739 (all X_a (linord581940658strict(X_a) -> (all A_3 all B_2 all 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_3),B_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B_2)))))))) # label(fact_485_mult__le__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 740 (all X_a (field_char_0(X_a) -> (all Z hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)),field_char_0_Rats(X_a)))))) # label(fact_3718_Rats__of__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 741 (all B all X all A_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),A_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),B)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)))))) # label(fact_3264_gcd__semilattice__nat_Ole__infI) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 742 (all X_b all X_a (metric_space(X_b) & metric_space(X_a) -> (all Ra all F all L_2 all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Ra)) -> (exists S_1 ((all X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X_1),A_3)),S_1)) & ti(X_a,X_1) != ti(X_a,A_3) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,hAPP(X_b,fun(X_b,real),dist_dist(X_b),hAPP(X_a,X_b,F,X_1)),L_2)),Ra)))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),S_1))))))))) # label(fact_4678_metric__LIM__D) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 743 (all R_3 all A_1 hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R_3),A_1) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,of_real(complex),R_3)),hAPP(real,complex,cis,A_1))) # label(fact_3198_rcis__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 744 (all X (X != zero_zero(real) -> hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,inverse_inverse(real),X)),X) = one_one(real))) # label(fact_2625_real__mult__inverse__left) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 745 (all A_1 all B hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),one_one(int)))) # label(fact_892_zcong__1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 746 (all X all Y hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,X)),hAPP(real,real,cos,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,sin,Y))) = hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y))) # label(fact_2252_sin__diff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 747 (all Ma all Ta all K (ti(int,K) != zero_zero(int) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Ma),Ta)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),Ma)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),Ta)))))) # label(fact_760_zdvd__mono) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 748 (all L all U hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),L),U)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,U)),L)) # label(fact_4563_card__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 749 (all M all N all R_4 all S_4 all S_3 all T_6 all T_5 all Rn all Sn_1 all Tn_1 all R_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),R_3)) -> (hAPP(product_prod(int,int),product_prod(int,product_prod(int,int)),hAPP(int,fun(product_prod(int,int),product_prod(int,product_prod(int,int))),product_Pair(int,product_prod(int,int)),Rn),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Sn_1),Tn_1)) = hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),hAPP(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))),hAPP(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))))),xzgcda,M),N),R_4),R_3),S_4),S_3),T_6),T_5) -> (ti(int,R_4) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),S_4),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),T_6),N)) -> (ti(int,R_3) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),S_3),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),T_5),N)) -> ti(int,Rn) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Sn_1),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Tn_1),N))))))) # label(fact_3110_xzgcda__linear) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 750 (all X_1 hAPP(real,real,inverse_inverse(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),X_1))) = hAPP(nat,real,hAPP(fun(nat,real),fun(nat,real),diffs(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat)))),X_1)) # label(fact_4146_exp__fdiffs) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 751 (all N hAPP(quickcheck_code_int,int,quickcheck_int_of,hAPP(int,quickcheck_code_int,quickcheck_of_int,N)) = ti(int,N)) # label(fact_3560_int__of__of__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 752 (all Z_1 all W (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),Z_1)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),W)))) # label(fact_205_zle__linear) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 753 (all X all Y all R_3 hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),R_3)),Y) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,X),Y)),hAPP(real,complex,of_real(complex),R_3))) # label(fact_2818_Complex__add__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 754 (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),hAPP(real,real,ln,X)),X)))) # label(fact_1707_ln__less__self) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 755 (all X_a (comm_semiring_1(X_a) -> (all A_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),one_one(X_a)) = ti(X_a,A_1)))) # label(fact_391_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 756 (all M all K_4 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K_4)),N)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)))) # label(fact_418_add__leD1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 757 (all X hAPP(int,real,real_1(int),hAPP(int,int,uminus_uminus(int),X)) = hAPP(real,real,uminus_uminus(real),hAPP(int,real,real_1(int),X))) # label(fact_3367_real__of__int__minus) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 758 (all A_1 all C_1 all D_3 all B (zero_zero(int) != ti(int,B) -> (ti(int,D_3) != zero_zero(int) -> hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),D_3)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C_1),B))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),D_3)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,C_1),D_3))))) # label(fact_3807_add__rat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 759 (all X_a all A_2 all B_1 (ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1) -> -(hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) -> -hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2))))) # label(fact_2202_equalityE) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 760 (all M all X hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),M) = hAPP(int,int,hAPP(int,fun(int,int),standardRes,M),X)) # label(fact_1274_StandardRes__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 761 (all X all Y (Y = hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) -> X = zero_zero(real))) # label(fact_2176_real__sqrt__sum__squares__eq__cancel2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.62 762 (all X_a (ring_1(X_a) -> (all A_1 all N hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N))))) # label(fact_2038_ring__1__class_Opower__minus__even) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 763 (all X_2 (hBOOL(hAPP(fun(nat,rat),bool,vanishes,X_2)) -> hBOOL(hAPP(fun(nat,rat),bool,vanishes,hAPP(fun(nat,rat),fun(nat,rat),hAPP(fun(rat,rat),fun(fun(nat,rat),fun(nat,rat)),combb(rat,rat,nat),uminus_uminus(rat)),X_2))))) # label(fact_4876_vanishes__minus) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 764 (all K_4 all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_4)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),N))))) # label(fact_863_dvd__mult__cancel) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 765 (all Z all Wa (hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),Z),Wa)) <-> Z != Wa & hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),Z),Wa)))) # label(fact_3735_less__rat__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 766 (all N zero_zero(real) != hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N))) # label(fact_2475_real__of__nat__fact__not__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 767 (all X_a (linordered_field(X_a) -> (all Z_1 all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),Y)),X)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)))))))) # label(fact_1872_mult__imp__less__div__pos) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 768 (all I_1 all J_2 -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),J_2)),I_1))) # label(fact_121_not__add__less1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 769 (all Xa all Ya (hAPP(quickcheck_code_int,int,quickcheck_int_of,Ya) = hAPP(quickcheck_code_int,int,quickcheck_int_of,Xa) <-> Xa = Ya)) # label(fact_3188_code__int_Oint__of__inject) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 770 (all U hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),zero_zero(int)),U)) = hAPP(int,nat,nat_1,U)) # label(fact_4285_card__atLeastZeroLessThan__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 771 (all X_a all X_b (comm_monoid_add(X_b) -> (all G all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2)) -> hAPP(X_a,X_b,G,X_1) = zero_zero(X_b))) -> zero_zero(X_b) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),A_2)))))) # label(fact_4334_setsum_Oneutral) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 772 (all X_a (archim1727834104eiling(X_a) -> (all Xa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Xa)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(X_a,int,archim2085082626_floor(X_a),Xa))))))) # label(fact_2873_one__less__floor) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 773 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) <-> hAPP(real,real,hAPP(real,fun(real,real),powr,Xa),one_one(real)) = Xa)) # label(fact_3404_powr__one__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 774 (all Z_1 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),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Z_1),X)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Z_1),Y))))) # label(fact_421_real__add__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 775 (all X_a all X_b all Xa all Ya all P_1 all Q_1 (hBOOL(hAPP(fun(X_a,fun(X_b,bool)),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),bool),ord_less_eq(fun(X_a,fun(X_b,bool))),P_1),Q_1)) -> (hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,Xa),Ya)) -> hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),Q_1,Xa),Ya))))) # label(fact_4367_predicate2D) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 776 (all X_a (linorder(X_a) -> (all A_1 all B all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),B),C_1))))) # label(fact_4837_min__max_Oinf_Oassoc) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 777 (all A_2 all K (hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less_eq(fun(nat,bool)),A_2),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),K),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),hAPP(fun(nat,bool),nat,finite_card(nat),A_2))))) -> hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),K),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),hAPP(fun(nat,bool),nat,finite_card(nat),A_2))) = A_2)) # label(fact_4259_subset__card__intvl__is__intvl) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 778 (all X_a (field_char_0(X_a) -> (all Na hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)),field_char_0_Rats(X_a)))))) # label(fact_3716_Rats__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 779 (all X_a (topological_space(X_a) -> (all F all L (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),suc)),L),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),L),sequentially)))))) # label(fact_4493_LIMSEQ__imp__Suc) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 780 (all Z1 all Z2 all W hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z1),W)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z2),W)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z1),Z2)),W)) # label(fact_408_zadd__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 781 (all X_a (comm_ring_1(X_a) -> (all X (hBOOL(hAPP(nat,bool,even_odd_even(nat),X)) -> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),X) = one_one(X_a))))) # label(fact_2493_minus__one__even__power) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 782 (all Xa all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)),one_one(real))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),one_one(real)))))) # label(fact_2705_real__root__lt__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 783 (all X_a (power(X_a) -> times_times(X_a) = ti(fun(X_a,fun(X_a,X_a)),times_times(X_a)))) # label(tsy_c_Groups_Otimes__class_Otimes_0_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 784 (all X_a all C all A_2 (-hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),A_2)) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2))))) # label(fact_2234_ComplI) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 785 (all X_a all X_b all P_1 hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),hilbert_Eps(product_prod(X_a,X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),hAPP(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,bool)),fun(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,bool))),combb(fun(X_b,product_prod(X_a,X_b)),fun(X_b,bool),X_a),hAPP(fun(product_prod(X_a,X_b),bool),fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,bool)),combb(product_prod(X_a,X_b),bool,X_b),P_1)),product_Pair(X_a,X_b)))) = hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),hilbert_Eps(product_prod(X_a,X_b)),P_1)) # label(fact_4806_split__paired__Eps) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 786 (all Ma all Z (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),Z)))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,semiring_1_of_nat(int),Ma)),Z)))) # label(fact_967_int__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 787 (all A_3 all B_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_3),B_2)) <-> (exists X_1 exists Y_1 (one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_3),X_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B_2),Y_1)) | one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B_2),X_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_3),Y_1)))))) # label(fact_2921_coprime__bezout) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 788 (all F all L all C (hBOOL(hAPP(filter(real),bool,hAPP(real,fun(filter(real),bool),hAPP(fun(real,real),fun(real,fun(filter(real),bool)),tendsto(real,real),F),L),hAPP(real,filter(real),at(real),C))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),L),zero_zero(real))) -> (exists R_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1)) & (all X_1 (C != X_1 & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),C),X_1))),R_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,X_1)),zero_zero(real)))))))))) # label(fact_4652_LIM__fun__less__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 789 (all X_a all X_b all P_1 all Q_1 all X_1 (hBOOL(hAPP(product_prod(X_a,X_b),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_a),hAPP(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),combb(bool,bool,X_b),hAPP(bool,fun(bool,bool),fconj,P_1))),Q_1)),X_1)) <-> hBOOL(P_1) & hBOOL(hAPP(product_prod(X_a,X_b),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),Q_1),X_1)))) # label(fact_4071_split__part) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 790 (all Xa ((exists N_1 (-hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1)) & Xa = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) | (exists N_1 (-hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1)) & hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = Xa)) <-> zero_zero(real) = hAPP(real,real,cos,Xa))) # label(fact_2467_cos__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 791 (all X_a (real_algebra_1(X_a) -> (all Z_1 hAPP(int,X_a,ring_1_of_int(X_a),Z_1) = hAPP(real,X_a,of_real(X_a),hAPP(int,real,ring_1_of_int(real),Z_1))))) # label(fact_3598_of__real__of__int__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 792 (all Na all Va (-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))) & (-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,Na)),hAPP(int,nat,number_number_of(nat),Va))))) # label(fact_3924_le__Suc__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 793 (all B_2 all A_3 all Ma (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,A_3),Ma))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),A_3)))) # label(fact_3231_Bnor__mem__zle) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 794 (all X_a (cancel_semigroup_add(X_a) -> (all A_1 all B all C_1 (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B) -> ti(X_a,B) = ti(X_a,C_1))))) # label(fact_292_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 795 (all X_a (ordere223160158up_add(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),B))))))) # label(fact_328_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 796 (all X_a (archim1727834104eiling(X_a) -> (all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),hAPP(X_a,int,archimedean_ceiling(X_a),Xa))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Xa)))))) # label(fact_3325_one__le__ceiling) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 797 (all Ma all Na all K (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Na)))))) # label(fact_662_nat__mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 798 (all X_a (archim1727834104eiling(X_a) -> (all Xa all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archim2085082626_floor(X_a),Xa)),Z)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)),one_one(X_a)))))))) # label(fact_3634_floor__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 799 (all X_b all X_a (semiring_1(X_a) -> (all F all A_2 hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(fun(X_b,bool),nat,hAPP(fun(X_b,nat),fun(fun(X_b,bool),nat),big_co1399186613setsum(X_b,nat),F),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,nat),fun(X_b,X_a),hAPP(fun(nat,X_a),fun(fun(X_b,nat),fun(X_b,X_a)),combb(nat,X_a,X_b),semiring_1_of_nat(X_a)),F)),A_2)))) # label(fact_4218_of__nat__setsum) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 800 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> hAPP(real,real,inverse_inverse(real),hAPP(real,real,sqrt,X)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sqrt,X)),X))) # label(fact_2635_sqrt__divide__self__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 801 (all A_1 all B hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,A_1),B))) # label(fact_3141_mod__int__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 802 (all X_a (linordered_field(X_a) -> (all Z_1 all W all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),W)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),W),Z_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Z_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Y),W)))))))))) # label(fact_1862_frac__less2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 803 (all Xa all A_2 all Ma (hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),A_2),hAPP(int,fun(fun(int,bool),bool),rsetR,Ma))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma)) -> (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,Xa),Ma) -> hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,int),fun(fun(int,bool),fun(int,bool)),image(int,int),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),times_times(int)),Xa)),A_2)),hAPP(int,fun(fun(int,bool),bool),rsetR,Ma))))))) # label(fact_3929_RsetR__zmult__mono) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 804 (all X_a (real_normed_algebra(X_a) -> (all Ya all A_3 hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),times_times(X_a)),Ya)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),Ya)),hAPP(X_a,filter(X_a),at(X_a),A_3)))))) # label(fact_4588_mult__left_Ocont) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 805 (all X_a (comm_semiring_1(X_a) -> (all X one_one(X_a) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),zero_zero(nat))))) # label(fact_196_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 806 (all N ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),N),zero_zero(int))) -> hAPP(int,int,zfact,N) = one_one(int)) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),N),zero_zero(int))) -> hAPP(int,int,zfact,N) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),N),hAPP(int,int,zfact,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),N),one_one(int))))))) # label(fact_1137_zfact_Osimps) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 807 (all A_1 all X hAPP(real,real,hAPP(real,fun(real,real),log,A_1),X) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,ln,X)),hAPP(real,real,ln,A_1))) # label(fact_1996_Log_Olog__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 808 (all P_4 (hBOOL(hAPP(int,bool,zprime,P_4)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),P_4),zOdd)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))))) # label(fact_1228_zprime__zOdd__eq__grt__2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 809 (all X (hBOOL(hAPP(nat,bool,even_odd_even(nat),X)) -> X = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),X),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))))) # label(fact_2516_even__nat__div__two__times__two) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 810 (all X all Y hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),hAPP(int,int,abs_abs(int),Y))) # label(fact_4392_gcd__abs2__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 811 (all X_a (real_normed_field(X_a) -> (all N all Z_1 all K_5 all H_2 (zero_zero(X_a) != ti(X_a,H_2) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),Z_1)),K_5)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z_1),H_2))),K_5)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z_1),H_2)),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1),N))),H_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(nat,nat,suc,zero_zero(nat)))))))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,semiring_1_of_nat(real),N)),hAPP(nat,real,semiring_1_of_nat(real),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(nat,nat,suc,zero_zero(nat)))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),K_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(X_a,real,norm_norm(X_a),H_2)))))))))) # label(fact_1895_lemma__termdiff3) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 812 (all K_4 all L_4 ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L_4)) -> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),K_4),L_4) = hAPP(int,int,hAPP(int,fun(int,int),z3mod,K_4),L_4)) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L_4)) -> hAPP(int,int,hAPP(int,fun(int,int),z3mod,K_4),L_4) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),K_4),hAPP(int,int,uminus_uminus(int),L_4))))) # label(fact_2187_z3mod__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 813 (all Z_1 all N (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Z_1),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_1),N))))) # label(fact_778_zdvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 814 (all X_a (semiring_1(X_a) -> -hBOOL(hAPP(X_a,bool,iszero(X_a),one_one(X_a))))) # label(fact_2152_not__iszero__1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 815 (all N -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N))) # label(fact_132_less__irrefl__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 816 (exists K_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2)) & (all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,cnj,X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_1)),K_2)))))) # label(fact_2991_cnj_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 817 (all X_a (real_normed_algebra(X_a) -> (all X all Y all Ya_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Ya_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Ya_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y)),Ya_1)))) # label(fact_1246_mult__left_Oadd) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 818 (all A_1 all B hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,expi,A_1)),hAPP(complex,complex,expi,B)) = hAPP(complex,complex,expi,hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A_1),B))) # label(fact_2828_expi__add) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 819 (all X_a (metric_space(X_a) -> (all Xa all Ya (ti(X_a,Xa) = ti(X_a,Ya) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Xa),Ya)),zero_zero(real))))))) # label(fact_3534_dist__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 820 (all Xa all A_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(real,int,archim2085082626_floor(real),Xa)),A_3)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),hAPP(int,real,real_1(int),A_3))))) # label(fact_3386_floor__less__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 821 (all X_a (archim1727834104eiling(X_a) -> (all X hAPP(int,int,uminus_uminus(int),hAPP(X_a,int,archimedean_ceiling(X_a),X)) = hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X))))) # label(fact_3318_floor__minus) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 822 (all X all Y hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,cnj,X)),hAPP(complex,complex,cnj,Y)) = hAPP(complex,complex,cnj,hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),X),Y))) # label(fact_2969_complex__cnj__mult) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 823 (all M all K_4 all N hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M)),N)) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),N)) # label(fact_4383_gcd__add__mult__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 824 (all X_a (semiring_1(X_a) -> ti(fun(X_a,bool),iszero(X_a)) = iszero(X_a))) # label(tsy_c_Int_Oiszero_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 825 (all X_b all X_a all A_1 all B hAPP(product_prod(X_b,X_a),X_a,product_snd(X_b,X_a),hAPP(X_a,product_prod(X_b,X_a),hAPP(X_b,fun(X_a,product_prod(X_b,X_a)),product_Pair(X_b,X_a),A_1),B)) = ti(X_a,B)) # label(fact_3156_snd__conv) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 826 (all C_1 all A_1 all B (A_1 = B -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),C_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C_1))))) # label(fact_952_dvd_Oord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 827 (all B all B_5 all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_5),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B_5)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B))))))) # label(fact_1104_zdiv__mono2__neg) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 828 (all M all N (hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N) = M -> N = zero_zero(nat))) # label(fact_156_add__eq__self__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 829 (all X all Y hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),Y)))) # label(fact_3371_powr__ge__pzero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 830 (all X_a (ring_1(X_a) -> (all C all X_1 hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),C),X_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,suc,X_1))),hAPP(nat,X_a,C,hAPP(nat,nat,suc,X_1)))))) # label(fact_4145_diffs__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 831 (all Xa all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Xa)) | zero_zero(nat) = Na <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Xa),Na))))) # label(fact_146_nat__zero__less__power__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 832 (all K_4 all I_1 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_4)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),I_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),J_2)))))) # label(fact_626_mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 833 (all X_a (real_normed_field(X_a) -> (all C all F all Xa all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),inverse_divide(X_a)),F)),C)),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),D),C))))))) # label(fact_3872_DERIV__cdivide) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 834 (all M all K_4 all N (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),K_4),N) -> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M)),N) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),N))) # label(fact_4413_gcd__mult__cancel__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 835 (all Xa all A_3 (hBOOL(hAPP(fun(nat,real),bool,monoseq(real),A_3)) -> (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A_3),Xa),sequentially)) -> (all M_2 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_2),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,A_3,M_2)),hAPP(nat,real,A_3,N_1))))) & (all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,A_3,N_1)),Xa))) | (all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),hAPP(nat,real,A_3,N_1)))) & (all M_2 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_2),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,A_3,N_1)),hAPP(nat,real,A_3,M_2)))))))) # label(fact_4515_monoseq__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 836 -(all M1 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),v),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),w),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) != hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),M1)) # label(fact_1155__096_B_Bthesis_O_A_I_B_Bm1_O_Av_A_094_A2_A_L_Aw_A_094_A2_A_061_A_I1_A_) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 837 (all X_a (number_ring(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls)))))) # label(fact_494_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 838 (all X_a (real_normed_vector(X_a) -> (all A_1 all B hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(X_a,real,norm_norm(X_a),A_1)),hAPP(X_a,real,norm_norm(X_a),B)))),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B))))))) # label(fact_2033_norm__triangle__ineq3) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 839 (all Xa all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Xa),Na))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Xa)) | zero_zero(nat) = Na)) # label(fact_150_zero__less__power__nat__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 840 (all X all Y hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,Y)),hAPP(real,real,sin,X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,Y)),hAPP(real,real,cos,X))) = hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y))) # label(fact_2253_sin__diff2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 841 (all X_a (linord20386208strict(X_a) -> (all C_1 all D_3 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D_3)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D_3)))))))))) # label(fact_482_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 842 (all Xa hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),sin)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),Xa),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,cos,Xa))),hAPP(real,real,sin,Xa))))) # label(fact_3937_DERIV__sin__realpow2a) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 843 (all Z all Wa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,nat_1,Wa)),hAPP(int,nat,nat_1,Z)))))) # label(fact_1032_nat__less__eq__zless) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 844 (all Z all Wa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z),Wa)) <-> ti(int,Z) != ti(int,Wa) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),Wa)))) # label(fact_223_zless__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 845 (all X_a (metric_space(X_a) -> (all X_2 hAPP(fun(X_a,bool),X_a,the(X_a),hAPP(filter(nat),fun(X_a,bool),hAPP(fun(X_a,fun(filter(nat),bool)),fun(filter(nat),fun(X_a,bool)),combc(X_a,filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2)),sequentially)) = hAPP(fun(nat,X_a),X_a,lim(X_a),X_2)))) # label(fact_4454_lim__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 846 (all X_b all X_a (comm_monoid_mult(X_a) -> (all A_2 one_one(X_a) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),hAPP(X_a,fun(X_b,X_a),combk(X_a,X_b),one_one(X_a))),A_2)))) # label(fact_3974_setprod__1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 847 (all X_a all X_b all F all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2)),B_1)) -> hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(fun(X_a,bool),bool),fun(fun(fun(X_a,bool),bool),bool),ord_less_eq(fun(fun(X_a,bool),bool)),hAPP(fun(fun(X_b,bool),bool),fun(fun(X_a,bool),bool),hAPP(fun(fun(X_b,bool),fun(X_a,bool)),fun(fun(fun(X_b,bool),bool),fun(fun(X_a,bool),bool)),image(fun(X_b,bool),fun(X_a,bool)),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F)),hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),pow(X_b),A_2))),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),B_1))))) # label(fact_3662_image__Pow__mono) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 848 (all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(nat,real,real_1(nat),hAPP(real,nat,natceiling,X))))) # label(fact_2390_real__natceiling__ge) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 849 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,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]. 7.62/7.63 850 (all A_3 all B_2 all C (zero_zero(real) != C -> (A_3 = B_2 <-> hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),B_2) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),A_3)))) # label(fact_609_real__mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 851 (all X_a (number_semiring(X_a) -> (all V (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V)) -> hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,succ,V)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),one_one(X_a)))))) # label(fact_270_semiring__number__of__add__1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 852 (all X_a all X_b (order(X_b) & order(X_a) -> (all C all A_3 all F all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),hAPP(X_b,X_a,F,B_2))) -> (hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),B_2),C)) -> ((all X_1 all Y_1 (hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_1),Y_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,X_1)),hAPP(X_b,X_a,F,Y_1))))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),hAPP(X_b,X_a,F,C))))))))) # label(fact_2854_order__le__less__subst1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 853 (all Y all X all U_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),X)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),U_1),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),U_1),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),U_1))))) # label(fact_2185_real__sqrt__sum__squares__less) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 854 (all X_a (linordered_field(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C_1)))))))) # label(fact_1838_divide__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 855 (all P_5 hAPP(rat,product_prod(int,int),quotient_of,hAPP(rat,rat,sgn_sgn(rat),P_5)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,sgn_sgn(int),hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(rat,product_prod(int,int),quotient_of,P_5)))),one_one(int))) # label(fact_3737_rat__sgn__code) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 856 (all Xa hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),arctan),Xa),hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) # label(fact_2656_DERIV__arctan) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 857 (all X_a (real_normed_field(X_a) -> (all F all Xa all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D)) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),inverse_divide(X_a)),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),minus_minus(X_a)),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),F),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Xa)))),hAPP(X_a,X_a,F,Xa)))),combi(X_a))),D),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a)))))))) # label(fact_4644_DERIV__D) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 858 (all X_b all X_a (topological_space(X_a) & real_normed_vector(X_b) -> (all F all L all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),hAPP(X_b,real,norm_norm(X_b),L)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4616_LIM__norm) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 859 (all X_a (linordered_idom(X_a) -> (all L_4 all K_4 (hAPP(X_a,X_a,abs_abs(X_a),L_4) = hAPP(X_a,X_a,abs_abs(X_a),K_4) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),L_4),K_4)))))) # label(fact_979_dvd__if__abs__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 860 (all Ma all Wa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,nat_1,Wa)),Ma)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),hAPP(nat,int,semiring_1_of_nat(int),Ma)))))) # label(fact_1055_nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 861 (all Y all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Y),X)) -> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y) = hAPP(int,int,abs_abs(int),Y))) # label(fact_4388_gcd__proj2__if__dvd__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 862 (all A_3 hBOOL(hAPP(complex,bool,hAPP(fun(complex,real),fun(complex,bool),isCont(complex,real),re),A_3))) # label(fact_4934_Re_OisCont) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 863 (all Ra hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),plus_plus(real),Ra)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),suc)))),Ra),sequentially))) # label(fact_4508_LIMSEQ__inverse__real__of__nat__add) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 864 (all X_a all C_2 all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),B_1),C_2)) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),C_2))))) # label(fact_2247_psubset__trans) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 865 (all Z_1 exists A_4 exists R_1 hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,of_real(complex),R_1)),hAPP(complex,complex,expi,A_4)) = Z_1) # label(fact_2939_complex__expi__Ex) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 866 (all Ta all B_1 all D_1 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),D)) -> (all X_1 ((all Xa_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D))) -> (all Xb (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1)) -> hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa_1) != ti(int,X_1))))) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),Ta))) -> -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),D)),Ta)))))))) # label(fact_4673_bset_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 867 (all X_a (ordered_ring(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))))))) # label(fact_438_split__mult__pos__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 868 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,K)),pls)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),pls)))) # label(fact_54_rel__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 869 (all X_a (linord581940658strict(X_a) -> (all C_1 all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B)))))))) # label(fact_456_mult__strict__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 870 (all X_a (real_normed_algebra(X_a) -> (all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),zero_zero(X_a)),Y) = zero_zero(X_a)))) # label(fact_1240_mult__left_Ozero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 871 (all X_a (linord1117847801e_zero(X_a) & number_ring(X_a) -> (all Ra (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ra),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Ra)))))) # label(fact_1807_half__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 872 (all X_a (semiring_div(X_a) -> (all A_1 all B all C_1 (ti(X_a,C_1) != zero_zero(X_a) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B))))) # label(fact_1086_div__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 873 (all Ta all A_2 all D_1 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),D)) -> (all X_1 ((all Xa_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D))) -> (all Xb (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2)) -> hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa_1) != ti(int,X_1))))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),Ta))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),D)),Ta)))))))) # label(fact_4675_aset_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 874 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 all C_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(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),B),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)))))))) # label(fact_227_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 875 (all X_a (group_add(X_a) -> (all A_1 all B ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),B)))) # label(fact_739_add__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 876 (all X_a all X_b (ab_group_add(X_b) -> (all F all B_1 all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2)) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),B_1))))))) # label(fact_4260_setsum__diff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 877 (all N hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),N))) # label(fact_357_le__refl) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 878 (all X_a (archim1727834104eiling(X_a) -> (all V hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(int,X_a,number_number_of(X_a),V)) = hAPP(int,int,number_number_of(int),V)))) # label(fact_3308_ceiling__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 879 (all A_1 all N all B hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A_1),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),N)) # label(fact_3274_gcd__exp__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 880 (all X all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y)) -> (exists R_1 exists A_4 (X = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_1),hAPP(real,real,cos,A_4)) & hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_1),hAPP(real,real,sin,A_4)) = Y)))) # label(fact_2560_polar__ex1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 881 (all M all N hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),M),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(nat,nat,suc,M)),hAPP(nat,nat,suc,N))) # label(fact_4829_min__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 882 (all X_a (comm_semiring_1(X_a) -> (all X hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X)))) # label(fact_548_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 883 (all X_a all A_2 ti(fun(X_a,bool),A_2) = hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),A_2))) # label(fact_2847_Union__Pow__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 884 (all N N = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,N)),one_one(nat))) # label(fact_1401_diff__Suc__1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 885 (all Na (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),Na)),zero_zero(real))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Na),zero_zero(int))))) # label(fact_3411_real__of__int__le__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 886 (all X_a (linordered_idom(X_a) -> (all X all Y all N (-hBOOL(hAPP(nat,bool,even_odd_even(nat),N)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),N)))))))) # label(fact_2476_power__mono__odd) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 887 (all X_a all A_3 all P_1 ((exists X_1 (hBOOL(hAPP(X_a,bool,P_1,X_1)) & (all Y_1 (hBOOL(hAPP(X_a,bool,P_1,Y_1)) -> ti(X_a,X_1) = ti(X_a,Y_1))))) -> (hBOOL(hAPP(X_a,bool,P_1,A_3)) -> ti(X_a,A_3) = hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1)))) # label(fact_4893_some1__equality) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 888 (all N all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(nat,real,real_1(nat),N))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(real,nat,natfloor,X)),N))))) # label(fact_2421_less__natfloor) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 889 (all X_a (group_add(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),B) = ti(X_a,A_1)))) # label(fact_740_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 890 (all K all L hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),K),L) = hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),hAPP(bool,fun(int,fun(int,int)),if(int),hAPP(int,bool,hAPP(int,fun(int,bool),fequal(int),L),zero_zero(int))),K),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),L),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,abs_abs(int),K)),hAPP(int,int,abs_abs(int),L)))))) # label(fact_4428_gcd__code__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 891 (all X all Y (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),hAPP(real,real,arctan,X)),hAPP(real,real,arctan,Y))))) # label(fact_2135_arctan__monotone) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 892 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,sqrt,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = X)) # label(fact_2168_real__sqrt__pow2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 893 (all X_a (comm_semiring_1(X_a) -> (all Lx all Rx all Ry hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),Ry)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Rx)),Ry)))) # label(fact_360_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 894 (all X_a (ring_1(X_a) -> (all P_1 all Q_2 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Q_2),ring_1_Ints(X_a))) -> ((all Z_2 hBOOL(hAPP(X_a,bool,P_1,hAPP(int,X_a,ring_1_of_int(X_a),Z_2)))) -> hBOOL(hAPP(X_a,bool,P_1,Q_2))))))) # label(fact_3838_Ints__induct) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 895 (all X_a (number_ring(X_a) -> (all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X)),Y)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) # label(fact_787_power2__diff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 896 (all X_a (comple44059174attice(X_a) -> (all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(X_a,bool),X_a,complete_Sup_Sup(X_a),A_2)),hAPP(fun(X_a,bool),X_a,complete_Sup_Sup(X_a),B_1))))))) # label(fact_1816_Sup__subset__mono) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 897 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,arcsin,Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) # label(fact_2231_arcsin__ubound) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 898 (all X_b all X_a (real_normed_vector(X_b) & topological_space(X_a) -> (all G all F all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),plus_plus(X_b)),F)),G)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3)))))))) # label(fact_4604_LIM__add__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 899 (all X_b all X_a all F all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),F),A_2))),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2))))) # label(fact_3666_card__image__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 900 (all X_a (real_algebra_1(X_a) -> of_real(X_a) = ti(fun(real,X_a),of_real(X_a)))) # label(tsy_c_RealVector_Oof__real_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 901 (all X all Y hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,abs_abs(int),X)),Y) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y)) # label(fact_4393_gcd__abs1__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 902 (all U_1 all M all N all I_1 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),I_1)),U_1)),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I_1),U_1)),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_2),U_1)),N)))) # label(fact_865_nat__diff__add__eq2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 903 (all X_a (linordered_idom(X_a) -> (all A_3 (hAPP(X_a,X_a,sgn_sgn(X_a),A_3) = hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a))))))) # label(fact_2317_sgn__1__neg) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 904 (all X_a (real_normed_algebra(X_a) -> (all Ya all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_2)) -> hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),X_2)),Ya))))))) # label(fact_4128_mult__left_OCauchy) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 905 (all X_a all One all Times all A_3 all Na hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),Times,A_3),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),hAPP(X_a,fun(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a))),power_1(X_a),One),Times),A_3),Na)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),hAPP(X_a,fun(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a))),power_1(X_a),One),Times),A_3),hAPP(nat,nat,suc,Na))) # label(fact_2867_power_Opower_Opower__Suc) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 906 (all X all Y all R_3 hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,X),Y)),hAPP(real,complex,of_real(complex),R_3)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),R_3)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Y),R_3))) # label(fact_2817_Complex__mult__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 907 (all X_b all X_a (uminus(X_a) -> (all A_2 all Xa hAPP(X_b,X_a,hAPP(fun(X_b,X_a),fun(X_b,X_a),uminus_uminus(fun(X_b,X_a)),A_2),Xa) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_b,X_a,A_2,Xa))))) # label(fact_2990_uminus__apply) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 908 (all X_a (ring_11004092258visors(X_a) -> (all A_3 (hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_3),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = one_one(X_a) <-> hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) = ti(X_a,A_3) | ti(X_a,A_3) = one_one(X_a))))) # label(fact_2037_power2__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 909 (all X_a (ordered_ring(X_a) -> (all A_3 all E_2 all C all B_2 all D_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B_2),A_3)),E_2)),D_1))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),E_2)),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),E_2)),D_1))))))) # label(fact_774_less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 910 (all X all A_1 hAPP(real,real,hAPP(real,fun(real,real),powr,X),hAPP(real,real,uminus_uminus(real),A_1)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1))) # label(fact_3406_powr__minus__divide) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 911 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),one_one(real))) -> hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(real,real,cos,hAPP(real,real,arcsin,X))))) # label(fact_2250_cos__arcsin) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 912 (all X_a (field(X_a) -> (all X all Y all Z_1 (zero_zero(X_a) != ti(X_a,Z_1) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Y),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),X)),Y)),Z_1))))) # label(fact_1880_add__divide__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 913 (all I_1 all K_4 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),J_2)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_4)),J_2) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),K_4)))) # label(fact_844_diff__diff__right) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 914 (all X_a (field_inverse_zero(X_a) -> (all A_1 all B all C_1 (ti(X_a,C_1) != zero_zero(X_a) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B))))) # label(fact_1851_mult__divide__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 915 (all Z_1 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_1)))) # label(fact_267_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 916 (all N all D_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),D_3)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_3),N)) -> hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,real_1(nat),N)),hAPP(nat,real,real_1(nat),D_3)) = hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N),D_3))))) # label(fact_2419_real__of__nat__div) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 917 (all I_1 all J_2 hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I_1),hAPP(int,int,uminus_uminus(int),J_2)) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I_1),J_2)) # label(fact_3469_zgcd__zminus2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 918 (all X 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),zero_zero(nat)),N)) -> hAPP(real,real,hAPP(nat,fun(real,real),root,M),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)) = hAPP(real,real,hAPP(nat,fun(real,real),root,N),hAPP(real,real,hAPP(nat,fun(real,real),root,M),X))))) # label(fact_2677_real__root__commute) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 919 (all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A_3),zero_zero(real)),sequentially)) -> (hBOOL(hAPP(fun(nat,real),bool,monoseq(real),A_3)) -> hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3)))))) # label(fact_4524_summable__Leibniz_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 920 (all X 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_eq(real),zero_zero(real)),X)) -> hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)),N) = X))) # label(fact_2713_real__root__pow__pos2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 921 (all X_a all A_2 all X_2 all C_2 (hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),X_2),C_2)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),X_2)) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),C_2)))))) # label(fact_1746_UnionI) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 922 (all Na (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Na)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(int,real,real_1(int),Na))))) # label(fact_3409_real__of__int__gt__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 923 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 all B hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B))))) # label(fact_975_abs__add__abs) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 924 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real))) -> Y = hAPP(real,real,cos,hAPP(real,real,arccos,Y)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,arccos,Y)),pi)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,arccos,Y)))))) # label(fact_2360_arccos) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 925 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),zero_zero(X_a)))))))) # label(fact_242_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 926 (all Z all Xa all X_2 (hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),Xa),X_2)) -> ((all X_1 (hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_1),X_2)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),Z)))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),hAPP(fun(real,bool),real,complete_Sup_Sup(real),X_2)))))) # label(fact_3722_SupInf_OSup__upper) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 927 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),Y)) -> (exists X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Y),one_one(real)))) & hAPP(real,real,exp(real),X_1) = Y & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X_1)))))) # label(fact_1820_lemma__exp__total) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 928 (all X_a (comm_monoid_add(X_a) & power(X_a) -> (all Xa all Ya all Na hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa)),hAPP(nat,fun(nat,nat),minus_minus(nat),Na)))),hAPP(X_a,fun(nat,X_a),power_power(X_a),Ya))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,Na))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Ya)),hAPP(nat,fun(nat,nat),minus_minus(nat),Na)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,Na)))))) # label(fact_4201_lemma__realpow__rev__sumr) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 929 (all Xa hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),cos_coeff)),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(real,real,cos,Xa)))) # label(fact_4124_cos__converges) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 930 (all X all Y (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),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) # label(fact_1809_real__less__half__sum) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 931 (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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)),N)))) # label(fact_1533_mod__less__divisor) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 932 (all X_a (division_ring(X_a) -> (all A_3 all B_2 all C (ti(X_a,C) != zero_zero(X_a) -> (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C) = ti(X_a,A_3) <-> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C) = ti(X_a,B_2)))))) # label(fact_1848_nonzero__eq__divide__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 933 (all A_1 all B all Qr_1 (hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)),Qr_1)) -> hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(product_prod(int,int),product_prod(int,int),negateSnd,Qr_1))))) # label(fact_3098_divmod__int__rel__neg) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 934 (all X_a (division_ring(X_a) -> (all A_1 (zero_zero(X_a) = hAPP(X_a,X_a,inverse_inverse(X_a),A_1) -> zero_zero(X_a) = ti(X_a,A_1))))) # label(fact_2577_inverse__zero__imp__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 935 (all Xa hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),cos)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),Xa),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,cos,Xa))),hAPP(real,real,sin,Xa)))))) # label(fact_3941_DERIV__cos__realpow2b) # label(axiom) # label(non_clause). [assumption]. 7.62/7.63 936 (all X_a (archim1727834104eiling(X_a) -> (all X all V hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(int,X_a,number_number_of(X_a),V))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(X_a,int,archimedean_ceiling(X_a),X)),hAPP(int,int,number_number_of(int),V))))) # label(fact_3331_ceiling__add__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 937 (all K_4 all L_4 hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),pdivmod,K_4),L_4) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,abs_abs(int),K_4)),hAPP(int,int,abs_abs(int),L_4))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,abs_abs(int),K_4)),hAPP(int,int,abs_abs(int),L_4)))) # label(fact_1580_pdivmod__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 938 (all X_a all X_b the(product_prod(X_a,X_b)) = ti(fun(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b)),the(product_prod(X_a,X_b)))) # label(tsy_c_HOL_OThe_2_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 939 (all M all N all K_4 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_4)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N)))) # label(fact_1168_nat__mult__div__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 940 (all X_b (ord(X_b) & number(X_b) -> (all U_1 all V ((-hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(int,X_b,number_number_of(X_b),U_1)),hAPP(int,X_b,number_number_of(X_b),V))) -> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),hAPP(int,X_b,number_number_of(X_b),U_1)),hAPP(int,X_b,number_number_of(X_b),V)) = hAPP(int,X_b,number_number_of(X_b),V)) & (hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(int,X_b,number_number_of(X_b),U_1)),hAPP(int,X_b,number_number_of(X_b),V))) -> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),hAPP(int,X_b,number_number_of(X_b),U_1)),hAPP(int,X_b,number_number_of(X_b),V)) = hAPP(int,X_b,number_number_of(X_b),U_1)))))) # label(fact_4856_min__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 941 (all C (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),C)),one_one(real))) -> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,abs_abs(real),C))),zero_zero(real)),sequentially)))) # label(fact_4462_LIMSEQ__rabs__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 942 (all M all K_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K_4)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,fact_fact(int),M)),hAPP(int,int,fact_fact(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),M),K_4)))))) # label(fact_3221_fact__mono__int__aux) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 943 (all X_a (ordere142940540dd_abs(X_a) -> abs_abs(X_a) = ti(fun(X_a,X_a),abs_abs(X_a)))) # label(tsy_c_Groups_Oabs__class_Oabs_1_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 944 (all X_a (boolean_algebra(X_a) -> (all X hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)) = ti(X_a,X)))) # label(fact_2979_double__compl) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 945 (all M all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),M)),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,suc,M)))),N)) # label(fact_1560_Suc__mod__eq__add3__mod) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 946 (all X all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5)) -> (hBOOL(hAPP(int,bool,zprime,P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_5)) -> -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),X)),zero_zero(int)),P_5)))))) # label(fact_1187_MultInv__prop3) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 947 (all V all W hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,int,number_number_of(int),W)) = hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),V),W))) # label(fact_420_times__numeral__code_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 948 (all X_b all X_a (metric_space(X_a) & real_normed_vector(X_b) -> (all L all F all G all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),plus_plus(X_b)),F)),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),uminus_uminus(X_b)),G))),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3)))))))) # label(fact_4625_LIM__trans) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 949 (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,hAPP(nat,fun(nat,nat),power_power(nat),X),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Y),N))))) # label(fact_829_divides__exp) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 950 (all X_a (idom(X_a) -> (all A_3 all B_2 (ti(X_a,A_3) = ti(X_a,B_2) | ti(X_a,A_3) = hAPP(X_a,X_a,uminus_uminus(X_a),B_2) <-> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),A_3) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),B_2))))) # label(fact_1904_square__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 951 (all Xa hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),fact_fact(nat)),hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),Xa)),hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,real,cos,Xa)))) # label(fact_4133_cos__paired) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 952 (all X_a (real_algebra_1(X_a) -> zero_zero(X_a) = hAPP(real,X_a,of_real(X_a),zero_zero(real)))) # label(fact_2792_of__real__0) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 953 (all X_b all X_a (topological_space(X_a) & power(X_b) & real_normed_algebra(X_b) -> (all Na all F all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(nat,fun(X_a,X_b),hAPP(fun(X_a,fun(nat,X_b)),fun(nat,fun(X_a,X_b)),combc(X_a,nat,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(nat,X_b)),hAPP(fun(X_b,fun(nat,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(nat,X_b))),combb(X_b,fun(nat,X_b),X_a),power_power(X_b)),F)),Na)),A_3)))))) # label(fact_4917_isCont__power) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 954 (all X all Y (X = hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) -> zero_zero(real) = Y)) # label(fact_2175_real__sqrt__sum__squares__eq__cancel) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 955 (all K_4 hAPP(int,int,pred,K_4) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K_4),one_one(int))) # label(fact_2930_pred__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 956 (all Ma all K all Na (Na = Ma <-> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),K) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),K))) # label(fact_129_nat__add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 957 (all X_a (ordere236663937imp_le(X_a) -> (all A_3 all C all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B_2),C))))))) # label(fact_331_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 958 (all M ((M != zero_zero(nat) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat)))) = hAPP(nat,nat,fact_fact(nat),M)) & (M = zero_zero(nat) -> one_one(nat) = hAPP(nat,nat,fact_fact(nat),M)))) # label(fact_2555_fact__num__eq__if__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 959 (all X_a (real_normed_vector(X_a) -> (all A_1 all B hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,norm_norm(X_a),A_1)),hAPP(X_a,real,norm_norm(X_a),B))))))) # label(fact_2010_norm__triangle__ineq4) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 960 (all L_4 all K_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L_4)) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),K_4),L_4)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),K_4),L_4)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,K_4),L_4)))) # label(fact_1655_posDivAlg__div__mod) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 961 (all F all A_3 all B_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),B_2)) -> ((all X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),X_1)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),B_2)) -> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),F),X_1)))) -> ((all X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),B_2)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),X_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),zero_zero(real))))) -> hAPP(real,real,F,B_2) = hAPP(real,real,F,A_3))))) # label(fact_4953_DERIV__isconst__end) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 962 (all M all N all Q_3 all R_3 (hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M),N),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Q_3),R_3))) -> R_3 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N))) # label(fact_3127_mod__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 963 (all P (ti(bool,P) = fFalse | fTrue = ti(bool,P))) # label(help_If_3_1_T) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 964 (all X_a all X_b all X_c all Y -(all A_4 all B_4 all C_4 hAPP(product_prod(X_b,X_c),product_prod(X_a,product_prod(X_b,X_c)),hAPP(X_a,fun(product_prod(X_b,X_c),product_prod(X_a,product_prod(X_b,X_c))),product_Pair(X_a,product_prod(X_b,X_c)),A_4),hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),B_4),C_4)) != ti(product_prod(X_a,product_prod(X_b,X_c)),Y))) # label(fact_3179_prod__cases3) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 965 (all Y all X 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_eq(real),zero_zero(real)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y)) -> hAPP(real,real,hAPP(nat,fun(real,real),root,N),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),Y)))))) # label(fact_2710_real__root__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 966 (all G all F all Xa all D_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_1)) -> ((all Z_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),Xa))),D_1)) -> Z_2 = hAPP(real,real,G,hAPP(real,real,F,Z_2)))) -> ((all Z_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),Xa))),D_1)) -> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),F),Z_2)))) -> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),G),hAPP(real,real,F,Xa))))))) # label(fact_4950_isCont__inv__fun) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 967 (all Ta all A_2 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D)) -> (all X_1 ((all Xa_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D))) -> (all Xb (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2)) -> ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa_1))))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Ta),X_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Ta),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),D)))))))) # label(fact_4655_aset_I8_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 968 (all X_a (abs_if(X_a) -> ti(fun(X_a,X_a),abs_abs(X_a)) = abs_abs(X_a))) # label(tsy_c_Groups_Oabs__class_Oabs_0_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 969 (all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> zero_zero(real) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),zero_zero(real)),N)))) # label(fact_2465_lemma__STAR__cos) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 970 (all X_a (division_ring(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,inverse_inverse(X_a),B))))) # label(fact_2599_divide__inverse) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 971 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) -> hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),B)) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)))) # label(fact_3143_mod__pos__pos) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 972 (all M one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),one_one(int))) # label(fact_4396_gcd__1__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 973 (all Xa hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),Xa)),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),fact_fact(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),Xa)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) = hAPP(real,real,exp(real),Xa)) # label(fact_4002_exp__first__two__terms) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 974 (all X_b all X_a (ring_1(X_a) & comm_monoid_mult(X_a) -> (all F all A_2 hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),hAPP(fun(X_b,int),fun(X_b,X_a),hAPP(fun(int,X_a),fun(fun(X_b,int),fun(X_b,X_a)),combb(int,X_a,X_b),ring_1_of_int(X_a)),F)),A_2) = hAPP(int,X_a,ring_1_of_int(X_a),hAPP(fun(X_b,bool),int,hAPP(fun(X_b,int),fun(fun(X_b,bool),int),big_co302546470etprod(X_b,int),F),A_2))))) # label(fact_3956_of__int__setprod) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 975 (all X_a (real_vector(X_a) -> (all A_1 all X hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,uminus_uminus(real),A_1)),X)))) # label(fact_4740_scaleR__minus__left) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 976 (all X all D_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),D_3)) -> hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),X),D_3))),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),D_3))),hAPP(nat,real,real_1(nat),D_3))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,real_1(nat),X)),hAPP(nat,real,real_1(nat),D_3)))) # label(fact_2432_real__of__nat__div__aux) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 977 (all X_a (metric_space(X_a) -> (all X all Y (ti(X_a,X) != ti(X_a,Y) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y))))))) # label(fact_3536_dist__pos__lt) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 978 (all P_1 all Na ((exists M_2 (hBOOL(hAPP(nat,bool,P_1,M_2)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),Na)))) <-> (exists X_1 (hBOOL(hAPP(nat,bool,P_1,X_1)) & hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_1),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))))))) # label(fact_4225_ex__nat__less__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 979 (all X_a (real_normed_vector(X_a) -> (all X_2 all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),X_2)),hAPP(X_a,real,norm_norm(X_a),A_3)),sequentially)))))) # label(fact_4496_LIMSEQ__norm) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 980 (all Ya all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya)) -> (exists X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),X_1)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),Ya)) & hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_1),field_char_0_Rats(real)))))))) # label(fact_3702_Rats__dense__in__nn__real) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 981 (all X_a (real_normed_field(X_a) -> (all Ya all X_2 all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),X_2),A_3)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),X_2)),Ya)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),Ya))))))) # label(fact_4113_divide_Osums) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 982 (all X_b all X_a (topological_space(X_a) & real_normed_vector(X_b) -> (all F all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,real),fun(X_a,bool),isCont(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),A_3)))))) # label(fact_4918_isCont__norm) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 983 (all N hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))))) # label(fact_2634_inv__real__of__nat__fact__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 984 (all X_a all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),fequal(X_a),X),Y)) | ti(X_a,Y) != ti(X_a,X))) # label(help_fequal_2_1_T) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 985 (all L_4 hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),min),hAPP(int,int,bit0,L_4)) = hAPP(int,int,bit1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),min),L_4))) # label(fact_782_diff__bin__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 986 (all N hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(nat,nat,fact_fact(nat),N)))) # label(fact_2551_fact__ge__Suc__0__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 987 (all Wa all Z (ti(int,Z) = ti(int,Wa) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z),one_one(int)))))) # label(fact_60_zless__add1__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 988 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X))),X))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) # label(fact_1694_abs__ln__one__plus__x__minus__x__bound__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 989 (all X_a (real_normed_vector(X_a) -> (all X_2 all K_3 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_3)) -> ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,X_2,N_1))),K_3))) -> hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2))))))) # label(fact_3523_BseqI) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 990 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> (zero_zero(real) = hAPP(real,real,cos,X) -> (exists N_1 (X = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) & -hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1))))))) # label(fact_2531_cos__zero__lemma) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 991 (all F all Ya all Xa all A_3 all B_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),B_2)) -> (hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),Xa),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_gr676655987ssThan(real),A_3),B_2))) -> (hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),Ya),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_gr676655987ssThan(real),A_3),B_2))) -> ((all X_1 (hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_1),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_gr676655987ssThan(real),A_3),B_2))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),zero_zero(real))))) -> hAPP(real,real,F,Xa) = hAPP(real,real,F,Ya)))))) # label(fact_3703_DERIV__isconst3) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 992 (all X_a (real_normed_vector(X_a) -> (all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),X_2))) <-> hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2)))))) # label(fact_3892_Bseq__minus__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 993 (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)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y)))))) # label(fact_469_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 994 (all Code_numeral_3 hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,Code_numeral_3) != zero_zero(code_code_numeral)) # label(fact_2972_code__numeral_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 995 (all Xa (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),hAPP(int,nat,nat_1,Xa)) = one_one(int)))) # label(fact_1691_EvenOdd_Oneg__one__even__power) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 996 (all X hAPP(real,real,uminus_uminus(real),hAPP(complex,real,im,X)) = hAPP(complex,real,im,hAPP(complex,complex,cnj,X))) # label(fact_2998_complex__Im__cnj) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 997 (all X_a (topological_space(X_a) -> (all G all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),zero_zero(complex)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),hAPP(fun(X_a,complex),fun(X_a,complex),hAPP(fun(complex,complex),fun(fun(X_a,complex),fun(X_a,complex)),combb(complex,complex,X_a),cnj),G)),zero_zero(complex)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4624_cnj_OLIM__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 998 (all Xa all A_3 all P_4 (hBOOL(hAPP(int,bool,zprime,P_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),zero_zero(int)),P_4)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),sRStar,P_4))) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),standardRes,P_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),Xa)))),hAPP(int,fun(int,bool),sRStar,P_4)))))))) # label(fact_1518_SRStar__mult__prop2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 999 (all X_a (real_normed_vector(X_a) -> (all X_2 all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),X_2)),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)),sequentially)))))) # label(fact_4469_LIMSEQ__minus) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1000 (all X_a (order(X_a) -> (all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) -> ti(X_a,X) != ti(X_a,Y))))) # label(fact_530_less__imp__neq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1001 (all X_a (number_semiring(X_a) -> (all V_2 all V (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_2)) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),V_2)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),V),V_2))))))) # label(fact_422_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1002 (all X_a (field_inverse_zero(X_a) -> (all X all Z_1 all Y (ti(X_a,Y) != zero_zero(X_a) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),Y))),Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),Z_1))))) # label(fact_1879_add__frac__num) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1003 (all Y all X (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = Y)) # label(fact_3262_gcd__proj2__if__dvd__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1004 (all X_a (comm_monoid_add(X_a) -> (all F all Na hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),ord_lessThan(nat),Na))),hAPP(nat,X_a,F,Na)) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),ord_lessThan(nat),hAPP(nat,nat,suc,Na)))))) # label(fact_4348_setsum__lessThan__Suc) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1005 (all X all Y hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(complex,real,im,X)),hAPP(complex,real,im,Y)) = hAPP(complex,real,im,hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),X),Y))) # label(fact_3006_Im_Odiff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1006 (all Xa all Na all Ya (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Xa),hAPP(nat,nat,suc,Na))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Ya),hAPP(nat,nat,suc,Na)))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Xa),Ya)))) # label(fact_1406_exp__mono__lt) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1007 (all X_a (ab_semigroup_mult(X_a) -> (all A_1 all B all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1))))) # label(fact_361_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1008 (all X_a all X_b (linordered_semidom(X_b) -> (all F all A_2 ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(X_a,X_b,F,X_1))))) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2))))))) # label(fact_4178_setprod__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1009 (all R_3 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),R_3)) -> (exists N_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_3),hAPP(nat,real,real_1(nat),N_1))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),one_one(nat)))),R_3)))))) # label(fact_2461_reals__Archimedean6) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1010 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,sqrt,X))))) # label(fact_2138_real__sqrt__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1011 (all X_a (linord581940658strict(X_a) -> (all C all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B_2))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C)) | 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),B_2),A_3)))))) # label(fact_442_mult__less__cancel__left__disj) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1012 (all B all A_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A_1)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(real,int,archim2085082626_floor(real),A_1)),hAPP(real,int,archim2085082626_floor(real),B))),hAPP(real,int,archim2085082626_floor(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_1),B))))))) # label(fact_2917_le__mult__floor) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1013 (all N N = hAPP(nat,nat,size_size(nat),N)) # label(fact_2948_nat__size) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1014 (all X_a (real_normed_vector(X_a) -> (all Ra all A_3 hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra),A_3)),hAPP(X_a,filter(X_a),at(X_a),A_3)))))) # label(fact_4759_scaleR__right_Ocont) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1015 (all X_a (real_normed_field(X_a) -> (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),Xa)),one_one(real))) -> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),one_one(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),one_one(X_a)),Xa)))))))) # label(fact_4125_geometric__sums) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1016 (all B_1 all A_2 (hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2)) -> (hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,B_1)) -> (hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less_eq(fun(nat,bool)),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2)),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),B_1))) <-> hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,bool),fun(fun(int,bool),bool),ord_less_eq(fun(int,bool)),A_2),B_1)))))) # label(fact_3691_transfer__int__nat__set__relations_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1017 (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),hAPP(int,int,bit1,K)),pls)))) # label(fact_275_rel__simps_I29_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1018 (all M all N all K_4 hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)),K_4)) # label(fact_601_nat__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1019 (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),hAPP(real,real,sqrt,X)),hAPP(real,real,sqrt,Y))))) # label(fact_2121_real__sqrt__le__mono) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1020 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)))))) # label(fact_972_abs__le__D1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1021 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,cos,X)))))) # label(fact_2344_cos__gt__zero__pi) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1022 (all X_a (linordered_semiring(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(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_1),B))))))) # label(fact_477_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1023 (all N hAPP(real,nat,natceiling,hAPP(int,real,number_number_of(real),N)) = hAPP(int,nat,number_number_of(nat),N)) # label(fact_1670_natceiling__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1024 (all X_a all A_2 ((exists N_1 exists F_3 hAPP(fun(nat,bool),fun(X_a,bool),hAPP(fun(nat,X_a),fun(fun(nat,bool),fun(X_a,bool)),image(nat,X_a),F_3),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),N_1))) = ti(fun(X_a,bool),A_2)) <-> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)))) # label(fact_4154_finite__conv__nat__seg__image) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1025 (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))),X))))) # label(fact_883_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1026 (all X_a (linordered_field(X_a) -> (all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),zero_zero(X_a)))))))) # label(fact_1865_divide__nonpos__pos) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1027 (all X_a all R_2 all F all Xa all Ya (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(X_a,nat,F,Xa)),hAPP(X_a,nat,F,Ya))) -> (hBOOL(hAPP(fun(product_prod(X_a,X_a),bool),bool,hAPP(product_prod(X_a,X_a),fun(fun(product_prod(X_a,X_a),bool),bool),member(product_prod(X_a,X_a)),hAPP(X_a,product_prod(X_a,X_a),hAPP(X_a,fun(X_a,product_prod(X_a,X_a)),product_Pair(X_a,X_a),Xa),Ya)),R_2)) -> hBOOL(hAPP(fun(product_prod(X_a,X_a),bool),bool,hAPP(product_prod(X_a,X_a),fun(fun(product_prod(X_a,X_a),bool),bool),member(product_prod(X_a,X_a)),hAPP(X_a,product_prod(X_a,X_a),hAPP(X_a,fun(X_a,product_prod(X_a,X_a)),product_Pair(X_a,X_a),Xa),Ya)),hAPP(fun(product_prod(X_a,X_a),bool),fun(product_prod(X_a,X_a),bool),hAPP(fun(X_a,nat),fun(fun(product_prod(X_a,X_a),bool),fun(product_prod(X_a,X_a),bool)),mlex_prod(X_a),F),R_2)))))) # label(fact_3821_mlex__leq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1028 (all X all Y (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_eq(real),zero_zero(real)),X)) -> (exists N_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_1))),Y))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),Y)),X))))))) # label(fact_2462_reals__Archimedean4) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1029 (all X_a (semiring_char_0(X_a) -> (all Ma all Na (hAPP(nat,X_a,semiring_1_of_nat(X_a),Na) = hAPP(nat,X_a,semiring_1_of_nat(X_a),Ma) <-> Na = Ma)))) # label(fact_161_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1030 (all X_a (ab_sem1668676832m_mult(X_a) -> (all X hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X) = ti(X_a,X)))) # label(fact_2976_mult__idem) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1031 (all T_2 all T_1 (minus(T_1) -> minus(fun(T_2,T_1)))) # label(arity_fun___Groups_Ominus) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1032 (all N all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) -> hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),hAPP(real,real,ln,X)) = hAPP(real,real,ln,hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N)))) # label(fact_2420_ln__realpow) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1033 (all X_a (ordere236663937imp_le(X_a) -> (all A_1 all C_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),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_1),B)))))) # label(fact_210_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1034 (all X_a (real_div_algebra(X_a) & divisi14063676e_zero(X_a) -> (all A_1 all X hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,inverse_inverse(real),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),X))))) # label(fact_4727_inverse__scaleR__distrib) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1035 (all X_c all X_b all X_a (metric_space(X_a) & topological_space(X_b) & topological_space(X_c) -> (all G all L all F all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3)) -> (hBOOL(hAPP(filter(X_b),bool,hAPP(X_c,fun(filter(X_b),bool),hAPP(fun(X_b,X_c),fun(X_c,fun(filter(X_b),bool)),tendsto(X_b,X_c),G),L),hAPP(X_b,filter(X_b),at(X_b),hAPP(X_a,X_b,F,A_3)))) -> ((exists D_2 ((all X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X_1),A_3)),D_2)) & ti(X_a,A_3) != ti(X_a,X_1) -> hAPP(X_a,X_b,F,X_1) != hAPP(X_a,X_b,F,A_3))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_2)))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_c,fun(filter(X_a),bool),hAPP(fun(X_a,X_c),fun(X_c,fun(filter(X_a),bool)),tendsto(X_a,X_c),hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c)),combb(X_b,X_c,X_a),G),F)),L),hAPP(X_a,filter(X_a),at(X_a),A_3))))))))) # label(fact_4944_metric__isCont__LIM__compose2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1036 (all N hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N))))) # label(fact_2485_real__of__nat__fact__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1037 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B all C_1 all D_3 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),D_3)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D_3))))) # label(fact_293_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1038 (all Xa all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)) <-> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Xa),Na)),zEven))))) # label(fact_1726_power__preserves__even) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1039 (all X all Y (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) -> Y != X)) # label(fact_940_dvd_Oless__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1040 (all Ya all Xa (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ya),zEven)) -> hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Xa),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Ya),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xa),Ya)),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) # label(fact_1738_even__sum__div__2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1041 (all Ma all Na (hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ma),Na) = zero_zero(nat) <-> Na = zero_zero(nat) | zero_zero(nat) = Ma)) # label(fact_604_mult__is__0) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1042 (all X_b all X_a (semiring_0(X_a) -> (all Ra all F all A_2 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ra),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ra)),F)),A_2)))) # label(fact_4242_setsum__right__distrib) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1043 (all X_b all X_a all Q_1 all Q_2 all P_1 all P_4 (hBOOL(hAPP(X_a,bool,P_1,P_4)) -> (hBOOL(hAPP(X_b,bool,Q_1,Q_2)) -> hBOOL(hAPP(X_b,bool,Q_1,hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),P_4),Q_2)))) & hBOOL(hAPP(X_a,bool,P_1,hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),P_4),Q_2))))))) # label(fact_3164_conjI__realizer) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1044 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),A_1)))))) # label(fact_728_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1045 (all X_a (group_add(X_a) -> (all A_3 (zero_zero(X_a) = ti(X_a,A_3) <-> zero_zero(X_a) = hAPP(X_a,X_a,uminus_uminus(X_a),A_3))))) # label(fact_1934_neg__equal__0__iff__equal) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1046 (all N all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,real_1(int),N)),one_one(real))),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(int,real,real_1(int),N))) -> ti(int,N) = hAPP(real,int,archimedean_ceiling(real),X)))) # label(fact_3440_ceiling__eq3) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1047 (all X_b all X_c all X_a (topological_space(X_a) & topological_space(X_c) & topological_space(X_b) -> (all G all F all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3)) -> (hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,X_c),fun(X_b,bool),isCont(X_b,X_c),G),hAPP(X_a,X_b,F,A_3))) -> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_c),fun(X_a,bool),isCont(X_a,X_c),hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c)),combb(X_b,X_c,X_a),G),F)),A_3))))))) # label(fact_4919_isCont__o2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1048 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,M)),N)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)))) # label(fact_1345_Suc__leD) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1049 (all A_1 all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> (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_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),inv,P_5),A_1))),one_one(int)),P_5)))))) # label(fact_1188_inv__is__inv) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1050 (all K_4 all L_4 all I_1 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),L_4)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),L_4)))))) # label(fact_416_add__le__mono) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1051 (all N hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,real_1(nat),N)))) # label(fact_2368_real__of__nat__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1052 (all X_a all F1 all F2 hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),F1),F2),zero_zero(nat)) = ti(X_a,F1)) # label(fact_4181_nat__rec__0) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1053 (all X_a (plus(X_a) & dvd(X_a) & linorder(X_a) -> (all D_1 all Sb exists Z_2 all X_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_2),X_1)) -> (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),Sb))) <-> -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),Sb)))))))) # label(fact_1215_pinf_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1054 (all Y exists X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X_1)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) & hAPP(real,real,tan,X_1) = Y)) # label(fact_2458_lemma__tan__total1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1055 (all W hAPP(int,nat,number_number_of(nat),W) = hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),W))) # label(fact_1005_nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1056 (all X_b all X_a (topological_space(X_a) & real_normed_vector(X_b) -> (all F all L all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),minus_minus(X_b)),F)),L)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4608_LIM__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1057 (all X_b all X_c all X_a all X_d all X_e all F all G all P_4 hAPP(product_prod(X_d,X_e),X_a,hAPP(fun(X_d,fun(X_e,X_a)),fun(product_prod(X_d,X_e),X_a),product_prod_case(X_d,X_e,X_a),hAPP(fun(X_d,fun(X_e,product_prod(X_b,X_c))),fun(X_d,fun(X_e,X_a)),hAPP(fun(fun(X_e,product_prod(X_b,X_c)),fun(X_e,X_a)),fun(fun(X_d,fun(X_e,product_prod(X_b,X_c))),fun(X_d,fun(X_e,X_a))),combb(fun(X_e,product_prod(X_b,X_c)),fun(X_e,X_a),X_d),hAPP(fun(product_prod(X_b,X_c),X_a),fun(fun(X_e,product_prod(X_b,X_c)),fun(X_e,X_a)),combb(product_prod(X_b,X_c),X_a,X_e),hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),product_prod_case(X_b,X_c,X_a),F))),G)),P_4) = hAPP(product_prod(X_b,X_c),X_a,hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),product_prod_case(X_b,X_c,X_a),F),hAPP(product_prod(X_d,X_e),product_prod(X_b,X_c),hAPP(fun(X_d,fun(X_e,product_prod(X_b,X_c))),fun(product_prod(X_d,X_e),product_prod(X_b,X_c)),product_prod_case(X_d,X_e,product_prod(X_b,X_c)),G),P_4))) # label(fact_4030_split__twice) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1058 (all X_a (number_semiring(X_a) -> (all Z_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z_1),Z_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) # label(fact_544_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1059 (all X_a all X_b all A_2 all B_1 (hBOOL(hAPP(fun(sum_sum(X_a,X_b),bool),bool,finite_finite_1(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_2),B_1))) -> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)))) # label(fact_3209_finite__PlusD_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1060 (all R_3 all S_3 hAPP(rat,real,ratreal,hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),hAPP(int,rat,number_number_of(rat),R_3)),hAPP(int,rat,number_number_of(rat),S_3))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,number_number_of(real),R_3)),hAPP(int,real,number_number_of(real),S_3))) # label(fact_3756_Ratreal__number__of__quotient2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1061 (all P_1 all A0 all A1 (hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),lazy_small_lazy_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))) -> ((all D_2 all I_2 (hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),lazy_small_lazy_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),D_2),I_2))) -> ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),D_2),I_2)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,D_2),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),I_2),one_one(int))))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,D_2),I_2))))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A0),A1))))) # label(fact_3583_small__lazy_H_Opinduct) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1062 (all X_a (division_ring(X_a) -> ti(fun(X_a,X_a),inverse_inverse(X_a)) = inverse_inverse(X_a))) # label(tsy_c_Fields_Oinverse__class_Oinverse_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1063 (all A_1 hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) # label(fact_549_cube__square) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1064 (all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> N = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(nat,nat,suc,zero_zero(nat)))))) # label(fact_1446_Suc__pred) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1065 (all X_a (ordered_ring(X_a) -> (all A_3 all E_2 all C all B_2 all D_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B_2),A_3)),E_2)),D_1))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),E_2)),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),E_2)),D_1))))))) # label(fact_772_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1066 (all A_1 all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),powr,hAPP(real,real,ln,X)),A_1)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),powr,A_1),A_1)),X)))))) # label(fact_3446_ln__powr__bound2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1067 (all A_3 all B_2 hAPP(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_prod_case(int,int,bool),hAPP(fun(int,bool),fun(int,fun(int,bool)),hAPP(fun(int,fun(fun(int,bool),fun(int,bool))),fun(fun(int,bool),fun(int,fun(int,bool))),combc(int,fun(int,bool),fun(int,bool)),hAPP(fun(int,fun(int,fun(bool,bool))),fun(int,fun(fun(int,bool),fun(int,bool))),hAPP(fun(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool))),fun(fun(int,fun(int,fun(bool,bool))),fun(int,fun(fun(int,bool),fun(int,bool)))),combb(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),int),combs(int,bool,bool)),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,fun(bool,bool))),hAPP(fun(fun(int,bool),fun(int,fun(bool,bool))),fun(fun(int,fun(int,bool)),fun(int,fun(int,fun(bool,bool)))),combb(fun(int,bool),fun(int,fun(bool,bool)),int),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj)),hAPP(fun(int,fun(int,int)),fun(int,fun(int,bool)),hAPP(fun(fun(int,int),fun(int,bool)),fun(fun(int,fun(int,int)),fun(int,fun(int,bool))),combb(fun(int,int),fun(int,bool),int),hAPP(fun(int,bool),fun(fun(int,int),fun(int,bool)),combb(int,bool,int),hAPP(int,fun(int,bool),fequal(int),A_3))),hAPP(fun(int,int),fun(int,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(int,int),fun(int,fun(int,int))),combb(int,fun(int,int),int),plus_plus(int)),hAPP(int,fun(int,int),times_times(int),B_2)))))),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(bool,bool),fun(fun(int,bool),fun(int,bool)),combb(bool,bool,int),hAPP(bool,fun(bool,bool),fimplies,hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2))),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),B_2))))),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(bool,bool),fun(fun(int,bool),fun(int,bool)),combb(bool,bool,int),hAPP(bool,fun(bool,bool),fimplies,hAPP(bool,bool,fNot,hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2)))),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),B_2))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less_eq(int)),zero_zero(int))))))) = hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_3),B_2)) # label(fact_4058_divmod__int__rel__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1068 (all X_a (linordered_semidom(X_a) -> (all A_1 all N all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,N))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),hAPP(nat,nat,suc,N)))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))))))) # label(fact_1425_power__le__imp__le__base) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1069 (all N all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M)) -> N != M)) # label(fact_133_less__not__refl2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1070 (all X_a (metric_space(X_a) & comm_monoid_add(X_a) -> (all F hAPP(fun(nat,X_a),X_a,suminf(X_a),F) = hAPP(fun(nat,X_a),X_a,lim(X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(fun(nat,bool),X_a),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combb(fun(nat,bool),X_a,nat),hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F)),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat))))))) # label(fact_4355_suminf__eq__lim) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1071 (all X_a (number_ring(X_a) & ring_char_0(X_a) -> (all Xa all Ya (hAPP(int,X_a,number_number_of(X_a),Xa) = hAPP(int,X_a,number_number_of(X_a),Ya) <-> ti(int,Xa) = ti(int,Ya))))) # label(fact_40_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1072 (all R_3 all Q_3 all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (ti(int,A_1) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),R_3),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),Q_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_3),A_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),Q_3)))))) # label(fact_577_self__quotient__aux1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1073 (all Z (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(int,nat,nat_1,Z))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z)))) # label(fact_1030_zero__less__nat__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1074 (all X_a (field_inverse_zero(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B))))) # label(fact_1940_minus__divide__right) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1075 (all Xa all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)),zero_zero(real))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),zero_zero(real)))))) # label(fact_2696_real__root__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1076 (all P_1 all A0 (hBOOL(hAPP(int,bool,hAPP(fun(int,fun(int,bool)),fun(int,bool),accp(int),quickc1265749348ro_rel),A0)) -> ((all I_2 (hBOOL(hAPP(int,bool,hAPP(fun(int,fun(int,bool)),fun(int,bool),accp(int),quickc1265749348ro_rel),I_2)) -> ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_2),zero_zero(int))) -> (ti(int,I_2) != zero_zero(int) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),I_2),one_one(int)))))) -> hBOOL(hAPP(int,bool,P_1,I_2))))) -> hBOOL(hAPP(int,bool,P_1,A0))))) # label(fact_3202_around__zero_Opinduct) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1077 (all P_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),P_4),zOdd)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) # label(fact_1217_Euler_Oaux__2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1078 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,sqrt,Xa)),one_one(real))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),one_one(real))))) # label(fact_2148_real__sqrt__le__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1079 (all A_9 all B_8 all Y all R_5 (ti(int,A_9) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_8),Y)),R_5) -> ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_8)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_8),R_5)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_5),zero_zero(int)))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_8)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_5),B_8)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_5))) -> (ti(int,B_8) != zero_zero(int) -> ti(int,Y) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_9),B_8))))) # label(fact_1125_divmod__int__rel__div__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1080 (all X_a (division_ring(X_a) -> (all A_1 (zero_zero(X_a) != ti(X_a,A_1) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),one_one(X_a)),A_1) = hAPP(X_a,X_a,inverse_inverse(X_a),A_1))))) # label(fact_2620_nonzero__inverse__eq__divide) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1081 (all N exists P_6 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),P_6)) & hBOOL(hAPP(nat,bool,prime,P_6)))) # label(fact_3130_euclid) # label(axiom) # label(non_clause). [assumption]. 7.62/7.64 1082 (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_758_zdvd__not__zless) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1083 (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)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(int,nat,nat_1,X)),hAPP(int,nat,nat_1,Y)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y))))) # label(fact_4433_transfer__nat__int__gcd_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1084 (all Ma all Na (Ma = Na <-> hAPP(nat,int,semiring_1_of_nat(int),Ma) = hAPP(nat,int,semiring_1_of_nat(int),Na))) # label(fact_48_int__int__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1085 (all X_a (linordered_semidom(X_a) -> (all N all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),N)))))))) # label(fact_251_power__mono) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1086 (all X_a (order(X_a) -> (all X_2 ((all M_2 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_2),N_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,X_2,N_1)),hAPP(nat,X_a,X_2,M_2))))) | (all M_2 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_2),N_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,X_2,M_2)),hAPP(nat,X_a,X_2,N_1))))) <-> hBOOL(hAPP(fun(nat,X_a),bool,monoseq(X_a),X_2)))))) # label(fact_3950_monoseq__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1087 (all Z_1 hAPP(complex,real,im,hAPP(complex,complex,sgn_sgn(complex),Z_1)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(complex,real,im,Z_1)),hAPP(complex,real,norm_norm(complex),Z_1))) # label(fact_3010_Im__sgn) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1088 (all X all Y hAPP(rat,real,ratreal,hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(rat,real,ratreal,X)),hAPP(rat,real,ratreal,Y))) # label(fact_3790_real__minus__code) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1089 (all X_a (ordered_ab_group_add(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,uminus_uminus(X_a),A_3))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),zero_zero(X_a))))))) # label(fact_1951_neg__0__le__iff__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1090 (all N_3 ((exists M_2 all X_1 (hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_1),N_3)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X_1),M_2)))) <-> hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),N_3)))) # label(fact_3683_finite__nat__set__iff__bounded__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1091 (all X all M hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),M)),M))) # label(fact_1320_mod__mod__is__mod) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1092 (all X_a (linorder(X_a) -> (all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) | ti(X_a,Y) = ti(X_a,X) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)))))) # label(fact_533_linorder__less__linear) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1093 (all Xa (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Xa)) <-> hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa)))) # label(fact_2520_nat__even__iff__2__dvd) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1094 (all Z1 all Z2 all Z3 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z2),Z3)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z1),Z2)),Z3)) # label(fact_45_zadd__assoc) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1095 (all N all M hAPP(int,quickcheck_code_int,quickcheck_of_int,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(quickcheck_code_int,int,quickcheck_int_of,N)),hAPP(quickcheck_code_int,int,quickcheck_int_of,M))) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),N),M)) # label(fact_3575_div__code__int__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1096 (all X_a (real_normed_vector(X_a) -> (all A_1 all X hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,abs_abs(real),A_1)),hAPP(X_a,real,norm_norm(X_a),X)) = hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X))))) # label(fact_4749_norm__scaleR) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1097 (all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> (exists M_2 hAPP(nat,nat,suc,M_2) = N))) # label(fact_1572_gr0__implies__Suc) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1098 (all P_5 all X hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),X)) # label(fact_1184_MultInv__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1099 (all X_b all X_a all F all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2))) -> (exists C_3 (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),C_3)) & ti(fun(X_a,bool),B_1) = hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),C_3) & hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),C_3),A_2))))))) # label(fact_3669_finite__subset__image) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1100 (all K_4 hAPP(int,int,succ,K_4) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),one_one(int))) # label(fact_303_succ__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1101 (all X_a (ring_1(X_a) -> (all C all X_1 hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),C)),X_1) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),C),X_1))))) # label(fact_4140_diffs__minus) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1102 (all Na (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),Na)),zero_zero(real))) <-> Na = zero_zero(nat))) # label(fact_2393_real__of__nat__le__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1103 (all X_a all X_b ti(fun(fun(X_a,bool),fun(fun(X_b,bool),fun(product_prod(X_a,X_b),bool))),prod(X_a,X_b)) = prod(X_a,X_b)) # label(tsy_c_Nitpick_Oprod_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1104 (all X_c all X_a all B_1_1 all B_2_1 ti(X_c,hAPP(X_a,X_c,B_1_1,B_2_1)) = hAPP(X_a,X_c,B_1_1,B_2_1)) # label(tsy_c_hAPP_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1105 (all Y all X all Z_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),Z_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z_1))))) # label(fact_3064_termination__basic__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1106 (all X_b all X_a all Q_1 all P_1 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),P_1))) -> ((all Y_1 (hBOOL(hAPP(X_a,bool,P_1,Y_1)) -> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(X_a,fun(X_b,bool),hAPP(fun(X_b,fun(X_a,bool)),fun(X_a,fun(X_b,bool)),combc(X_b,X_a,bool),Q_1),Y_1)))))) <-> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,fun(X_a,bool)),fun(X_b,bool),hAPP(fun(fun(X_a,bool),bool),fun(fun(X_b,fun(X_a,bool)),fun(X_b,bool)),combb(fun(X_a,bool),bool,X_b),fEx(X_a)),hAPP(fun(X_b,fun(X_a,bool)),fun(X_b,fun(X_a,bool)),hAPP(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_b,fun(X_a,bool)),fun(X_b,fun(X_a,bool))),combb(fun(X_a,bool),fun(X_a,bool),X_b),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),P_1))),Q_1)))))))) # label(fact_4448_finite__Collect__bounded__ex) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1107 (all X_a (real_normed_field(X_a) -> (all Y exists K_2 all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X_1),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_1)),K_2)))))) # label(fact_2109_divide_Obounded) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1108 (all X_a all X_b all B_2 all F all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2))) -> -(all X_1 (hAPP(X_b,X_a,F,X_1) = ti(X_a,B_2) -> -hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_2)))))) # label(fact_3991_imageE) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1109 (all N all M ((M != zero_zero(nat) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)) & (zero_zero(nat) = M -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N) = zero_zero(nat)))) # label(fact_876_mult__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1110 (all Diff all F all Na all H (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),H),zero_zero(real))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F -> ((all M_2 all T_4 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_4),zero_zero(real))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),H),T_4)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),Na)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_2)),T_4),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),T_4))))) -> (exists T_4 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),T_4),zero_zero(real))) & hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),H))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,Na),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),H),Na))) = hAPP(real,real,F,H) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),H),T_4))))))))) # label(fact_4308_Maclaurin__minus) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1111 (all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) = N)) # label(fact_1465_Suc__pred_H) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1112 (all X_a all X_b (ord(X_b) -> (all F all G ((all X_1 hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,G,X_1)))) -> hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G)))))) # label(fact_2837_le__funI) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1113 (all M all K_4 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),N)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),M)),K_4))))) # label(fact_846_le__add__diff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1114 (all X_a (semiring_div(X_a) -> (all A_1 all B all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C_1)),B)),C_1)))) # label(fact_1284_mod__mult__left__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1115 (all X_a (field_inverse_zero(X_a) & number_ring(X_a) -> (all X hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),one_one(X_a)),X)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(int,X_a,number_number_of(X_a),min)),X)))) # label(fact_2032_minus1__divide) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1116 (all A_1 all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M)) -> (exists X_1 ((all Y_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),Y_1),M)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y_1),M)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_1)) -> ti(int,X_1) = ti(int,Y_1))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),X_1),M)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),M)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)))))) # label(fact_929_zcong__zless__unique) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1117 (all X_a (group_add(X_a) -> (all A_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),A_1) = zero_zero(X_a)))) # label(fact_735_diff__self) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1118 (all M all N (M = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N) -> N = one_one(nat) | zero_zero(nat) = M)) # label(fact_637_mult__eq__self__implies__10) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1119 (all X_2 (hBOOL(hAPP(fun(nat,complex),bool,summable(complex),X_2)) -> hBOOL(hAPP(fun(nat,complex),bool,summable(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(complex,complex),fun(fun(nat,complex),fun(nat,complex)),combb(complex,complex,nat),cnj),X_2))))) # label(fact_4072_cnj_Osummable) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1120 (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_181_zero__less__one) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1121 (all X all Y hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),X),Y)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,semiring_1_of_nat(int),X)),hAPP(nat,int,semiring_1_of_nat(int),Y))) # label(fact_177_Nat__Transfer_Otransfer__int__nat__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1122 (all X_a (linordered_idom(X_a) -> (all X ti(X_a,X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,sgn_sgn(X_a),X)),hAPP(X_a,X_a,abs_abs(X_a),X))))) # label(fact_2282_mult__sgn__abs) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1123 (all R_3 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),R_3)) -> (exists N_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),N_1)),R_3)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_3),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_1)))))))) # label(fact_2463_reals__Archimedean6a) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1124 (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_131_linorder__neqE__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1125 (all Xa all Na all Ya (hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Xa),hAPP(nat,nat,suc,Na)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Ya),hAPP(nat,nat,suc,Na)) <-> Xa = Ya)) # label(fact_1349_exp__mono__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1126 (all X_a all B_1 all A_2 ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2)) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),B_1)))) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)))) # label(fact_2249_subsetI) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1127 (all X_a (division_ring(X_a) -> (all A_1 all B all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C_1))))) # label(fact_1829_add__divide__distrib) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1128 (all D_1 all Ta (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),Ta)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,uminus_uminus(int),D_1)),Ta)))) # label(fact_2065_uminus__dvd__conv_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1129 (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)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),M),N)))))))) # label(fact_459_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1130 (all X_a (number_semiring(X_a) -> (all N hAPP(int,X_a,number_number_of(X_a),hAPP(nat,int,semiring_1_of_nat(int),N)) = hAPP(nat,X_a,semiring_1_of_nat(X_a),N)))) # label(fact_97_number__of__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1131 (all X_a all X_b (order(X_b) & order(X_a) -> (all F all C all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) -> (hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,B_2)),C)) -> ((all X_1 all Y_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,F,Y_1))))) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,A_3)),C)))))))) # label(fact_2860_order__less__le__subst2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1132 (all X_a all F all A_2 hAPP(fun(X_a,bool),real,hAPP(fun(X_a,real),fun(fun(X_a,bool),real),big_co302546470etprod(X_a,real),hAPP(fun(X_a,nat),fun(X_a,real),hAPP(fun(nat,real),fun(fun(X_a,nat),fun(X_a,real)),combb(nat,real,X_a),real_1(nat)),F)),A_2) = hAPP(nat,real,real_1(nat),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co302546470etprod(X_a,nat),F),A_2))) # label(fact_3969_real__of__nat__setprod) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1133 (all M all N all K_4 hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M)),N)),hAPP(nat,nat,suc,K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),K_4)) # label(fact_1348_Suc__diff__diff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1134 (all X_a (order(X_a) -> (all Z_1 all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z_1),Y)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),X))))))) # label(fact_554_xt1_I7_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1135 (all X hAPP(complex,complex,uminus_uminus(complex),X) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,uminus_uminus(real),hAPP(complex,real,re,X))),hAPP(real,real,uminus_uminus(real),hAPP(complex,real,im,X)))) # label(fact_3040_complex__minus__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1136 (all X_b all X_a (minus(X_a) -> (all A_2 all B_1 all Xa hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_b,X_a,A_2,Xa)),hAPP(X_b,X_a,B_1,Xa)) = hAPP(X_b,X_a,hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_b,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),minus_minus(fun(X_b,X_a)),A_2),B_1),Xa)))) # label(fact_2988_minus__apply) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1137 (all X all Y hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y))) # label(fact_3275_gcd__red__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1138 (all X_a all X_b (comm_monoid_mult(X_b) -> (all G all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2)) -> one_one(X_b) = hAPP(X_a,X_b,G,X_1))) -> one_one(X_b) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),A_2)))))) # label(fact_4175_setprod_Oneutral) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1139 (all X_a (archim1727834104eiling(X_a) -> (all Xa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),zero_zero(X_a))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archimedean_ceiling(X_a),Xa)),one_one(int))))))) # label(fact_3324_ceiling__less__one) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1140 (all X_a (ordere236663937imp_le(X_a) -> (all C all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),A_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),B_2))))))) # label(fact_214_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1141 (all J all I all P_1 all P_4 ((all I_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),P_4)) -> (hBOOL(hAPP(nat,bool,P_1,I_2)) -> hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,I_2)),P_4)))))) -> (hBOOL(hAPP(nat,bool,P_1,I)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),P_4)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J),P_4)) -> hBOOL(hAPP(nat,bool,P_1,J))))))) # label(fact_1577_mod__induct) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1142 (all X_a (linordered_field(X_a) -> (all A_3 all B_2 all 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(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)),B_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)))))))) # label(fact_1869_pos__less__divide__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1143 (all X_a (real_normed_algebra(X_a) -> (all Xa all X_2 all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa)),X_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),A_3)),sequentially)))))) # label(fact_4482_mult__right_OLIMSEQ) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1144 (all M all N all P_5 (hBOOL(hAPP(nat,bool,prime,P_5)) -> (N != zero_zero(nat) -> (hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),P_5),M) = N -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)))))) # label(fact_3091_prime__factor__lt) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1145 (all X all Y hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y)) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y)) # label(fact_4389_abs__gcd__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1146 (all X_a (semiring_1(X_a) -> ti(fun(fun(X_a,X_a),fun(nat,fun(X_a,X_a))),semiri532925092at_aux(X_a)) = semiri532925092at_aux(X_a))) # label(tsy_c_Nat_Osemiring__1__class_Oof__nat__aux_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1147 (all P_1 ((all A_4 all B_4 all C_4 (hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_4),B_4))) & hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),B_4),C_4))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B_4),C_4)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_4),B_4)) -> hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_4),C_4))))) & (all X_1 exists D_2 ((all A_4 all B_4 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),B_4)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B_4),A_4)),D_2)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_4),X_1)) -> hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_4),B_4))))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_2)))) -> (all A_4 all B_4 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_4),B_4)) -> hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_4),B_4))))))) # label(fact_3230_lemma__BOLZANO2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1148 (all X_a (linord219039673up_add(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_3)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)),A_3)))))) # label(fact_1955_neg__less__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1149 (all X_a all A_2 hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(fun(X_a,bool),bool),fun(fun(fun(X_a,bool),bool),bool),ord_less_eq(fun(fun(X_a,bool),bool)),A_2),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),A_2))))) # label(fact_2850_subset__Pow__Union) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1150 (all X_a complex_rec(X_a) = ti(fun(fun(real,fun(real,X_a)),fun(complex,X_a)),complex_rec(X_a))) # label(tsy_c_Complex_Ocomplex_Ocomplex__rec_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1151 (all A_1 all B hAPP(real,complex,cis,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A_1),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),hAPP(real,complex,cis,A_1)),hAPP(real,complex,cis,B))) # label(fact_3056_cis__divide) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1152 (all M all N all K_4 hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),K_4)) # label(fact_822_diff__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1153 (all U all Ma all Na all J all I (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),I)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I),U)),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J),U)),Na))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),Ma)),Na))))) # label(fact_877_nat__less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1154 (all B all A_1 all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> (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),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)),zero_zero(int)),P_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),P_5)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,B),zero_zero(int)),P_5)))))) # label(fact_921_zcong__zprime__prod__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1155 (all T_2 all T_1 (ord(T_1) -> ord(fun(T_2,T_1)))) # label(arity_fun___Orderings_Oord) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1156 (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),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,sin,X)))))) # label(fact_2336_sin__gt__zero2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1157 (all X_a (number_ring(X_a) -> (all Z_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),min)),Z_1) = hAPP(X_a,X_a,uminus_uminus(X_a),Z_1)))) # label(fact_2013_mult__minus1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1158 (all X_a (metric_space(X_a) -> (all X all Z_1 all Y all E_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Z_1)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Y),Z_1))),E_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y)),E_1)))))) # label(fact_3528_dist__triangle__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1159 (all X_a (group_add(X_a) -> (all A_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),A_1) = zero_zero(X_a)))) # label(fact_1959_left__minus) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1160 (all Xa all P_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4)) -> (hBOOL(hAPP(int,bool,zprime,P_4)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),sRStar,P_4))) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),standardRes,P_4),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),Xa))),hAPP(int,fun(int,bool),sRStar,P_4))))))) # label(fact_1567_StandardRes__SRStar__prop2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1161 (all P_1 all Na all K ((zero_zero(nat) = K -> hBOOL(hAPP(nat,bool,P_1,zero_zero(nat)))) & (zero_zero(nat) != K -> (all I_2 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J_1),K)) -> (hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),I_2)),J_1) = Na -> hBOOL(hAPP(nat,bool,P_1,I_2)))))) <-> hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),Na),K))))) # label(fact_1170_split__div) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1162 (all X_b all X_a all Q_1 all P_1 all Xa all Ya (hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,Xa),Ya)) -> (hBOOL(hAPP(fun(X_a,fun(X_b,bool)),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),bool),ord_less_eq(fun(X_a,fun(X_b,bool))),P_1),Q_1)) -> hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),Q_1,Xa),Ya))))) # label(fact_4368_rev__predicate2D) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1163 (all X_a ti(fun(product_prod(fun(X_a,bool),fun(X_a,bool)),bool),finite_psubset(X_a)) = finite_psubset(X_a)) # label(tsy_c_Wellfounded_Ofinite__psubset_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1164 (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),hAPP(int,int,bit1,K))))) # label(fact_276_rel__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1165 (all I_1 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_2)) -> (exists K_1 (hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_1) = J_2 & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_1)))))) # label(fact_194_less__imp__add__positive) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1166 (all X_a (number_ring(X_a) & real_n2089651433ebra_1(X_a) -> (all W hAPP(real,real,abs_abs(real),hAPP(int,real,number_number_of(real),W)) = hAPP(X_a,real,norm_norm(X_a),hAPP(int,X_a,number_number_of(X_a),W))))) # label(fact_2019_norm__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1167 (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),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,nat_1,X))),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,nat_1,X))))))) # label(fact_1063_of__nat__double) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1168 (all K_4 all N all M hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N)),M))),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,M)),N)) # label(fact_1544_mod__mult__self4) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1169 (all X all Y hAPP(complex,real,re,hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(complex,real,re,X)),hAPP(complex,real,re,Y))) # label(fact_3027_Re_Oadd) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1170 (all Ma all Na (hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),Na))) <-> (hBOOL(hAPP(nat,bool,even_odd_even(nat),Na)) <-> hBOOL(hAPP(nat,bool,even_odd_even(nat),Ma))))) # label(fact_2469_even__add) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1171 (all R_3 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),R_3),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),hAPP(real,int,archim2085082626_floor(real),R_3))),one_one(real))))) # label(fact_3421_real__of__int__floor__add__one__ge) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1172 (all Sb all Ta all A_3 all B_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_3),B_2)) -> hBOOL(hAPP(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),bool,hAPP(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),hAPP(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),hAPP(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A_3),Sb)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B_2),Ta))),pair_less)))) # label(fact_3511_pair__lessI1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1173 (all X_a (linorder(X_a) -> (all Xa all Ya all Z (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Xa),Ya)),Z)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Z)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ya),Z)))))) # label(fact_4983_max__less__iff__conj) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1174 (all X_a (real_normed_algebra(X_a) -> (all Xa all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2)) -> hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa)),X_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),hAPP(fun(nat,X_a),X_a,suminf(X_a),X_2)))))) # label(fact_4014_mult__right_Osuminf) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1175 (all Na all Q_1 all P_1 ((all X_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,Q_1,X_1)),hAPP(nat,nat,P_1,X_1)))) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),P_1),hAPP(nat,fun(nat,bool),ord_lessThan(nat),Na))),hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),Q_1),hAPP(nat,fun(nat,bool),ord_lessThan(nat),Na))) = hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,nat)),combs(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),minus_minus(nat)),P_1)),Q_1)),hAPP(nat,fun(nat,bool),ord_lessThan(nat),Na)))) # label(fact_4353_sum__diff__distrib) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1176 (all M hAPP(nat,nat,suc,M) != zero_zero(nat)) # label(fact_1316_Zero__neq__Suc) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1177 (all N all A_1 hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),A_1)) = hAPP(complex,real,im,hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(real,complex,cis,A_1)),N))) # label(fact_3049_sin__n__Im__cis__pow__n) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1178 (all X_a (field_inverse_zero(X_a) -> (all A_1 all B all C_1 (ti(X_a,C_1) != zero_zero(X_a) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1)))))) # label(fact_1850_mult__divide__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1179 (all W hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),W),W))) # label(fact_599_real__le__refl) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1180 (all X_a (group_add(X_a) -> hAPP(X_a,X_a,uminus_uminus(X_a),zero_zero(X_a)) = zero_zero(X_a))) # label(fact_1937_minus__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1181 (all X_a (real_normed_vector(X_a) -> (all Y_2 all B_2 all X_2 all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially)) -> (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),Y_2),B_2),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),plus_plus(X_a)),X_2)),Y_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_2)),sequentially))))))) # label(fact_4505_LIMSEQ__add) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1182 (all Y all X all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Y),zero_zero(int)),P_5)) -> -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y)),zero_zero(int)),P_5)))))) # label(fact_919_zcong__zmult__prop3) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1183 (all X_a (banach(X_a) -> (all F all C all N_3 ((all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_3),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,F,hAPP(nat,nat,suc,N_1)))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,F,N_1))))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),C)) | hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)))))) # label(fact_4082_ratio__test__lemma2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1184 (all X hAPP(real,real,tan,X) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sin,X)),hAPP(real,real,cos,X))) # label(fact_2404_tan__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1185 (all Xa hAPP(nat,nat,suc,Xa) = hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less_eq(nat)),Xa)))) # label(fact_3869_card__bdd__nat__set__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1186 (all A_1 all B hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)) # label(fact_3248_gcd__semilattice__nat_Oinf_Oleft__idem) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1187 (all X_a (ring_1(X_a) -> ti(fun(fun(nat,X_a),fun(nat,X_a)),diffs(X_a)) = diffs(X_a))) # label(tsy_c_Transcendental_Odiffs_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1188 (all M hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),pi),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),M))))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = zero_zero(real)) # label(fact_2455_cos__pi__eq__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1189 (all A_1 all N all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),N))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),A_1))))) # label(fact_769_zprime__zdvd__power) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1190 (all X_1 hAPP(real,real,uminus_uminus(real),hAPP(nat,real,sin_coeff,X_1)) = hAPP(nat,real,hAPP(fun(nat,real),fun(nat,real),diffs(real),cos_coeff),X_1)) # label(fact_4148_cos__fdiffs) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1191 (all X_a all X_b (comm_monoid_add(X_b) -> (all F all P_1 all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),A_2))),P_1))) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),P_1)),F)),zero_zero(X_b))),A_2))))) # label(fact_4216_setsum__restrict__set_H_H) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1192 (all A_3 all P_4 (hBOOL(hAPP(int,bool,zprime,P_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),P_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int)))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,A_3),P_4))),one_one(int)),P_4)))))) # label(fact_3961_wset__zcong__prod__1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1193 (all P all Q (-hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fimplies,P),Q)) | hBOOL(Q) | -hBOOL(P))) # label(help_fimplies_3_1_U) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1194 (all X_a (linordered_field(X_a) -> (all B_2 all A_3 all 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(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)),A_3)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)))))))) # label(fact_1870_pos__divide__less__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1195 (all K hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),K))))) # label(fact_3843_finite__Collect__less__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1196 (all N one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),N)) # label(fact_3291_coprime__plus__one__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1197 (all X_a (no_zero_divisors(X_a) -> ti(fun(X_a,fun(X_a,X_a)),times_times(X_a)) = times_times(X_a))) # label(tsy_c_Groups_Otimes__class_Otimes_3_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1198 (all X_a (comple1803745165norder(X_a) -> (all A_3 all S (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),hAPP(fun(X_a,bool),X_a,complete_Sup_Sup(X_a),S))) <-> (exists X_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),X_1)) & hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),S)))))))) # label(fact_1818_less__Sup__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1199 (all X hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(complex,real,re,X)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,re,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,im,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(complex,real,re,hAPP(complex,complex,inverse_inverse(complex),X))) # label(fact_3044_complex__Re__inverse) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1200 (all X_a all X_b (comm_monoid_add(X_b) -> (all F all S all T_3 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),T_3)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),S),T_3)) -> ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),T_3),S))) -> hAPP(X_a,X_b,F,X_1) = zero_zero(X_b))) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),S) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),T_3))))))) # label(fact_4321_setsum__mono__zero__left) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1201 (all X_b (number(X_b) & semiring(X_b) -> (all A_1 all B all V hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_1),B)),hAPP(int,X_b,number_number_of(X_b),V)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_1),hAPP(int,X_b,number_number_of(X_b),V))),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),B),hAPP(int,X_b,number_number_of(X_b),V)))))) # label(fact_461_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1202 (all X_a all A_2 all X_1 (hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),X_1),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),A_2))) <-> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),powp(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),A_2)),X_1)))) # label(fact_4896_Powp__Pow__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1203 (all Ya hAPP(real,real,arcsin,Ya) = hAPP(fun(real,bool),real,the(real),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),ord_less_eq(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(real,real),fun(real,fun(real,bool))),combb(real,fun(real,bool),real),fequal(real)),sin)),Ya))))) # label(fact_4160_arcsin__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1204 (all N all K_4 all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),M)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),N)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)))))) # label(fact_800_dvd__diff__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1205 (all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),zero_zero(nat)),N) = zero_zero(nat)) # label(fact_4828_min__0L) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1206 (all Xa all F (hBOOL(hAPP(fun(nat,real),bool,summable(real),F)) -> ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N_1))),Xa))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,real),real,suminf(real),F)),Xa))))) # label(fact_4317_suminf__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1207 (all I (hAPP(int,nat,nat_1,I) = zero_zero(nat) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I),zero_zero(int))))) # label(fact_1016_nat__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1208 (all X_a all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),hAPP(bool,fun(X_a,fun(X_a,X_a)),if(X_a),fTrue),X),Y) = ti(X_a,X)) # label(help_If_1_1_T) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1209 (all X_a (metric_space(X_a) -> (all X all Y hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y) = hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Y),X)))) # label(fact_3543_dist__commute) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1210 (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_731_one__dvd) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1211 (all X_a (division_ring(X_a) -> (all A_1 all B all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C_1))))) # label(fact_1831_diff__divide__distrib) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1212 (all X_a (linordered_idom(X_a) -> (all X all W (hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,number_number_of(nat),W))) -> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,abs_abs(X_a),X)),hAPP(int,nat,number_number_of(nat),W)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),W)))))) # label(fact_2499_power__even__abs__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1213 (all X_a (boolean_algebra(X_a) -> (all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Y)),X)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),Y)))))) # label(fact_2985_compl__less__swap2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1214 (all Na all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A_3),zero_zero(real)),sequentially)) -> ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,A_3,N_1)))) -> ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,A_3,hAPP(nat,nat,suc,N_1))),hAPP(nat,real,A_3,N_1)))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Na)))),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3)))))))) # label(fact_4541_summable__Leibniz_H_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1215 (all Z_1 all X all Y all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),Y),M)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),Z_1)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Y),Z_1)),M)))) # label(fact_902_zcong__zpower) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1216 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),one_one(real))) -> hBOOL(hAPP(fun(nat,real),bool,bseq(real),hAPP(real,fun(nat,real),power_power(real),Xa)))))) # label(fact_3847_Bseq__realpow) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1217 (all N N != hAPP(nat,nat,suc,N)) # label(fact_1277_Suc__n__not__n) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1218 (all X_a all X_b all T_5 hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),T_5)),hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),T_5)) = ti(product_prod(X_a,X_b),T_5)) # label(fact_3148_surjective__pairing) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1219 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,arctan,Y))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,arctan,Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) & hAPP(real,real,tan,hAPP(real,real,arctan,Y)) = Y)) # label(fact_2452_arctan) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1220 (all X_a (real_normed_algebra(X_a) -> (all Y exists K_2 all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_1)),K_2)))))) # label(fact_2110_mult__left_Obounded) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1221 (all X_a (real_normed_field(X_a) -> (all K all Xa hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(X_a,fun(X_a,X_a),combk(X_a,X_a),K)),Xa),zero_zero(X_a)))))) # label(fact_3889_DERIV__const) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1222 (all X_a (ring_1(X_a) -> (all Z_1 all N hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)),N) = hAPP(int,X_a,ring_1_of_int(X_a),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Z_1),N))))) # label(fact_3612_of__int__power) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1223 (all B all A_1 all K_4 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B))),K_4) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),K_4)) # label(fact_1433_zdiv__zmod__equality) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1224 (all R_3 hAPP(real,real,norm_norm(real),R_3) = hAPP(real,real,abs_abs(real),R_3)) # label(fact_1947_real__norm__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1225 (all X_a all F all A_2 hAPP(int,real,real_1(int),hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co302546470etprod(X_a,int),F),A_2)) = hAPP(fun(X_a,bool),real,hAPP(fun(X_a,real),fun(fun(X_a,bool),real),big_co302546470etprod(X_a,real),hAPP(fun(X_a,int),fun(X_a,real),hAPP(fun(int,real),fun(fun(X_a,int),fun(X_a,real)),combb(int,real,X_a),real_1(int)),F)),A_2)) # label(fact_3968_real__of__int__setprod) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1226 (all X_a (division_ring(X_a) -> (all A_1 all B (ti(X_a,B) != zero_zero(X_a) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(X_a,X_a,uminus_uminus(X_a),B)))))) # label(fact_1973_nonzero__minus__divide__divide) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1227 (all X hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),pi))) = hAPP(real,real,sin,X)) # label(fact_2325_sin__periodic) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1228 (all Xa all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),hAPP(nat,fun(real,real),root,Na)),Xa)))) # label(fact_4904_isCont__real__root) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1229 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),Ma)),hAPP(nat,int,semiring_1_of_nat(int),Na))))) # label(fact_471_zle__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1230 (all X_a all X_b all X_c all X_d all X_e all X_f all Xa all P_1 ((all A_4 all B_4 all C_4 all D_2 all E all F_3 hBOOL(hAPP(product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),bool,P_1,hAPP(product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),hAPP(X_a,fun(product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))))),product_Pair(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),A_4),hAPP(product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))),product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),hAPP(X_b,fun(product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))),product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),product_Pair(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),B_4),hAPP(product_prod(X_d,product_prod(X_e,X_f)),product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))),hAPP(X_c,fun(product_prod(X_d,product_prod(X_e,X_f)),product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),product_Pair(X_c,product_prod(X_d,product_prod(X_e,X_f))),C_4),hAPP(product_prod(X_e,X_f),product_prod(X_d,product_prod(X_e,X_f)),hAPP(X_d,fun(product_prod(X_e,X_f),product_prod(X_d,product_prod(X_e,X_f))),product_Pair(X_d,product_prod(X_e,X_f)),D_2),hAPP(X_f,product_prod(X_e,X_f),hAPP(X_e,fun(X_f,product_prod(X_e,X_f)),product_Pair(X_e,X_f),E),F_3)))))))) -> hBOOL(hAPP(product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),bool,P_1,Xa)))) # label(fact_3172_prod__induct6) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1231 (all M all K_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K_4)) -> (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),hAPP(int,int,fact_fact(int),M)),hAPP(int,int,fact_fact(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),M),one_one(int))),K_4))))))) # label(fact_3225_fact__less__mono__int__aux) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1232 (all D_3 all C_1 all A_1 all B all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M)) -> (ti(int,B) = ti(int,C_1) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,C_1),D_3),M)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),D_3),M)))))) # label(fact_872_zcong__eq__trans) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1233 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B)),A_1)))) # label(fact_4423_gcd__le1__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1234 (all X_a all A_1 all B hAPP(product_prod(int,int),X_a,abs_Frac(X_a),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,A_1),B)) = hAPP(int,X_a,hAPP(int,fun(int,X_a),frac(X_a),A_1),B)) # label(fact_4445_Nitpick_Ofrac__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1235 (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)),hAPP(real,real,sqrt,X))))) # label(fact_2144_real__sqrt__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1236 (all X_a (semiring_div(X_a) -> (all B all B_5 all A_1 all C_1 all A_6 (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_6),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C_1) -> (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B_5),C_1) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_6),B_5)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C_1)))))) # label(fact_1289_mod__mult__cong) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1237 (all X_a (real_vector(X_a) -> (all A_1 all B all X hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A_1),B)),X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),B),X))))) # label(fact_4698_scaleR__left__diff__distrib) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1238 (all Xa (zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Xa),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))) <-> hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa)))) # label(fact_2505_even__nat__equiv__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1239 (all R_3 all A_1 hAPP(complex,real,im,hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R_3),A_1)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),hAPP(real,real,sin,A_1))) # label(fact_3195_Im__rcis) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1240 (all X_b all X_a (real_normed_vector(X_b) & real_normed_vector(X_a) -> (all F all L_2 all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3))) <-> (all R_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1)) -> (exists S_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),S_1)) & (all X_1 (ti(X_a,A_3) != ti(X_a,X_1) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_1),A_3))),S_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_a,X_b,F,X_1)),L_2))),R_1)))))))))))) # label(fact_4669_LIM__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1241 (all X_a all C all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),A_2)) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),B_1))))) # label(fact_2246_psubsetD) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1242 (all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A_3),zero_zero(real)),sequentially)) -> (hBOOL(hAPP(fun(nat,real),bool,monoseq(real),A_3)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,A_3,zero_zero(nat))),zero_zero(real))) -> (all N_1 hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastAtMost(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_1)),one_one(nat))))),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_1))))))))))) # label(fact_4548_summable__Leibniz_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1243 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,zero_zero(nat))),Na)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,zero_zero(nat))),Ma)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ma),Na))))) # label(fact_1447_one__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1244 (all X_a (archim1727834104eiling(X_a) -> (all Z all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z),hAPP(X_a,int,archim2085082626_floor(X_a),Xa))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)),one_one(X_a))),Xa)))))) # label(fact_3633_less__floor__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1245 (all X_a (real_normed_vector(X_a) -> (all F all G all L_2 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),G),L_2),sequentially)) -> (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),F)),G)),zero_zero(X_a)),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),L_2),sequentially))))))) # label(fact_4510_LIMSEQ__diff__approach__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1246 (all A_1 all B all P_5 (hBOOL(hAPP(nat,bool,prime,P_5)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),A_1)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),B))))) # label(fact_3078_prime__divprod) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1247 (all Xa hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),sin),Xa),hAPP(real,real,cos,Xa)))) # label(fact_2650_DERIV__sin) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1248 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),zero_zero(X_a)))))))) # label(fact_243_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1249 (all A_3 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),zero_zero(int)),Ma)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Ma),A_3)))) # label(fact_910_zcong__zero__equiv__div) # label(axiom) # label(non_clause). [assumption]. 7.62/7.65 1250 (all X_a (semiring_div(X_a) -> (all Y all Z_1 all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Z_1),X)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Z_1),Y)) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),X),Z_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),Y),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y)),Z_1)))))) # label(fact_1096_div__add) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1251 (all X_b all X_c all X_a all F1 all A_3 all B_2 hAPP(X_c,X_a,hAPP(X_b,fun(X_c,X_a),F1,A_3),B_2) = hAPP(product_prod(X_b,X_c),X_a,hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),product_prod_rec(X_b,X_c,X_a),F1),hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),A_3),B_2))) # label(fact_3180_prod_Orecs) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1252 (all R_3 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),R_3),one_one(real))),hAPP(int,real,real_1(int),hAPP(real,int,archim2085082626_floor(real),R_3))))) # label(fact_3426_real__of__int__floor__gt__diff__one) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1253 (all X_a (group_add(X_a) -> (all A_3 (hAPP(X_a,X_a,uminus_uminus(X_a),A_3) = zero_zero(X_a) <-> ti(X_a,A_3) = zero_zero(X_a))))) # label(fact_1936_neg__0__equal__iff__equal) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1254 (all X_a all C all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2))) -> -hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),A_2)))) # label(fact_2235_ComplE) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1255 (all Nat_3 zero_zero(nat) != hAPP(nat,nat,suc,Nat_3)) # label(fact_1315_nat_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1256 (all X_a (linorder(X_a) -> (all B all D_3 all A_1 all C_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),C_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),D_3)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),C_1),D_3)))))))) # label(fact_4853_min__max_Oinf__mono) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1257 (all X all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,exp(real),X)),hAPP(real,real,exp(real),Y))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y)))) # label(fact_1755_exp__less__cancel) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1258 (all Q_3 all B all R_3 all C_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_3),zero_zero(int))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Q_3),C_1))),R_3)),zero_zero(int))))))) # label(fact_1477_zmult2__lemma__aux2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1259 (all W hAPP(int,int,number_number_of(int),hAPP(int,int,uminus_uminus(int),W)) = hAPP(int,int,uminus_uminus(int),hAPP(int,int,number_number_of(int),W))) # label(fact_2063_minus__numeral__code_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1260 (all X1_2 all X2_2 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma)) -> (hAPP(int,int,hAPP(int,fun(int,int),standardRes,Ma),X1_2) = hAPP(int,int,hAPP(int,fun(int,int),standardRes,Ma),X2_2) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X1_2),X2_2),Ma))))) # label(fact_1470_StandardRes__prop2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1261 (all X_a (archim1727834104eiling(X_a) -> (all X hAPP(X_a,int,archimedean_ceiling(X_a),X) = hAPP(int,int,uminus_uminus(int),hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)))))) # label(fact_3320_ceiling__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1262 (all X_a all A_3 all F all Va all Na hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),hAPP(bool,fun(X_a,fun(X_a,X_a)),if(X_a),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),A_3),F),Na)),hAPP(X_a,X_a,hAPP(nat,fun(X_a,X_a),F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),Na)),hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),A_3),F),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),Na)))) = hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),A_3),F),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),Va)),Na))) # label(fact_4179_nat__rec__add__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1263 (all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(complex,real,norm_norm(complex),X))),hAPP(complex,real,norm_norm(complex),X)))) # label(fact_1903_complex__mod__minus__le__complex__mod) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1264 (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,hAPP(int,fun(nat,int),power_power(int),X),N))))) # label(fact_268_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1265 (all X_a all F all G all Ma all S ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),S)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(X_a,int,F,X_1)),hAPP(X_a,int,G,X_1)),Ma)))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co302546470etprod(X_a,int),F),S)),hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co302546470etprod(X_a,int),G),S)),Ma)))) # label(fact_4444_setprod__same__function__zcong) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1266 (all Z_1 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),Z_1)),Z_1) != zero_zero(int)) # label(fact_96_odd__nonzero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1267 (all X_b all X_c all X_a all F all A_3 all B_2 hAPP(product_prod(X_b,X_c),X_a,hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),product_prod_case(X_b,X_c,X_a),F),hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),A_3),B_2)) = hAPP(X_c,X_a,hAPP(X_b,fun(X_c,X_a),F,A_3),B_2)) # label(fact_4021_split__conv) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1268 (all X_a (linorder(X_a) -> (all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ya),Xa)) | ti(X_a,Xa) = ti(X_a,Ya) <-> -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya)))))) # label(fact_534_not__less__iff__gr__or__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1269 (all P_1 all Ma all Na (zero_zero(nat) = Na & hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) | (exists Q_4 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),hAPP(nat,nat,suc,Q_4)))) & hBOOL(hAPP(nat,bool,P_1,Q_4)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),Q_4)),Ma)))) <-> hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),Ma),Na))))) # label(fact_1496_split__div_H) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1270 (all Ma all K all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ma),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),K))) <-> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na))))) # label(fact_643_mult__le__cancel2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1271 (all P_1 all Na all A_3 all B_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2)) -> hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(product_prod(real,real),real,product_snd(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),Na))),hAPP(product_prod(real,real),real,product_fst(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),Na))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B_2),A_3)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Na)))) # label(fact_3162_Bolzano__bisect__diff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1272 (all Xa all Ya (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Xa),Ya)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),Xa)),hAPP(int,real,real_1(int),Ya))))) # label(fact_3356_real__of__int__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1273 (all K all P_1 all D_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_1)) -> ((all X_1 (hBOOL(hAPP(int,bool,P_1,X_1)) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),D_1))))) -> (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,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),D_1)))))))))) # label(fact_579_incr__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1274 (all J all A_3 all P_4 (hBOOL(hAPP(int,bool,zprime,P_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),zero_zero(int)),P_4)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_4),A_3)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J),zero_zero(int)),P_4)) -> hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) = hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),hAPP(int,fun(int,fun(int,fun(int,bool))),multInvPair,A_3),P_4),J)))))))) # label(fact_1682_MultInvPair__card__two) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1275 (all X_a combi(X_a) = ti(fun(X_a,X_a),combi(X_a))) # label(tsy_c_COMBI_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1276 (all D_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,zero_zero(nat)),D_1)) <-> D_1 = one_one(nat))) # label(fact_2893_coprime__0_H) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1277 (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_962_dvd_Oless__irrefl) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1278 (all T_2 all T_1 (preorder(T_1) -> preorder(fun(T_2,T_1)))) # label(arity_fun___Orderings_Opreorder) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1279 (all X hAPP(real,real,tan,hAPP(real,real,uminus_uminus(real),X)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,tan,X))) # label(fact_2369_tan__minus) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1280 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X))),X)))) # label(fact_1731_ln__add__one__self__le__self) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1281 (all N all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> (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(int),N),P_5)) -> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,N),P_5) = one_one(int))))) # label(fact_3494_zless__zprime__imp__zrelprime) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1282 (all Z_1 all X all Y all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),Y)),one_one(int)),P_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Y),Z_1))),one_one(int)),P_5)))) # label(fact_762_zcong__zpower__zmult) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1283 (all L_4 all K_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K_4),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),L_4)) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,K_4),L_4) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),K_4),L_4)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),K_4),L_4))))) # label(fact_1637_negDivAlg__div__mod) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1284 (all U hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(nat,fun(nat,bool),ord_lessThan(nat),U)) = U) # label(fact_4342_card__lessThan) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1285 (all K_4 all L_4 hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),hAPP(int,rat,ring_1_of_int(rat),K_4)),hAPP(int,rat,ring_1_of_int(rat),L_4)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,K_4),L_4)) # label(fact_3772_Fract__of__int__quotient) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1286 (all I (ti(int,I) != zero_zero(int) -> hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),dvd_dvd(int)),I)))))) # label(fact_3897_finite__divisors__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1287 (all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) <-> -hBOOL(hAPP(int,bool,nat_neg,Xa)))) # label(fact_1605_not__neg__eq__ge__0) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1288 (all X (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))) -> hAPP(nat,nat,suc,zero_zero(nat)) = X | zero_zero(nat) = X)) # label(fact_1373_nat__lt__two__imp__zero__or__one) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1289 (all Y all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),X)),one_one(real))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),Y)),one_one(real))) -> hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,arctan,X)),hAPP(real,real,arctan,Y)) = hAPP(real,real,arctan,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y))))))) # label(fact_2117_arctan__add) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1290 (all X_a all X_c (comm_monoid_add(X_c) -> ti(fun(fun(X_a,X_c),fun(fun(X_a,bool),X_c)),big_co1399186613setsum(X_a,X_c)) = big_co1399186613setsum(X_a,X_c))) # label(tsy_c_Big__Operators_Ocomm__monoid__add__class_Osetsum_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1291 (all X_a (number_ring(X_a) -> -hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls)))))) # label(fact_2160_not__iszero__Numeral1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1292 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,M)),N)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)))) # label(fact_1386_Suc__le__lessD) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1293 (all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) -> hAPP(nat,nat,suc,zero_zero(nat)) = N | N = zero_zero(nat))) # label(fact_1502_less__2__cases) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1294 (all K all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),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_644_mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1295 (all X_a (number_ring(X_a) -> (all Z_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Z_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z_1),Z_1)))) # label(fact_545_mult__2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1296 (all X_a (real_n2089651433ebra_1(X_a) -> (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),Xa)),one_one(real))) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa)),zero_zero(X_a)),sequentially)))))) # label(fact_4460_LIMSEQ__power__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1297 (all Na all P_1 (-hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) -> (hBOOL(hAPP(nat,bool,P_1,Na)) -> (exists K_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_1),Na)) & hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_1),one_one(nat)))) & (all I_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),K_1)) -> -hBOOL(hAPP(nat,bool,P_1,I_2))))))))) # label(fact_578_ex__least__nat__less) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1298 (all Ma (Ma != zero_zero(nat) -> hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),dvd_dvd(nat)),Ma)))))) # label(fact_3898_finite__divisors__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1299 (all N all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N),X))),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,real_1(nat),N)),hAPP(nat,real,real_1(nat),X))))) # label(fact_2398_real__of__nat__div4) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1300 (all X_a (archim1727834104eiling(X_a) -> (all X all V hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(int,X_a,number_number_of(X_a),V))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),hAPP(int,int,number_number_of(int),V))))) # label(fact_2914_floor__add__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1301 (all A_1 all B ((zero_zero(int) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) -> zero_zero(int) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),hAPP(int,int,uminus_uminus(int),B))) & (zero_zero(int) != hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) -> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),hAPP(int,int,uminus_uminus(int),B)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)),B)))) # label(fact_2090_zmod__zminus2__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1302 (all X_a (linordered_field(X_a) -> (all B_2 all A_3 all C (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_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)),A_3)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)),B_2))))))) # label(fact_1890_neg__divide__le__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1303 (all X_a (field_char_0(X_a) -> (all A_3 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),field_char_0_Rats(X_a))) -> (ti(X_a,A_3) != zero_zero(X_a) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_3)),field_char_0_Rats(X_a)))))))) # label(fact_3704_nonzero__Rats__inverse) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1304 (all Xa all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa),Ya))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),Xa)),Ya)))) # label(fact_2020_real__0__le__add__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1305 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N) = zero_zero(nat))) # label(fact_836_diff__is__0__eq_H) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1306 (all W (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(int,int,number_number_of(int),W))) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,one_one(int)),hAPP(int,int,number_number_of(int),W)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(int,int,number_number_of(int),W))) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,one_one(int)),hAPP(int,int,number_number_of(int),W)) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,hAPP(int,int,number_number_of(int),W)),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,one_one(int)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,number_number_of(int),W))))))) # label(fact_1648_posDivAlg__eqn__1__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1307 (all X_a (topological_space(X_a) -> (all F all L (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),L),sequentially)) <-> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),suc)),L),sequentially)))))) # label(fact_4495_LIMSEQ__Suc__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1308 (all X_a (real_normed_vector(X_a) -> (all K all F (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) -> hAPP(fun(nat,X_a),X_a,suminf(X_a),F) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K))),hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K)))))))) # label(fact_4195_suminf__split__initial__segment) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1309 (all X_a all A_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2))) -> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)))) # label(fact_2746_card__ge__0__finite) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1310 (all X_a (ord(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (ti(X_a,C_1) = ti(X_a,B) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),C_1))))))) # label(fact_506_ord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1311 (all M all V hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),M)),hAPP(int,nat,number_number_of(nat),V)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,suc,M)))),hAPP(int,nat,number_number_of(nat),V))) # label(fact_1492_Suc__div__eq__add3__div__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1312 (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)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Y))))))) # label(fact_486_sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1313 (all A_1 all X hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A_1),X) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_1),X)) # label(fact_4717_real__scaleR__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1314 (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),hAPP(nat,nat,suc,N))))) # label(fact_1329_less__SucI) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1315 (all X_a (order(X_a) -> (all Na all N_6 all F ((all N_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,F,N_1)),hAPP(nat,X_a,F,hAPP(nat,nat,suc,N_1))))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),N_6)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,F,Na)),hAPP(nat,X_a,F,N_6)))))))) # label(fact_1573_lift__Suc__mono__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1316 (all X_a all X_b (comm_monoid_mult(X_b) -> (all F all S all T_3 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),T_3)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),S),T_3)) -> ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),T_3),S))) -> hAPP(X_a,X_b,F,X_1) = one_one(X_b))) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),S) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),T_3))))))) # label(fact_3988_setprod__mono__one__left) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1317 (all I all J all K (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K)),J)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),K))))) # label(fact_837_less__diff__conv) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1318 (all K all Ma all Na (Na = Ma | K = zero_zero(nat) <-> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Na) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Ma))) # label(fact_693_nat__mult__eq__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1319 (all M hAPP(int,int,abs_abs(int),M) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,zero_zero(int)),M)) # label(fact_3477_zgcd__0__left) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1320 (all Code_numeral_2 hAPP(code_code_numeral,nat,code_c271388182l_size,hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,Code_numeral_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(code_code_numeral,nat,code_c271388182l_size,Code_numeral_2)),hAPP(nat,nat,suc,zero_zero(nat)))) # label(fact_2951_code__numeral_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1321 (all X_a (archim1727834104eiling(X_a) -> (all X hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),one_one(int)) = hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),one_one(X_a)))))) # label(fact_2913_floor__add__one) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1322 (all D all Xa all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (zero_zero(real) != Xa -> ((hBOOL(hAPP(nat,bool,even_odd_even(nat),Na)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) -> D = hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),Na)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,zero_zero(nat)))))))) -> ((hBOOL(hAPP(nat,bool,even_odd_even(nat),Na)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),zero_zero(real))) -> hAPP(real,real,uminus_uminus(real),hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),Na)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,zero_zero(nat))))))) = D)) -> ((-hBOOL(hAPP(nat,bool,even_odd_even(nat),Na)) -> D = hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),Na)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,zero_zero(nat))))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),root,Na)),Xa),D)))))))) # label(fact_2670_DERIV__real__root__generic) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1323 (all Xa all Ya (Xa = zero_zero(nat) & Ya = zero_zero(nat) <-> zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Xa),Ya))) # label(fact_3282_gcd__lcm__complete__lattice__nat_Oinf__eq__top__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1324 (all Z_1 hAPP(complex,real,re,hAPP(real,complex,of_real(complex),Z_1)) = Z_1) # label(fact_3018_Re__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1325 (all X hAPP(rat,real,ratreal,hAPP(rat,rat,inverse_inverse(rat),X)) = hAPP(real,real,inverse_inverse(real),hAPP(rat,real,ratreal,X))) # label(fact_3794_real__inverse__code) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1326 (all X_a (division_ring(X_a) -> (all A_3 all B_2 (ti(X_a,B_2) != zero_zero(X_a) -> (ti(X_a,A_3) = ti(X_a,B_2) <-> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),B_2) = one_one(X_a)))))) # label(fact_1855_right__inverse__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1327 (all Xa all Ya (hAPP(complex,complex,cnj,Ya) = hAPP(complex,complex,cnj,Xa) <-> Ya = Xa)) # label(fact_2960_complex__cnj__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1328 (all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),hAPP(nat,fun(real,real),root,Na)),zero_zero(real))))) # label(fact_4938_isCont__root__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1329 (all V_2 all V ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_2),pls)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_2)) = hAPP(int,nat,number_number_of(nat),V)) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_2),pls)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_2)) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),V_2)))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls)) -> hAPP(int,nat,number_number_of(nat),V_2) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_2))))) # label(fact_36_add__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1330 (all X_a (archim1727834104eiling(X_a) -> (all Xa all Va (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archim2085082626_floor(X_a),Xa)),hAPP(int,int,number_number_of(int),Va))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),hAPP(int,X_a,number_number_of(X_a),Va))))))) # label(fact_2911_floor__less__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1331 (all S (hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),S)) <-> (exists K_1 hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less_eq(fun(nat,bool)),S),hAPP(nat,fun(nat,bool),ord_atMost(nat),K_1)))))) # label(fact_4963_finite__nat__iff__bounded__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1332 (all A_1 all B all C_1 all D_3 hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,C_1),D_3)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A_1),C_1)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),D_3))) # label(fact_2765_complex__diff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1333 (all X_a (real_normed_vector(X_a) -> (all X hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,sgn_sgn(X_a),X)) = hAPP(X_a,X_a,sgn_sgn(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X))))) # label(fact_2272_sgn__minus) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1334 (all Na all Ma (ti(int,Ma) = ti(int,Na) <-> hAPP(int,quickcheck_code_int,quickcheck_of_int,Ma) = hAPP(int,quickcheck_code_int,quickcheck_of_int,Na))) # label(fact_3563_Quickcheck__Narrowing_Oof__int__inject) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1335 (all A_2 all B_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,bool),fun(fun(int,bool),bool),ord_less_eq(fun(int,bool)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B_1))) <-> hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less_eq(fun(nat,bool)),A_2),B_1)))) # label(fact_3676_transfer__nat__int__set__relations_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1336 (all X_a (linordered_ring(X_a) -> (all A_1 -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),A_1)),zero_zero(X_a)))))) # label(fact_440_not__square__less__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1337 (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),X),pi)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,sin,X)))))) # label(fact_2285_sin__gt__zero__pi) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1338 (all X_a (linord1117847801e_zero(X_a) -> (all Xa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),zero_zero(X_a))) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),Xa)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),Xa)),one_one(X_a))))))) # label(fact_2611_inverse__le__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1339 (all M all N all R_4 all S_4 all S_3 all T_6 all T_5 all R_3 ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_3),zero_zero(int))) -> hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),hAPP(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))),hAPP(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))))),xzgcda,M),N),R_4),R_3),S_4),S_3),T_6),T_5) = hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),hAPP(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))),hAPP(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))))),xzgcda,M),N),R_3),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),R_4),R_3)),S_3),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),S_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),R_4),R_3)),S_3))),T_5),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),T_6),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),R_4),R_3)),T_5)))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_3),zero_zero(int))) -> hAPP(product_prod(int,int),product_prod(int,product_prod(int,int)),hAPP(int,fun(product_prod(int,int),product_prod(int,product_prod(int,int))),product_Pair(int,product_prod(int,int)),R_4),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),S_4),T_6)) = hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),hAPP(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))),hAPP(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))))),xzgcda,M),N),R_4),R_3),S_4),S_3),T_6),T_5)))) # label(fact_3108_xzgcda_Osimps) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1340 (all X_a (linordered_idom(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,sgn_sgn(X_a),A_1)),hAPP(X_a,X_a,sgn_sgn(X_a),B)) = hAPP(X_a,X_a,sgn_sgn(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))))) # label(fact_2264_sgn__times) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1341 (all X_a (real_normed_vector(X_a) -> (all X_2 ((exists K_1 ((exists X_1 all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,X_2,N_1)),hAPP(X_a,X_a,uminus_uminus(X_a),X_1)))),K_1))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_1)))) <-> hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2)))))) # label(fact_3515_Bseq__iff2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1342 (all X -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,abs_abs(real),X)),one_one(real))),X))) # label(fact_1114_abs__add__one__not__less__self) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1343 (all A_3 all 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),A_3),B_2)) <-> hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_2)),one_one(rat)))))) # label(fact_3816_Fract__le__one__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1344 (all Va all V_1 ((hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),Va))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),V_1)),zero_zero(int)))) & (-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),Va))) -> (hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V_1))) -> hAPP(int,int,number_number_of(int),Va) = zero_zero(int)) & (-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V_1))) -> ti(int,Va) = ti(int,V_1))) <-> hAPP(int,nat,number_number_of(nat),V_1) = hAPP(int,nat,number_number_of(nat),Va))) # label(fact_1620_eq__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1345 (all X_a (real_n2089651433ebra_1(X_a) -> (all F hAPP(nat,X_a,F,zero_zero(nat)) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),F)),hAPP(X_a,fun(nat,X_a),power_power(X_a),zero_zero(X_a))))))) # label(fact_4000_powser__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1346 (all A_1 all B hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),hAPP(int,int,uminus_uminus(int),B)) = hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),A_1)),B))) # label(fact_2071_zmod__zminus2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1347 (all X_a (real_normed_vector(X_a) -> (all Y_2 all B_2 all X_2 all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially)) -> (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),Y_2),B_2),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),plus_plus(X_a)),X_2)),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),Y_2))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),hAPP(X_a,X_a,uminus_uminus(X_a),B_2))),sequentially))))))) # label(fact_4509_LIMSEQ__add__minus) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1348 (all B_5 all Q_5 all R_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_5),Q_5)),R_4)),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_5),zero_zero(int))))))) # label(fact_582_q__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1349 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,arcsin,Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) & hAPP(real,real,sin,hAPP(real,real,arcsin,Y)) = Y & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,arcsin,Y)))))) # label(fact_2251_arcsin) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1350 (all B_5 all A_6 all A_1 all B (zero_zero(nat) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B) -> (A_1 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_6),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)) -> (hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)) = B -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_6),B_5) = one_one(nat))))) # label(fact_3311_gcd__coprime__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1351 (all X_a (archim1727834104eiling(X_a) -> (all R_3 hAPP(real,int,archimedean_ceiling(real),hAPP(int,real,real_1(int),hAPP(X_a,int,archim2085082626_floor(X_a),R_3))) = hAPP(X_a,int,archim2085082626_floor(X_a),R_3)))) # label(fact_3382_ceiling__floor) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1352 (all X_a (semiring_div(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),B) = zero_zero(X_a)))) # label(fact_1351_mod__mult__self2__is__0) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1353 (all X_a (topological_space(X_a) -> (all G all L all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,complex),fun(X_a,real),hAPP(fun(complex,real),fun(fun(X_a,complex),fun(X_a,real)),combb(complex,real,X_a),re),G)),hAPP(complex,real,re,L)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4621_Re_OLIM) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1354 (all X_b all X_a (topological_space(X_a) & real_normed_vector(X_b) -> (all F all L all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> (zero_zero(X_b) != ti(X_b,L) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),sgn_sgn(X_b)),F)),hAPP(X_b,X_b,sgn_sgn(X_b),L)),hAPP(X_a,filter(X_a),at(X_a),A_3)))))))) # label(fact_4615_LIM__sgn) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1355 (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_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)),N)))) # label(fact_1543_mod__le__divisor) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1356 (all X_a all F1 all F2 all Nat hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),F1),F2),hAPP(nat,nat,suc,Nat)) = hAPP(X_a,X_a,hAPP(nat,fun(X_a,X_a),F2,Nat),hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),F1),F2),Nat))) # label(fact_4182_nat__rec__Suc) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1357 (all X_a (linord219039673up_add(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),hAPP(X_a,X_a,uminus_uminus(X_a),A_3))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),zero_zero(X_a))))))) # label(fact_1950_le__minus__self__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1358 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B)) -> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) = zero_zero(int)))) # label(fact_1107_div__pos__pos__trivial) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1359 (all A_1 all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int)))) -> hAPP(int,int,hAPP(int,fun(int,int),inv,P_5),A_1) != ti(int,A_1))))) # label(fact_1179_inv__distinct) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1360 (all X_a (real_normed_vector(X_a) -> (all B_2 all F all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),A_3),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),plus_plus(X_a)),F)),B_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_2)),sequentially)))))) # label(fact_4506_LIMSEQ__add__const) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1361 (all A_1 (ti(int,A_1) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),zero_zero(int)) & zero_zero(int) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),zero_zero(int)))) # label(fact_1366_DIVISION__BY__ZERO) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1362 (all W hAPP(complex,complex,cnj,hAPP(int,complex,number_number_of(complex),W)) = hAPP(int,complex,number_number_of(complex),W)) # label(fact_2968_complex__cnj__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1363 (all R_3 all P_5 all Q_3 (hAPP(product_prod(int,int),product_prod(int,int),normalize,R_3) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_5),Q_3) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Q_3)))) # label(fact_3749_normalize__denom__pos) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1364 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,K1)),hAPP(int,int,bit0,K2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)))) # label(fact_277_less__eq__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1365 (all X_a (semiring_div(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B)) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),B),A_1)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1)),A_1))))) # label(fact_1094_dvd__div__mult) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1366 (all X_c all X_b all X_a (topological_space(X_b) & topological_space(X_c) & metric_space(X_a) -> (all G all C all F all B_2 all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),B_2),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> (hBOOL(hAPP(filter(X_b),bool,hAPP(X_c,fun(filter(X_b),bool),hAPP(fun(X_b,X_c),fun(X_c,fun(filter(X_b),bool)),tendsto(X_b,X_c),G),C),hAPP(X_b,filter(X_b),at(X_b),B_2))) -> ((exists D_2 ((all X_1 (ti(X_a,X_1) != ti(X_a,A_3) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X_1),A_3)),D_2)) -> hAPP(X_a,X_b,F,X_1) != ti(X_b,B_2))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_2)))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_c,fun(filter(X_a),bool),hAPP(fun(X_a,X_c),fun(X_c,fun(filter(X_a),bool)),tendsto(X_a,X_c),hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c)),combb(X_b,X_c,X_a),G),F)),C),hAPP(X_a,filter(X_a),at(X_a),A_3))))))))) # label(fact_4627_metric__LIM__compose2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1367 (all X_a all B_1 all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1))))) # label(fact_2737_finite__Diff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1368 (all Xa (ti(int,Xa) = zero_zero(int) <-> zero_zero(real) = hAPP(int,real,real_1(int),Xa))) # label(fact_3347_real__of__int__zero__cancel) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1369 (all X_a all A_2 hAPP(nat,real,real_1(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)) = hAPP(fun(X_a,bool),real,hAPP(fun(X_a,real),fun(fun(X_a,bool),real),big_co1399186613setsum(X_a,real),hAPP(real,fun(X_a,real),combk(real,X_a),one_one(real))),A_2)) # label(fact_4256_real__of__card) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1370 (all X_a (real_algebra_1(X_a) & real_normed_vector(X_a) -> (all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(real,X_a,of_real(X_a),X)),hAPP(real,X_a,of_real(X_a),Y)) = hAPP(real,X_a,of_real(X_a),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y))))) # label(fact_2798_of__real_Oadd) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1371 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),zero_zero(X_a)))))))) # label(fact_226_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1372 (all X_a (comm_semiring_1(X_a) -> (all M hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),M),M) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a))),M)))) # label(fact_462_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1373 (all X_a (one(X_a) -> ti(X_a,one_one(X_a)) = one_one(X_a))) # label(tsy_c_Groups_Oone__class_Oone_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1374 (all Xa all Ya all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa) = hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Ya) <-> Xa = Ya))) # label(fact_2676_real__root__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1375 (all Xa all Ya all Z (hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Ya),Z) = Xa <-> Ya = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa),Z))) # label(fact_1211_eq__diff__eq_H) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1376 (all M all K_4 all N (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),K_4),N) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),N))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),M))))) # label(fact_4378_coprime__dvd__mult__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1377 (all X_a (linorder(X_a) -> (all Xa all Ya (ti(X_a,Ya) = ti(X_a,Xa) <-> hAPP(X_a,fun(X_a,bool),ord_lessThan(X_a),Ya) = hAPP(X_a,fun(X_a,bool),ord_lessThan(X_a),Xa))))) # label(fact_4343_lessThan__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1378 (all X_a (boolean_algebra(X_a) -> (all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),hAPP(X_a,X_a,uminus_uminus(X_a),X))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,uminus_uminus(X_a),Y))))))) # label(fact_2982_compl__le__swap1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1379 (all I all P_1 all P_4 ((all I_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),P_4)) -> (hBOOL(hAPP(nat,bool,P_1,I_2)) -> hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,I_2)),P_4)))))) -> (hBOOL(hAPP(nat,bool,P_1,I)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),P_4)) -> hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))))))) # label(fact_1571_mod__induct__0) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1380 (all V_2 all K_4 all V ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),V),V_2))),K_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),V_2)),K_4))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),V_2)),K_4)) = zero_zero(nat)))) # label(fact_648_nat__number__of__mult__left) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1381 (all M all N (N = M -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)))) # label(fact_355_eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1382 (all R_3 all A_1 hAPP(real,real,abs_abs(real),R_3) = hAPP(complex,real,norm_norm(complex),hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R_3),A_1))) # label(fact_3191_complex__mod__rcis) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1383 (all X_a (linorder(X_a) -> (all B all A_1 all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),B),C_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),C_1))))) # label(fact_4835_min__max_Oinf_Oleft__commute) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1384 (all X_a (divisi14063676e_zero(X_a) -> (all A_1 all B (hAPP(X_a,X_a,inverse_inverse(X_a),B) = hAPP(X_a,X_a,inverse_inverse(X_a),A_1) -> ti(X_a,B) = ti(X_a,A_1))))) # label(fact_2568_inverse__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1385 (all X_a all R1_1 all R2_1 (hBOOL(hAPP(fun(X_a,fun(X_a,bool)),bool,hAPP(fun(X_a,fun(X_a,bool)),fun(fun(X_a,fun(X_a,bool)),bool),ord_less_eq(fun(X_a,fun(X_a,bool))),R1_1),R2_1)) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,bool),accp(X_a),R2_1)),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,bool),accp(X_a),R1_1))))) # label(fact_2938_accp__subset) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1386 (all X all Y hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),hAPP(complex,complex,cnj,X)),hAPP(complex,complex,cnj,Y)) = hAPP(complex,complex,cnj,hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),X),Y))) # label(fact_2955_complex__cnj__divide) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1387 (all N hAPP(nat,nat,suc,N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) # label(fact_1398_Suc__eq__plus1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1388 (all V_2 ((-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V_2))) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(int,nat,number_number_of(nat),V_2)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),one_one(int)),hAPP(int,int,number_number_of(int),V_2)))) & (hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V_2))) -> hAPP(nat,nat,suc,zero_zero(nat)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(int,nat,number_number_of(nat),V_2))))) # label(fact_1582_one__mod__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1389 (all A_3 all A_2 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Ma)) -> (hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),Ma)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),A_2)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),hAPP(int,int,hAPP(int,fun(int,int),hAPP(fun(int,bool),fun(int,fun(int,int)),rRset2norRR,A_2),Ma),A_3)),Ma)))))) # label(fact_3649_RRset2norRR__correct1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1390 (all X_a (archim1727834104eiling(X_a) -> (all N hAPP(nat,int,semiring_1_of_nat(int),N) = hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N))))) # label(fact_2901_floor__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1391 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) <-> (exists K_1 Na = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),K_1)))) # label(fact_411_le__iff__add) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1392 (all X_a (real_normed_vector(X_a) -> (all K all F (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) -> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K)))))))) # label(fact_4040_summable__ignore__initial__segment) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1393 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,arcsin,Y)))))) # label(fact_2229_arcsin__lbound) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1394 (all A_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)),B)))) # label(fact_1419_pos__mod__conj) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1395 (all N ti(int,N) = hAPP(real,int,archim2085082626_floor(real),hAPP(int,real,real_1(int),N))) # label(fact_3352_floor__real__of__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1396 (all X_a all X_b (comm_monoid_mult(X_b) -> big_co302546470etprod(X_a,X_b) = ti(fun(fun(X_a,X_b),fun(fun(X_a,bool),X_b)),big_co302546470etprod(X_a,X_b)))) # label(tsy_c_Big__Operators_Ocomm__monoid__mult__class_Osetprod_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1397 (all X (hBOOL(hAPP(int,bool,even_odd_even(int),X)) -> hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = ti(int,X))) # label(fact_1630_two__times__even__div__two) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1398 (all N all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),N)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),X)),one_one(real)))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),N),X)))) # label(fact_3418_lemma__floor2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1399 (all X_a (group_add(X_a) -> (all A_1 all B (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B) = zero_zero(X_a) -> hAPP(X_a,X_a,uminus_uminus(X_a),A_1) = ti(X_a,B))))) # label(fact_1961_minus__unique) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1400 (all X_a (semiring_div(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),zero_zero(X_a))))) # label(fact_1281_mod__by__0) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1401 (all A_3 all P_4 all X_1 (hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),X_1),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,A_3),P_4))) -> hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),X_1)))) # label(fact_2731_SetS__elems__finite) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1402 (all Ma all K all Na (hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),K),Na) = one_one(int) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),Ma)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Ma),Na)))))) # label(fact_4377_coprime__dvd__mult__iff__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1403 (all X_a (real_vector(X_a) -> (all A_1 all B all X hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),B),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),B),X))))) # label(fact_4713_scaleR__left__commute) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1404 (all X_a (linord1117847801e_zero(X_a) -> (all A_3 all B_2 all C (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),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(X_a),C),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)))) & (-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),A_3),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)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)),B_2))))))) # label(fact_1867_less__divide__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1405 (all N hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),hAPP(nat,nat,fact,N)))) # label(fact_3572_fact__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1406 (all X_a all Xa hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),hAPP(X_a,fun(X_a,bool),fequal(X_a),Xa)) = ti(X_a,Xa)) # label(fact_4810_some__sym__eq__trivial) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1407 (all X all Y all M (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y)),M) -> one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),M))) # label(fact_4420_invertible__coprime__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1408 (all X_a (banach(X_a) & real_normed_field(X_a) -> (all Xa hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))),hAPP(X_a,X_a,exp(X_a),Xa)))))) # label(fact_4777_exp__converges) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1409 (all X_a (preorder(X_a) -> (all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)))))) # label(fact_524_order__less__asym_H) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1410 (all X_b all X_a (ring_1(X_a) -> (all F all A_2 hAPP(int,X_a,ring_1_of_int(X_a),hAPP(fun(X_b,bool),int,hAPP(fun(X_b,int),fun(fun(X_b,bool),int),big_co1399186613setsum(X_b,int),F),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,int),fun(X_b,X_a),hAPP(fun(int,X_a),fun(fun(X_b,int),fun(X_b,X_a)),combb(int,X_a,X_b),ring_1_of_int(X_a)),F)),A_2)))) # label(fact_4217_of__int__setsum) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1411 (all K_4 all I_1 all J_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),J_2)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),I_1),K_4)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),J_2),K_4))))) # label(fact_35_zadd__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1412 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all Ya (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),Ya)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Ya))))))) # label(fact_236_le__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1413 (all A_2 hAPP(fun(int,bool),nat,finite_card(int),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2)) = hAPP(fun(nat,bool),nat,finite_card(nat),A_2)) # label(fact_3671_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1414 (all B all M all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),A_1)) -> -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M))))))) # label(fact_905_zcong__not) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1415 (all A_3 hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),abs_abs(real)),A_3))) # label(fact_4907_isCont__abs) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1416 (all X_a (linord20386208strict(X_a) -> (all A_1 all C_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),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_eq(X_a),A_1),B))))))) # label(fact_475_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1417 (all X_a (ring(X_a) -> (all A_3 all E_2 all C all B_2 all D_1 (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),E_2)),D_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),E_2)),C) <-> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2)),E_2)),C) = ti(X_a,D_1))))) # label(fact_753_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.66 1418 (all X_a (ring_char_0(X_a) & number_ring(X_a) -> (all W -hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,W))))))) # label(fact_2155_iszero__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1419 (all X_a (archim1727834104eiling(X_a) -> (all X hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),one_one(X_a))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(X_a,int,archimedean_ceiling(X_a),X)),one_one(int))))) # label(fact_3330_ceiling__add__one) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1420 (all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,real_1(nat),Na))))) # label(fact_2405_real__of__nat__gt__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1421 (all X_a ti(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),pow(X_a)) = pow(X_a)) # label(tsy_c_Set_OPow_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1422 (all N hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N)))),zero_zero(int)))) # label(fact_2095_negative__zless__0) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1423 (all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,zfact,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int)))),hAPP(int,int,number_number_of(int),min)),P_5)))) # label(fact_1070_Wilson__Russ) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1424 (all X_a (semiring_div(X_a) -> (all A_1 all C_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)),C_1)))) # label(fact_1287_mod__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1425 (all X_a (real_normed_field(X_a) -> (all Y_2 all B_2 all X_2 all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),X_2),A_3)) -> (hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),Y_2),B_2)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),X_2)),Y_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2)))))))) # label(fact_4114_sums__diff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1426 (all X_b all X_a (real_normed_vector(X_b) & real_normed_vector(X_a) -> (all F all L_2 all A_3 ((all R_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1)) -> (exists S_1 ((all X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_1),A_3))),S_1)) & ti(X_a,X_1) != ti(X_a,A_3) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_a,X_b,F,X_1)),L_2))),R_1)))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),S_1)))))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4688_LIM__I) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1427 (all X_a (linordered_idom(X_a) -> (all Xa all Wa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),hAPP(int,nat,number_number_of(nat),Wa)))) <-> ti(X_a,Xa) != zero_zero(X_a) & hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,number_number_of(nat),Wa))) | -hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,number_number_of(nat),Wa))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Xa)) | hAPP(int,nat,number_number_of(nat),Wa) = zero_zero(nat))))) # label(fact_2519_zero__less__power__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1428 (all X_a (semiring_div(X_a) -> (all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B)) -> zero_zero(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),A_1))))) # label(fact_1356_dvd__imp__mod__0) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1429 (all X_a (no_zero_divisors(X_a) -> (all A_1 all B (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B) = zero_zero(X_a) -> zero_zero(X_a) = ti(X_a,B) | ti(X_a,A_1) = zero_zero(X_a))))) # label(fact_377_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1430 (all A_1 all B (zero_zero(int) != ti(int,B) -> (zero_zero(int) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) -> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,uminus_uminus(int),B)) = hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B))) & (zero_zero(int) != hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) -> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,uminus_uminus(int),B)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B))),one_one(int))))) # label(fact_2098_zdiv__zminus2__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1431 (all K_4 all L_4 all I_1 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),L_4)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_2),L_4)))))) # label(fact_617_mult__le__mono) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1432 (all X_a (comm_monoid_add(X_a) -> (all F all Na all Ma ((-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,Na)),Ma)) -> hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),hAPP(nat,nat,suc,Na))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),Na))),hAPP(nat,X_a,F,hAPP(nat,nat,suc,Na)))) & (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,Na)),Ma)) -> zero_zero(X_a) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),hAPP(nat,nat,suc,Na)))))))) # label(fact_4628_setsum__cl__ivl__Suc) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1433 (all X_a all X_b in_rel(X_a,X_b) = ti(fun(fun(product_prod(X_a,X_b),bool),fun(X_a,fun(X_b,bool))),in_rel(X_a,X_b))) # label(tsy_c_FunDef_Oin__rel_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1434 (all A_3 all Na all B_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A_3),hAPP(nat,nat,suc,Na))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B_2),hAPP(nat,nat,suc,Na)))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_3),B_2)))) # label(fact_2900_coprime__exp2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1435 (all N all M all R_3 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),M)),R_3)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_3),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),N)),one_one(real)))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),M),N))))) # label(fact_3433_lemma__floor) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1436 (all Xa all Ya (hBOOL(hAPP(nat,bool,even_odd_even(nat),Ya)) | hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa)) <-> hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Xa),Ya))))) # label(fact_2474_even__product__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1437 (all B all N all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) -> hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),hAPP(real,real,hAPP(real,fun(real,real),log,B),X)) = hAPP(real,real,hAPP(real,fun(real,real),log,B),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N)))) # label(fact_2427_log__nat__power) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1438 (all X_a (monoid_mult(X_a) -> (all A_1 hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),A_1)))) # label(fact_547_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1439 (all X_a (ordere779506340up_add(X_a) -> (all C_1 all D_3 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),D_3)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D_3)))))))) # label(fact_211_add__mono) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1440 (all X_a (ordered_ab_group_add(X_a) -> (all Xa all Ya hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,X_a),fun(fun(X_a,bool),fun(X_a,bool)),image(X_a,X_a),uminus_uminus(X_a)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_gr676655987ssThan(X_a),Xa),Ya)) = hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_gr676655987ssThan(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Ya)),hAPP(X_a,X_a,uminus_uminus(X_a),Xa))))) # label(fact_3696_image__uminus__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1441 (all A_2 all B_1 (hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less(fun(nat,bool)),A_2),B_1)) <-> hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,bool),fun(fun(int,bool),bool),ord_less(fun(int,bool)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B_1))))) # label(fact_3675_transfer__nat__int__set__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1442 (all Xa (-hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd)) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)))) # label(fact_1701_not__odd__impl__even) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1443 (all X hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),zero_zero(int)),X) = hAPP(int,int,abs_abs(int),X)) # label(fact_4382_gcd__0__left__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1444 (all Na all Ma (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,real_1(nat),Na)),hAPP(nat,real,real_1(nat),Ma))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),Ma)))) # label(fact_2383_real__of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1445 (all M all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,suc,N)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),N))) # label(fact_1494_div__Suc__eq__div__add3) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1446 (all X hAPP(rat,int,archim2085082626_floor(rat),X) = hAPP(real,int,archim2085082626_floor(real),hAPP(rat,real,ratreal,X))) # label(fact_3768_real__floor__code) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1447 (all X_a all X_b bijR(X_a,X_b) = ti(fun(fun(X_a,fun(X_b,bool)),fun(product_prod(fun(X_a,bool),fun(X_b,bool)),bool)),bijR(X_a,X_b))) # label(tsy_c_BijectionRel_ObijR_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1448 (all X_a (boolean_algebra(X_a) -> (all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Y)),X)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),Y)))))) # label(fact_2981_compl__le__swap2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1449 (all X_a (real_div_algebra(X_a) -> (all X (X != zero_zero(real) -> hAPP(real,X_a,of_real(X_a),hAPP(real,real,inverse_inverse(real),X)) = hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(real,X_a,of_real(X_a),X)))))) # label(fact_2815_nonzero__of__real__inverse) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1450 (all X_a all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1))) <-> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))))) # label(fact_2743_finite__Diff2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1451 (all X_a (topological_space(X_a) -> (all F all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,real),fun(X_a,real),hAPP(fun(real,real),fun(fun(X_a,real),fun(X_a,real)),combb(real,real,X_a),abs_abs(real)),F)),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4617_LIM__rabs__zero__cancel) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1452 (all X_a (no_zero_divisors(X_a) -> (all B all A_1 (ti(X_a,A_1) != zero_zero(X_a) -> (zero_zero(X_a) != ti(X_a,B) -> zero_zero(X_a) != hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)))))) # label(fact_376_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1453 (all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1)) -> ti(int,Z_1) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,nat_1,Z_1)))) # label(fact_1022_nat__0__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1454 (all M all N hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M),N),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M),N)))) # label(fact_3124_divmod__nat__rel__divmod__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1455 (all X_a (group_add(X_a) -> (all A_1 hAPP(X_a,X_a,uminus_uminus(X_a),A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),zero_zero(X_a)),A_1)))) # label(fact_1964_diff__0) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1456 (all A_3 hBOOL(hAPP(complex,bool,hAPP(fun(complex,real),fun(complex,bool),isCont(complex,real),im),A_3))) # label(fact_4933_Im_OisCont) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1457 (all X_a all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) -> hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(fun(X_a,bool),bool),fun(fun(fun(X_a,bool),bool),bool),ord_less_eq(fun(fun(X_a,bool),bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),A_2)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),B_1))))) # label(fact_2848_Pow__mono) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1458 (all Xa all Na all Diff all F (F = hAPP(nat,fun(real,real),Diff,zero_zero(nat)) -> ((all M_2 all T_4 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),Na)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),T_4)),hAPP(real,real,abs_abs(real),Xa))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_2)),T_4),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),T_4))))) -> (exists T_4 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),T_4)),hAPP(real,real,abs_abs(real),Xa))) & hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,Na),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),Na))) = hAPP(real,real,F,Xa)))))) # label(fact_4305_Maclaurin__bi__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1459 (all Xa all Na all Ya (hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Xa),Na) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Ya),Na) <-> (exists Q1 exists Q2 hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Xa),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),Q1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ya),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),Q2))))) # label(fact_1575_nat__mod__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1460 (all X_b all X_a (real_normed_vector(X_b) & topological_space(X_a) -> (all Xa all G all L all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,real),fun(X_a,fun(X_b,X_b)),hAPP(fun(real,fun(X_b,X_b)),fun(fun(X_a,real),fun(X_a,fun(X_b,X_b))),combb(real,fun(X_b,X_b),X_a),scaleR_scaleR(X_b)),G)),Xa)),hAPP(X_b,X_b,hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),L),Xa)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4763_scaleR__left_OLIM) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1461 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),N)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),M)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)))))) # label(fact_1438_one__less__mult) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1462 (all X_a (semiri456707255roduct(X_a) -> (all Wa all Ya all Xa all Z (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Wa),Z)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),Ya)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Wa),Ya)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),Z)) <-> ti(X_a,Wa) = ti(X_a,Xa) | ti(X_a,Ya) = ti(X_a,Z))))) # label(fact_386_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1463 (all Xa all Na all Ya (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Xa),hAPP(nat,nat,suc,Na))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Ya),hAPP(nat,nat,suc,Na)))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Xa),Ya)))) # label(fact_1408_exp__mono__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1464 (all X_a (linordered_idom(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),hAPP(X_a,X_a,uminus_uminus(X_a),A_3))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a))))))) # label(fact_1952_less__minus__self__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1465 (all X all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5)) -> (hBOOL(hAPP(int,bool,zprime,P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),X))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),X))),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),X)))),P_5)))))) # label(fact_1194_Int2_Oaux____1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1466 (all R_3 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),hAPP(real,int,archimedean_ceiling(real),R_3))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),R_3),one_one(real))))) # label(fact_3427_real__of__int__ceiling__le__add__one) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1467 (all X_a (preorder(X_a) -> (all Xa all Ya (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ya),Xa)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya)))))) # label(fact_568_less__le__not__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1468 (all X_a (real_normed_algebra(X_a) -> (all Y exists K_2 ((all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_1)),K_2)))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2)))))) # label(fact_2106_mult__left_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1469 (all Na all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),Ma)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,fact_fact(nat),Ma)),hAPP(nat,nat,fact_fact(nat),Na)) = hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co302546470etprod(nat,nat),combi(nat)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))),Ma)))) # label(fact_4637_fact__div__fact) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1470 (all X_a (topological_space(X_a) -> (all K hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(X_a,fun(nat,X_a),combk(X_a,nat),K)),K),sequentially))))) # label(fact_4492_LIMSEQ__const) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1471 (all Xa all A_3 all P_4 (hBOOL(hAPP(int,bool,zprime,P_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),zero_zero(int)),P_4)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_4),A_3)) -> (hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),Xa),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,A_3),P_4))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),Xa)),A_3),P_4)))))))) # label(fact_3960_SetS__setprod__prop) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1472 (all X_a (comm_semiring_1(X_a) -> (all X all N hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N))) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)))))) # label(fact_1504_comm__semiring__1__class_Onormalizing__semiring__rules_I37_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1473 (all X_a (cancel_semigroup_add(X_a) -> (all B_2 all A_3 all C (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),A_3) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B_2),A_3) <-> ti(X_a,C) = ti(X_a,B_2))))) # label(fact_294_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1474 (all X_a (linorder(X_a) -> (all Ya all Xa (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ya),Xa)) -> (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya)) <-> ti(X_a,Ya) = ti(X_a,Xa)))))) # label(fact_532_linorder__antisym__conv3) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1475 (all X_a (metric_space(X_a) -> (all X all Y hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y)))))) # label(fact_3535_zero__le__dist) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1476 (all X_2 (hBOOL(hAPP(fun(nat,complex),bool,cauchy(complex),X_2)) -> hBOOL(hAPP(fun(nat,complex),bool,cauchy(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(complex,complex),fun(fun(nat,complex),fun(nat,complex)),combb(complex,complex,nat),cnj),X_2))))) # label(fact_3879_cnj_OCauchy) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1477 (all X_a (semiring_div(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)),B)))) # label(fact_1279_mod__mod__trivial) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1478 (all X_a (group_add(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)) = ti(X_a,B)))) # label(fact_1913_minus__add__cancel) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1479 (all M all N (hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),N)) = one_one(int) -> one_one(int) = hAPP(int,int,abs_abs(int),M))) # label(fact_999_abs__zmult__eq__1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1480 (all M all N hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),M))) # label(fact_821_Nat_Odiff__le__self) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1481 (all P_1 all I all J (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J)) -> (hBOOL(hAPP(nat,bool,P_1,J)) -> ((all I_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),J)) -> (hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,suc,I_2))) -> hBOOL(hAPP(nat,bool,P_1,I_2))))) -> hBOOL(hAPP(nat,bool,P_1,I)))))) # label(fact_1523_inc__induct) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1482 (all Ya all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),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),hAPP(real,real,ln,Xa)),hAPP(real,real,ln,Ya))))))) # label(fact_1717_ln__le__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1483 (all X_a (real_normed_field(X_a) -> (all F all Xa all L (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),L)) <-> (exists G_1 (hAPP(X_a,X_a,G_1,Xa) = ti(X_a,L) & hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_a),fun(X_a,bool),isCont(X_a,X_a),G_1),Xa)) & (all Z_2 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,G_1,Z_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Z_2),Xa)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,F,Z_2)),hAPP(X_a,X_a,F,Xa))))))))) # label(fact_4903_CARAT__DERIV) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1484 (all X all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5)) -> (hBOOL(hAPP(int,bool,zprime,P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),X))),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),X)))),X),P_5)))))) # label(fact_1193_Int2_Oaux____2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1485 (all X_a ti(fun(fun(X_a,fun(X_a,bool)),fun(X_a,bool)),accp(X_a)) = accp(X_a)) # label(tsy_c_Wellfounded_Oaccp_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1486 (all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) # label(fact_1061_Int2_Oaux__2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1487 (all X_a (comm_monoid_mult(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),one_one(X_a)),A_1)))) # label(fact_389_mult__1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1488 (all A_2 (hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2))) <-> hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),A_2)))) # label(fact_3670_transfer__nat__int__set__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1489 (all Y all X (hBOOL(hAPP(int,bool,nat_is_nat,X)) -> (hBOOL(hAPP(int,bool,nat_is_nat,Y)) -> hBOOL(hAPP(int,bool,nat_is_nat,hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),Y)))))) # label(fact_4870_Divides_Otransfer__int__nat__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1490 (all N all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) -> hAPP(real,real,hAPP(real,fun(real,real),powr,X),hAPP(nat,real,real_1(nat),N)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N))) # label(fact_3429_powr__realpow) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1491 (all X_a (semiring_1(X_a) -> (all Inc all Na all I hAPP(X_a,X_a,hAPP(nat,fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(nat,fun(X_a,X_a)),semiri532925092at_aux(X_a),Inc),Na),hAPP(X_a,X_a,Inc,I)) = hAPP(X_a,X_a,hAPP(nat,fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(nat,fun(X_a,X_a)),semiri532925092at_aux(X_a),Inc),hAPP(nat,nat,suc,Na)),I)))) # label(fact_4188_of__nat__aux_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1492 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),X)),one_one(real))) -> (exists Z_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),Z_2)) & hAPP(real,real,tan,Z_2) = X & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Z_2),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))))) # label(fact_2456_tan__total__pi4) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1493 (all Na hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),Na))) = Na) # label(fact_3894_card__Collect__less__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1494 (all X one_one(real) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,cos,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,sin,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) # label(fact_2297_sin__cos__squared__add2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1495 (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_151_Nat__Transfer_Otransfer__int__nat__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1496 (all Xa hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),arctan),Xa))) # label(fact_4931_isCont__arctan) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1497 (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(nat),M),hAPP(nat,nat,suc,N))))) # label(fact_1389_le__imp__less__Suc) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1498 (all X_c all X_b all X_a all P_4 all F all G all Q_2 ((all X_1 all Y_1 (ti(product_prod(X_a,X_b),Q_2) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X_1),Y_1) -> hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),F,X_1),Y_1) = hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),G,X_1),Y_1))) -> (ti(product_prod(X_a,X_b),Q_2) = ti(product_prod(X_a,X_b),P_4) -> hAPP(product_prod(X_a,X_b),X_c,hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),G),Q_2) = hAPP(product_prod(X_a,X_b),X_c,hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),F),P_4)))) # label(fact_4796_split__cong) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1499 (all Ma all Na (hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),Na) = hAPP(nat,nat,suc,zero_zero(nat)) <-> Na = zero_zero(nat) & hAPP(nat,nat,suc,zero_zero(nat)) = Ma | zero_zero(nat) = Ma & Na = hAPP(nat,nat,suc,zero_zero(nat)))) # label(fact_1380_add__is__1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1500 (all X hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),pi))) = hAPP(real,real,cos,X)) # label(fact_2332_cos__periodic) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1501 (all X_a (ring_1(X_a) -> (all Z_1 hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)) = hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,uminus_uminus(int),Z_1))))) # label(fact_3609_of__int__minus) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1502 (all X hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),X))) # label(fact_932_dvd_Oorder__refl) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1503 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,arccos,Y))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,arccos,Y)),pi))))) # label(fact_2358_arccos__bounded) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1504 (all Xa all Ya (Ya = Xa <-> hAPP(nat,int,semiring_1_of_nat(int),Xa) = hAPP(nat,int,semiring_1_of_nat(int),Ya))) # label(fact_162_Nat__Transfer_Otransfer__int__nat__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1505 (all X_a all A_2 all F all Na (hAPP(fun(nat,bool),fun(X_a,bool),hAPP(fun(nat,X_a),fun(fun(nat,bool),fun(X_a,bool)),image(nat,X_a),F),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),Na))) = ti(fun(X_a,bool),A_2) -> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)))) # label(fact_3899_nat__seg__image__imp__finite) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1506 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,uminus_uminus(real),pi)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),zero_zero(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,sin,X)),zero_zero(real)))))) # label(fact_2341_sin__less__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1507 (all N hAPP(real,int,archimedean_ceiling(real),hAPP(nat,real,real_1(nat),N)) = hAPP(nat,int,semiring_1_of_nat(int),N)) # label(fact_3303_ceiling__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1508 (all X all N hAPP(int,real,real_1(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),N)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,real_1(int),X)),N)) # label(fact_3361_real__of__int__power) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1509 (all X_b all X_a (real_n2089651433ebra_1(X_b) & topological_space(X_a) -> (all F all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,real),fun(X_a,bool),isCont(X_a,real),F),A_3)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,real),fun(X_a,X_b),hAPP(fun(real,X_b),fun(fun(X_a,real),fun(X_a,X_b)),combb(real,X_b,X_a),of_real(X_b)),F)),A_3)))))) # label(fact_4915_isCont__of__real) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1510 (all X_a (ordered_ab_group_add(X_a) -> (all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2)),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)))))) # label(fact_752_less__iff__diff__less__0) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1511 (all B_5 all Q_5 all R_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_5),Q_5)),R_4))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_4),B_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Q_5)))))) # label(fact_581_q__pos__lemma) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1512 (all A_3 all B_2 all Q_2 all Ra (hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_3),B_2),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_2),Ra))) -> hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,hAPP(int,int,uminus_uminus(int),A_3)),B_2),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),hAPP(bool,fun(int,fun(int,int)),if(int),hAPP(int,bool,hAPP(int,fun(int,bool),fequal(int),Ra),zero_zero(int))),hAPP(int,int,uminus_uminus(int),Q_2)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,uminus_uminus(int),Q_2)),one_one(int)))),hAPP(int,int,hAPP(int,fun(int,int),hAPP(bool,fun(int,fun(int,int)),if(int),hAPP(int,bool,hAPP(int,fun(int,bool),fequal(int),Ra),zero_zero(int))),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B_2),Ra))))))) # label(fact_2102_zminus1__lemma) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1513 (all Na all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A_3),zero_zero(real)),sequentially)) -> ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,A_3,N_1)))) -> ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,A_3,hAPP(nat,nat,suc,N_1))),hAPP(nat,real,A_3,N_1)))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Na)),one_one(nat)))))))))) # label(fact_4528_summable__Leibniz_H_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1514 (all Q_3 -(all A_4 all B_4 (hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_4),B_4) = Q_3 -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_4)) -> one_one(int) != hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_4),B_4))))) # label(fact_4440_Rat__cases) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1515 (all A_1 all J_2 all K_4 all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),K_4),P_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),J_2))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),K_4))),P_5))))) # label(fact_1183_MultInv__zcong__prop1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1516 (all N -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat)))) # label(fact_118_less__zeroE) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1517 (all X_a (comm_semiring_1(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),zero_zero(X_a)),A_1)))) # label(fact_323_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1518 (all X X = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),X)) # label(fact_3251_gcd__idem__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1519 (exists X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int)))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,s1),X_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int)))) & (all Y_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_1)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,s1),Y_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int)))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int)))) -> ti(int,Y_1) = ti(int,X_1))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)))) # label(fact_710__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]. 7.62/7.67 1520 (all X_a (linordered_idom(X_a) -> (all Wa all Z (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Wa)),hAPP(int,X_a,ring_1_of_int(X_a),Z))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Wa),Z)))))) # label(fact_3604_of__int__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1521 (all X_a all X_b (real_normed_vector(X_b) -> (all G all B_2 all F all A_3 all A_2 (hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),A_3),A_2)) -> (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),B_2),A_2)) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,real),fun(X_a,fun(X_b,X_b)),hAPP(fun(real,fun(X_b,X_b)),fun(fun(X_a,real),fun(X_a,fun(X_b,X_b))),combb(real,fun(X_b,X_b),X_a),scaleR_scaleR(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),A_3),B_2)),A_2))))))) # label(fact_4753_scaleR_Otendsto) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1522 (all X_a (banach(X_a) & real_normed_field(X_a) -> (all Xa hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),exp(X_a)),Xa),hAPP(X_a,X_a,exp(X_a),Xa)))))) # label(fact_2645_DERIV__exp) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1523 (all X_a (real_normed_field(X_a) -> (all X all Y all Ya_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y)),Ya_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Ya_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Y),Ya_1))))) # label(fact_1751_divide_Oadd) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1524 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) -> X = hAPP(real,real,arcsin,hAPP(real,real,sin,X))))) # label(fact_2345_arcsin__sin) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1525 (all X_a (linorder(X_a) -> (all X all Y (ti(X_a,Y) != ti(X_a,X) -> (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X))))))) # label(fact_531_linorder__neqE) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1526 (all X_a (topological_space(X_a) -> (all K all F all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),A_3),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),A_3),sequentially)))))) # label(fact_4507_LIMSEQ__ignore__initial__segment) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1527 (all X all A_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(int,real,real_1(int),A_1))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archimedean_ceiling(real),X)),A_1)))) # label(fact_3393_ceiling__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1528 (all Y all X (hBOOL(hAPP(int,bool,twoSqu1091635170sum2sq,X)) -> (hBOOL(hAPP(int,bool,twoSqu1091635170sum2sq,Y)) -> hBOOL(hAPP(int,bool,twoSqu1091635170sum2sq,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y)))))) # label(fact_642_is__mult__sum2sq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1529 (all Q_3 all P_5 (hBOOL(hAPP(nat,bool,prime,P_5)) -> (hBOOL(hAPP(nat,bool,prime,Q_3)) -> (P_5 != Q_3 -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,P_5),Q_3)))))) # label(fact_3074_distinct__prime__coprime) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1530 (all Xa all Ma (hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Xa),Ma) = hAPP(nat,nat,suc,zero_zero(nat)) <-> Ma = zero_zero(nat) | Xa = hAPP(nat,nat,suc,zero_zero(nat)))) # label(fact_1405_nat__power__eq__Suc__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1531 (all W hAPP(int,complex,ring_1_of_int(complex),W) = hAPP(int,complex,number_number_of(complex),W)) # label(fact_3595_complex__number__of__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1532 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),zero_zero(int))) -> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A_1)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),one_one(int))),A_1))) # label(fact_1131_neg__zdiv__mult__2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1533 (all X_a (semiring(X_a) -> ti(fun(X_a,fun(X_a,X_a)),times_times(X_a)) = times_times(X_a))) # label(tsy_c_Groups_Otimes__class_Otimes_2_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1534 (all C all D_1 all A_3 all B_2 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),Ma)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,C),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),D_1),A_3)),Ma)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,C),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),D_1),B_2)),Ma))))) # label(fact_896_zcong__zmult__prop2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1535 (all M all N all K_4 ((zero_zero(nat) = K_4 -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N)) = zero_zero(nat)) & (K_4 != zero_zero(nat) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N))))) # label(fact_1160_nat__mult__div__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1536 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),pi)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,sin,X)))))) # label(fact_2284_sin__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1537 (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]. 7.62/7.67 1538 (all K_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K_4)) -> (exists N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) & hAPP(nat,int,semiring_1_of_nat(int),N_1) = ti(int,K_4))))) # label(fact_193_zero__less__imp__eq__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1539 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx all Ry hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ly),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),Ry))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ly)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),Ry))))) # label(fact_364_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1540 (all X_a (real_normed_field(X_a) -> (all Xa (zero_zero(X_a) != ti(X_a,Xa) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),inverse_inverse(X_a)),Xa),hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),Xa)),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))))))))) # label(fact_3917_DERIV__inverse) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1541 (all X hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,fact_fact(nat),X)) = hAPP(int,int,fact_fact(int),hAPP(nat,int,semiring_1_of_nat(int),X))) # label(fact_3218_transfer__int__nat__factorial) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1542 (all X_a all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),B_1)),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2))))) # label(fact_2195_Compl__anti__mono) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1543 (all X_a (real_normed_vector(X_a) -> (all A_1 all B hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),A_1)) = hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B))))) # label(fact_1929_norm__minus__commute) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1544 (all X all Y all Z_1 hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Z_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Y),Z_1))) # label(fact_3245_gcd__semilattice__nat_Oinf__left__commute) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1545 (all V_2 all V ((hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V))) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_2)) = zero_zero(nat)) & (-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V))) -> (hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V_2))) -> hAPP(int,nat,number_number_of(nat),V) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_2))) & (-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V_2))) -> hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,int,number_number_of(int),V_2))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_2)))))) # label(fact_1634_mod__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1546 (all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M)) -> ti(int,M) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),M))) # label(fact_3472_zgcd__self) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1547 (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),hAPP(int,int,bit0,K1)),hAPP(int,int,bit1,K2))))) # label(fact_257_less__eq__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1548 (all X_a (linord581940658strict(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)))))))) # label(fact_451_mult__neg__neg) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1549 (all X_b all X_a (real_normed_vector(X_b) & topological_space(X_a) -> (all C all F all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,real),fun(X_a,fun(X_b,X_b)),hAPP(fun(real,fun(X_b,X_b)),fun(fun(X_a,real),fun(X_a,fun(X_b,X_b))),combb(real,fun(X_b,X_b),X_a),scaleR_scaleR(X_b)),F)),C)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4772_scaleR_OLIM__left__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1550 (all X_a (ordered_ab_group_add(X_a) -> (all B_2 all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),B_2)),hAPP(X_a,X_a,uminus_uminus(X_a),A_3))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2)))))) # label(fact_1900_neg__le__iff__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1551 (all K_4 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_eq(nat),M),N)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),K_4),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),K_4),M)))))) # label(fact_1165_div__le__mono2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1552 (all K_4 all L_4 (hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),K_4)),L_4) != zero_zero(int) -> zero_zero(int) != hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),K_4),L_4))) # label(fact_2080_zmod__zminus1__not__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1553 (all X_a (real_normed_field(X_a) -> (all Y_2 all F all Xa all X_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),X_2)) -> (ti(X_a,X_2) = ti(X_a,Y_2) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),Y_2))))))) # label(fact_2647_DERIV__cong) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1554 (all M hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,M))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) # label(fact_1489_div2__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1555 (all X_a all A_2 hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2)) = ti(fun(X_a,bool),A_2)) # label(fact_2239_double__complement) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1556 (all X_a (ordere216010020id_add(X_a) -> (all B all C_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),B),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)))))))) # label(fact_245_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1557 (all Y all U_1 all X1_1 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)),X1_1)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X1_1),Y)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),U_1))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,inverse_inverse(real),X)),Y)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,inverse_inverse(real),X1_1)),U_1))))))) # label(fact_2624_real__mult__inverse__cancel) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1558 (all X_a (real_normed_field(X_a) -> (all R_3 all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),X)),Y) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y))))) # label(fact_4704_divide_OscaleR) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1559 (all X_a all A_2 (-hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> zero_zero(nat) = hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2))) # label(fact_2745_card__infinite) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1560 (all X_a (archim1727834104eiling(X_a) -> (all Va all Xa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),Va)),one_one(X_a))),Xa)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),Va)),hAPP(X_a,int,archim2085082626_floor(X_a),Xa))))))) # label(fact_2919_number__of__less__floor) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1561 (all X all Y hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),X),Y) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),X),hAPP(complex,complex,inverse_inverse(complex),Y))) # label(fact_2772_complex__divide__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1562 (all A_1 all J_2 all K_4 all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5)) -> (hBOOL(hAPP(int,bool,zprime,P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,K_4),zero_zero(int)),P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),zero_zero(int)),P_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),K_4))),P_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,K_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),J_2))),P_5)))))))) # label(fact_1191_MultInv__zcong__prop2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1563 (all X_b all X_a (topological_space(X_b) & topological_space(X_a) -> isCont(X_a,X_b) = ti(fun(fun(X_a,X_b),fun(X_a,bool)),isCont(X_a,X_b)))) # label(tsy_c_Lim_OisCont_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1564 (all X_a (real_normed_algebra(X_a) -> (all X all Y hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X)),hAPP(X_a,real,norm_norm(X_a),Y))))))) # label(fact_2004_norm__mult__ineq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1565 (all X_a all A_2 hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),A_2))) # label(fact_2210_subset__refl) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1566 (all X all N all P_5 (hBOOL(hAPP(nat,bool,prime,P_5)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),N))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),X))))) # label(fact_3083_prime__divexp) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1567 (all X_b all X_a (comm_monoid_mult(X_a) -> (all H all G all S all R_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_2,one_one(X_a)),one_one(X_a))) -> ((all X1 all Y1 all X2 all Y2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_2,Y1),Y2)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_2,X1),X2)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_2,hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X1),Y1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X2),Y2))))) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),S)) -> ((all X_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),S)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_2,hAPP(X_b,X_a,H,X_1)),hAPP(X_b,X_a,G,X_1))))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_2,hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),H),S)),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),G),S)))))))))) # label(fact_4804_setprod_Orelated) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1568 (all Xa all Na all Ya (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Na),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xa),Ya))) <-> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Ya),Na) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Xa),Na))) # label(fact_1378_zmod__eq__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1569 (all X_a (linorder(X_a) -> (all X all Y hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),Y))))) # label(fact_4844_min__max_Oinf__le2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1570 (all M hAPP(nat,int,semiring_1_of_nat(int),M) = hAPP(int,int,abs_abs(int),hAPP(nat,int,semiring_1_of_nat(int),M))) # label(fact_983_abs__int__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1571 (all X_a (linordered_semidom(X_a) -> (all A_1 all N all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),N))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))))))) # label(fact_272_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1572 (all N hAPP(nat,nat,suc,N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),N)) # label(fact_1399_Suc__eq__plus1__left) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1573 (all X_a (linordered_idom(X_a) -> hAPP(X_a,X_a,abs_abs(X_a),one_one(X_a)) = one_one(X_a))) # label(fact_976_abs__one) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1574 (all Xa hAPP(fun(real,bool),real,complete_Sup_Sup(real),hAPP(real,fun(real,bool),ord_lessThan(real),Xa)) = Xa) # label(fact_4354_Sup__lessThan) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1575 (all X_a all X_b ti(fun(fun(X_a,nat),fun(fun(X_b,nat),fun(product_prod(X_a,X_b),nat))),product_prod_size(X_a,X_b)) = product_prod_size(X_a,X_b)) # label(tsy_c_Product__Type_Oprod_Oprod__size_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1576 (all Z_1 hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),Z_1),hAPP(complex,complex,cnj,Z_1)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,of_real(complex),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(complex,real,im,Z_1)))),ii)) # label(fact_2995_complex__diff__cnj) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1577 (all X all Y hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),X),Y)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,X),Y)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_lcm,X),Y)) # label(fact_3744_Nitpick_Onat__lcm__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1578 (all J all K all I (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),K)),I)))) # label(fact_845_le__diff__conv) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1579 (all F all Na zero_zero(real) = hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),F)),hAPP(real,fun(nat,real),power_power(real),zero_zero(real)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),hAPP(nat,nat,suc,zero_zero(nat))),Na))) # label(fact_4214_sumr__one__lb__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1580 (all X_a (comm_monoid_mult(X_a) -> (all A_1 all B all N hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),N) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),N))))) # label(fact_394_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1581 (all L_4 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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),L_4),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),L_4),M))))) # label(fact_820_diff__le__mono2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1582 (all X all Y hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X)),Y))) # label(fact_655_real__sum__squared__expand) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1583 (all A_3 all 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(int),B_2),A_3)) <-> hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_2)))))) # label(fact_3809_one__less__Fract__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1584 (all Z1 all Z2 all Z3 hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z1),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z2),Z3)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z1),Z2)),Z3)) # label(fact_664_real__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1585 (all X_a (real_field(X_a) -> (all Na all Xa all Ya hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa)))),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),fact_fact(nat)),hAPP(nat,fun(nat,nat),minus_minus(nat),Na)))))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Ya)),hAPP(nat,fun(nat,nat),minus_minus(nat),Na))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)),Na)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Xa),Ya)),Na))))) # label(fact_4689_exp__series__add) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1586 (all X_a (linord20386208strict(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),A_1)),zero_zero(X_a)))))))) # label(fact_446_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1587 (all X_a (mult_zero(X_a) -> times_times(X_a) = ti(fun(X_a,fun(X_a,X_a)),times_times(X_a)))) # label(tsy_c_Groups_Otimes__class_Otimes_4_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.67 1588 (all X_b all X_a (topological_space(X_a) & real_normed_vector(X_b) -> (all G all M_1 all F all L_2 all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),M_1),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,real),fun(X_a,fun(X_b,X_b)),hAPP(fun(real,fun(X_b,X_b)),fun(fun(X_a,real),fun(X_a,fun(X_b,X_b))),combb(real,fun(X_b,X_b),X_a),scaleR_scaleR(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),L_2),M_1)),hAPP(X_a,filter(X_a),at(X_a),A_3)))))))) # label(fact_4762_LIM__scaleR) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1589 (all U hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),zero_zero(int)),U)))) # label(fact_4279_finite__atLeastZeroLessThan__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1590 (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]. 7.62/7.68 1591 (all X hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,cos,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,sin,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) # label(fact_2299_cos__squared__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1592 (all K_4 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),pls),K_4) = ti(int,K_4)) # label(fact_79_add__Pls) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1593 (all F all L all C (hBOOL(hAPP(filter(real),bool,hAPP(real,fun(filter(real),bool),hAPP(fun(real,real),fun(real,fun(filter(real),bool)),tendsto(real,real),F),L),hAPP(real,filter(real),at(real),C))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),L)) -> (exists R_1 ((all X_1 (X_1 != C & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),C),X_1))),R_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,F,X_1))))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1))))))) # label(fact_4653_LIM__fun__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1594 (all X_a (linorder(X_a) -> (all B 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),X),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)))))))) # label(fact_4851_min__max_Ole__infI) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1595 (all A_3 all B_2 (A_3 = one_one(real) & B_2 = zero_zero(real) <-> one_one(complex) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_3),B_2))) # label(fact_2761_Complex__eq__1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1596 (all B all Bq all Br all A_1 all C_1 all Aq all Ar (hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),C_1),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Aq),Ar))) -> (hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,B),C_1),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Bq),Br))) -> (ti(int,C_1) != zero_zero(int) -> hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)),C_1),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Aq),Bq)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Ar),Br)),C_1))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Ar),Br)),C_1)))))))) # label(fact_1628_zadd1__lemma) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1597 (all Xa hAPP(fun(real,bool),real,the(real),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(real,real),fun(real,fun(real,bool))),combb(real,fun(real,bool),real),fequal(real)),exp(real))),Xa)) = hAPP(real,real,ln,Xa)) # label(fact_4161_ln__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1598 (all X_a (linordered_field(X_a) -> (all Z_1 all W 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),X),Y)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),W)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),W),Z_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Z_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Y),W)))))))))) # label(fact_1864_frac__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1599 (all X all Y hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Y),X)) # label(fact_3249_gcd__semilattice__nat_Oinf__commute) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1600 (all W ii != hAPP(int,complex,number_number_of(complex),W)) # label(fact_2779_complex__i__not__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1601 (all X_a (comm_monoid_mult(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),one_one(X_a))))) # label(fact_392_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1602 (all L all U hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_gr676655987ssThan(int),L),U)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),U),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),L),one_one(int))))) # label(fact_3694_card__greaterThanLessThan__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1603 (all A_1 all B all C_1 hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A_1),C_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),C_1))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A_1),C_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),B),C_1))),C_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A_1),B)),C_1)) # label(fact_1539_div__add1__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1604 (all Z_1 all Z_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_4),Z_1)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,nat_1,Z_1)),hAPP(int,nat,nat_1,Z_4)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z_1),Z_4))))) # label(fact_1046_nat__diff__distrib) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1605 (all M all N hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N)),M))) # label(fact_1157_div__le__dividend) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1606 (all X_b all X_a (topological_space(X_a) & real_normed_algebra(X_b) -> (all G all F all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3)) -> (hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),G),A_3)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),F)),G)),A_3))))))) # label(fact_4912_isCont__mult) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1607 (all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A_3),zero_zero(real)),sequentially)) -> ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,A_3,N_1)))) -> ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,A_3,hAPP(nat,nat,suc,N_1))),hAPP(nat,real,A_3,N_1)))) -> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,fun(nat,bool)),fun(nat,real),hAPP(fun(fun(nat,bool),real),fun(fun(nat,fun(nat,bool)),fun(nat,real)),combb(fun(nat,bool),real,nat),hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat))))),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))),sequentially)))))) # label(fact_4529_summable__Leibniz_H_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1608 (all X 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)),X)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)))))) # label(fact_2700_real__root__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1609 (all L all U hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_gr676655987ssThan(nat),L),U)))) # label(fact_3699_finite__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1610 (all X_a (linorder(X_a) -> (all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya)) <-> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),ord_lessThan(X_a),Xa)),hAPP(X_a,fun(X_a,bool),ord_lessThan(X_a),Ya))))))) # label(fact_4346_lessThan__subset__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1611 (all Va all V_1 ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,number_number_of(nat),Va)),hAPP(int,nat,number_number_of(nat),V_1)) = hAPP(int,nat,number_number_of(nat),Va)) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,number_number_of(nat),Va)),hAPP(int,nat,number_number_of(nat),V_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Va),hAPP(int,int,uminus_uminus(int),V_1))))),zero_zero(nat)),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Va),hAPP(int,int,uminus_uminus(int),V_1)))))))) # label(fact_3938_diff__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1612 (all X_a all X_b all F all A_2 hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),bool),nat_tr1645093318rphism(X_a,X_b),F),A_2))) # label(fact_1268_transfer__morphismI) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1613 (all T_2 all T_1 (finite_finite(T_2) & finite_finite(T_1) -> finite_finite(fun(T_2,T_1)))) # label(arity_fun___Finite__Set_Ofinite) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1614 (all M all V hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,suc,M)))),hAPP(int,nat,number_number_of(nat),V)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),M)),hAPP(int,nat,number_number_of(nat),V))) # label(fact_1561_Suc__mod__eq__add3__mod__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1615 (all Xa all Ya (hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Xa),Ya))) <-> hBOOL(hAPP(int,bool,even_odd_even(int),Xa)) | hBOOL(hAPP(int,bool,even_odd_even(int),Ya)))) # label(fact_1593_Parity_Oeven__product) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1616 (all Y hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,arctan,Y)))) # label(fact_2226_arctan__lbound) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1617 (all X hAPP(real,real,abs_abs(real),hAPP(nat,real,real_1(nat),X)) = hAPP(nat,real,real_1(nat),X)) # label(fact_2373_abs__real__of__nat__cancel) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1618 (all X_a all X_b (comm_monoid_add(X_b) -> (all G all A_2 (-hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),A_2) = zero_zero(X_b))))) # label(fact_4235_setsum_Oinfinite) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1619 (all A_3 all B_2 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Ma),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_3),B_2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),Ma)))) # label(fact_911_zcong__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1620 (all B all A_1 all C_1 hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),B),C_1)) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),B),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),C_1))) # label(fact_4406_gcd__int_Oleft__commute) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1621 (all X_a (ordere236663937imp_le(X_a) -> (all C all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),A_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),B_2))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)))))) # label(fact_330_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1622 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) -> hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M),N) = hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),M))) # label(fact_3118_divmod__nat__base) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1623 (all A_1 all B all C_1 (ti(int,C_1) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),B) -> hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),C_1),B) = ti(int,A_1))) # label(fact_891_Int2_Oaux1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1624 (all X_a all X_b all F all G ((all X_1 hAPP(X_a,X_b,G,X_1) = hAPP(X_a,X_b,F,X_1)) -> ti(fun(X_a,X_b),F) = ti(fun(X_a,X_b),G))) # label(fact_73_ext) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1625 (all I_1 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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(nat,nat,suc,I_1))),N)))) # label(fact_1445_diff__Suc__less) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1626 (all X_a (field_inverse_zero(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B)) = hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))))) # label(fact_2571_inverse__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1627 (all X_a (real_normed_vector(X_a) & real_algebra_1(X_a) -> (all A_3 hBOOL(hAPP(filter(real),bool,hAPP(X_a,fun(filter(real),bool),hAPP(fun(real,X_a),fun(X_a,fun(filter(real),bool)),tendsto(real,X_a),of_real(X_a)),hAPP(real,X_a,of_real(X_a),A_3)),hAPP(real,filter(real),at(real),A_3)))))) # label(fact_4586_of__real_Ocont) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1628 (all X_a (ring_1(X_a) -> ti(fun(X_a,bool),ring_1_Ints(X_a)) = ring_1_Ints(X_a))) # label(tsy_c_Int_Oring__1__class_OInts_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1629 (all X_a (archim1727834104eiling(X_a) -> (all R_3 hAPP(real,int,archim2085082626_floor(real),hAPP(int,real,real_1(int),hAPP(X_a,int,archimedean_ceiling(X_a),R_3))) = hAPP(X_a,int,archimedean_ceiling(X_a),R_3)))) # label(fact_3380_floor__ceiling) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1630 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> one_one(X_a) = hAPP(X_a,X_a,abs_abs(X_a),hAPP(int,X_a,number_number_of(X_a),min)))) # label(fact_1015_abs__minus__one) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1631 (all V all W hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),W)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,int,number_number_of(int),W))) # label(fact_17_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1632 (all X_a all A_2 all B_1 (hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),B_1))) <-> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)))) # label(fact_2844_Pow__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1633 (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_356_nat__le__linear) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1634 (all X_a (number_semiring(X_a) -> hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls)) = one_one(X_a))) # label(fact_104_semiring__numeral__1__eq__1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1635 (all A_3 hBOOL(hAPP(filter(complex),bool,hAPP(real,fun(filter(complex),bool),hAPP(fun(complex,real),fun(real,fun(filter(complex),bool)),tendsto(complex,real),re),hAPP(complex,real,re,A_3)),hAPP(complex,filter(complex),at(complex),A_3)))) # label(fact_4571_Re_Ocont) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1636 (all A_1 all B (hAPP(rat,product_prod(int,int),quotient_of,B) = hAPP(rat,product_prod(int,int),quotient_of,A_1) -> B = A_1)) # label(fact_3739_quotient__of__inject) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1637 (all Z1 all Z2 all W hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z1),Z2)),W) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z1),W)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z2),W))) # label(fact_744_zdiff__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1638 (all X_a (archim1727834104eiling(X_a) -> (all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(X_a,int,archimedean_ceiling(X_a),X)))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(X_a,int,archimedean_ceiling(X_a),X))),one_one(X_a))),X)))))) # label(fact_3632_ceiling__correct) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1639 (all X all Z_1 all D_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_3)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),Z_1))),one_one(int))),D_3))),Z_1)))) # label(fact_1058_decr__lemma) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1640 (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N))))))) # label(fact_750_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1641 (all X_a (semiring_div(X_a) -> (all A_1 all B all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B)),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B))),C_1)))) # label(fact_1414_div__mod__equality) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1642 (all M hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),M)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,M))) # label(fact_1450_int__Suc) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1643 (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)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(int,nat,nat_1,X)),hAPP(int,nat,nat_1,Y)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),Y))))) # label(fact_1556_nat__mod__distrib) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1644 (all Z_1 exists R_1 exists A_4 hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,of_real(complex),R_1)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,cos,A_4)),hAPP(real,real,sin,A_4))) = Z_1) # label(fact_2831_complex__split__polar) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1645 (all K_4 all L_4 hAPP(int,int,bit0,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),hAPP(int,int,succ,L_4))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,K_4)),hAPP(int,int,bit1,L_4))) # label(fact_309_add__Bit1__Bit1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1646 (all X_a (linord1117847801e_zero(X_a) & number(X_a) -> (all B_2 all C all Wa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)),hAPP(int,X_a,number_number_of(X_a),Wa))) <-> (-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),C),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),C)),B_2))) & (-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),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Wa))))) & (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),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),C)))))))) # label(fact_1794_divide__less__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1647 (all X_a all C all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1))) <-> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),A_2)) & -hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),B_1)))) # label(fact_2244_Diff__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1648 (all X_a (group_add(X_a) -> (all A_3 all B_2 (ti(X_a,B_2) = ti(X_a,A_3) <-> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2) = zero_zero(X_a))))) # label(fact_733_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1649 (all A_3 (one_one(nat) = A_3 <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_3),one_one(nat))))) # label(fact_1259_gcd__lcm__complete__lattice__nat_Obot__unique) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1650 (all A_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B))))) # label(fact_1418_pos__mod__sign) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1651 (all X_a all X_b (real_normed_vector(X_b) & real_algebra_1(X_b) -> (all G all A_3 all A_2 (hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),G),A_3),A_2)) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,real),fun(X_a,X_b),hAPP(fun(real,X_b),fun(fun(X_a,real),fun(X_a,X_b)),combb(real,X_b,X_a),of_real(X_b)),G)),hAPP(real,X_b,of_real(X_b),A_3)),A_2)))))) # label(fact_4488_of__real_Otendsto) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1652 (all M hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,M))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) # label(fact_1558_mod2__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1653 (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)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,nat_1,X)),hAPP(int,nat,nat_1,Y)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y))))) # label(fact_1040_Nat__Transfer_Otransfer__nat__int__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1654 (all Wa all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Wa),Z)) <-> (exists N_1 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Wa),hAPP(nat,int,semiring_1_of_nat(int),N_1)) = ti(int,Z)))) # label(fact_269_zle__iff__zadd) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1655 (all B (ti(int,B) != zero_zero(int) -> hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,zero_zero(int)),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),zero_zero(int)))))) # label(fact_1615_divmod__int__rel__0) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1656 (all A_1 all C_1 all D_3 all B (ti(int,B) != zero_zero(int) -> (ti(int,D_3) != zero_zero(int) -> hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,C_1),D_3)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),D_3)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C_1),B))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),D_3))))) # label(fact_3808_diff__rat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1657 (all N all 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),hAPP(nat,nat,fact_fact(nat),N)),hAPP(nat,nat,fact_fact(nat),M))))) # label(fact_2547_fact__dvd) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1658 (all I all M_1 (-hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),zero_zero(nat)),M_1)) -> hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),combs(nat,bool,bool),hAPP(fun(nat,bool),fun(nat,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(fun(nat,bool),bool)),fun(fun(nat,bool),fun(nat,bool)),combc(nat,fun(nat,bool),bool),member(nat)),M_1))),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),hAPP(nat,nat,suc,I))))) = hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),combs(nat,bool,bool),hAPP(fun(nat,bool),fun(nat,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(fun(nat,bool),bool)),fun(fun(nat,bool),fun(nat,bool)),combc(nat,fun(nat,bool),bool),hAPP(fun(nat,nat),fun(nat,fun(fun(nat,bool),bool)),hAPP(fun(nat,fun(fun(nat,bool),bool)),fun(fun(nat,nat),fun(nat,fun(fun(nat,bool),bool))),combb(nat,fun(fun(nat,bool),bool),nat),member(nat)),suc)),M_1))),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),I)))))) # label(fact_3902_card__less__Suc2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1659 (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)) -> hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),Y)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(int,nat,nat_1,X)),hAPP(int,nat,nat_1,Y))))) # label(fact_1557_Divides_Otransfer__nat__int__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1660 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) | M = N -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)))) # label(fact_404_less__or__eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1661 (all X_a all X_b all Z all F all A_2 ((exists X_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_2)) & hAPP(X_b,X_a,F,X_1) = ti(X_a,Z))) <-> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Z),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2))))) # label(fact_3659_image__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1662 (all X_2 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma)) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,bool),fun(fun(int,bool),bool),ord_less_eq(fun(int,bool)),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,int),fun(fun(int,bool),fun(int,bool)),image(int,int),hAPP(int,fun(int,int),standardRes,Ma)),X_2)),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),Ma))))))) # label(fact_3926_SR__pos) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1663 (all M all N (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N)) -> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,N),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M),N)))) # label(fact_3485_zgcd__non__0) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1664 (all X_a (linord1278240602ring_1(X_a) -> (all V 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)) -> (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),U_1),V) = one_one(X_a) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),U_1),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),V),Y))),A_1)))))))))) # label(fact_537_convex__bound__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1665 (all R_3 all X all Y hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),R_3),X)),Y) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(real,complex,of_real(complex),R_3)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,X),Y))) # label(fact_2819_complex__of__real__add__Complex) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1666 (all K_4 hAPP(int,rat,hAPP(int,fun(int,rat),fract,zero_zero(int)),K_4) = zero_zero(rat)) # label(fact_3764_rat__number__collapse_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1667 (all X_a (real_normed_vector(X_a) -> (all F all L (hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),suc)),L)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),L),hAPP(nat,X_a,F,zero_zero(nat))))))))) # label(fact_4122_sums__Suc) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1668 (all A_1 all P_5 (hBOOL(hAPP(nat,bool,prime,P_5)) -> (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),A_1)) -> (exists X_1 exists Y_1 hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),X_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),P_5),Y_1)),one_one(nat)))))) # label(fact_3115_bezout__prime) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1669 (all X_a all C all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1))) -> -(hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),A_2)) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),B_1))))) # label(fact_2232_DiffE) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1670 (all X_a (archim1727834104eiling(X_a) -> (all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(X_a,int,archimedean_ceiling(X_a),Xa))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Xa)))))) # label(fact_3322_zero__less__ceiling) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1671 (all N zero_zero(real) = hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(nat,real,real_1(nat),N))),pi))) # label(fact_2444_sin__2npi) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1672 (all X_a (real_vector(X_a) -> (all A_1 all B all X (ti(X_a,X) != zero_zero(X_a) -> (hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),B),X) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X) -> B = A_1))))) # label(fact_4733_scaleR__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1673 (all Na (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Na)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,nat_1,Na)),one_one(nat)) = hAPP(fun(int,bool),nat,finite_card(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),Na)))))) # label(fact_3912_card__bdd__int__set__l__l) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1674 (all X_b all X_c all X_a (metric_space(X_c) & metric_space(X_b) & topological_space(X_a) -> (all G all Ma all F all L all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> ((all X_1 (ti(X_a,A_3) != ti(X_a,X_1) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_c,real,hAPP(X_c,fun(X_c,real),dist_dist(X_c),hAPP(X_a,X_c,G,X_1)),Ma)),hAPP(X_b,real,hAPP(X_b,fun(X_b,real),dist_dist(X_b),hAPP(X_a,X_b,F,X_1)),L))))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_c,fun(filter(X_a),bool),hAPP(fun(X_a,X_c),fun(X_c,fun(filter(X_a),bool)),tendsto(X_a,X_c),G),Ma),hAPP(X_a,filter(X_a),at(X_a),A_3)))))))) # label(fact_4681_metric__LIM__imp__LIM) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1675 (all M all K_4 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K_4)),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_4),N))))) # label(fact_419_add__leE) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1676 (all A_1 hAPP(int,int,uminus_uminus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))) # label(fact_2094_power3__minus) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1677 (all X_a (archim1727834104eiling(X_a) -> (all X all V hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),hAPP(int,X_a,number_number_of(X_a),V))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(X_a,int,archimedean_ceiling(X_a),X)),hAPP(int,int,number_number_of(int),V))))) # label(fact_3333_ceiling__diff__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1678 (all M hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),M) = hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M),zero_zero(nat))) # label(fact_3117_divmod__nat__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1679 (all N all X (hBOOL(hAPP(int,bool,nat_is_nat,X)) -> hBOOL(hAPP(int,bool,nat_is_nat,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),N))))) # label(fact_4869_Nat__Transfer_Otransfer__int__nat__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1680 (all X_a collect(X_a) = ti(fun(fun(X_a,bool),fun(X_a,bool)),collect(X_a))) # label(tsy_c_Set_OCollect_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1681 (all X_a (plus(X_a) & dvd(X_a) & linorder(X_a) -> (all D_1 all Sb exists Z_2 all X_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_2),X_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),Sb))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),Sb)))))))) # label(fact_1213_pinf_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1682 (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,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),M))))) # label(fact_904_zcong__not__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1683 (all F all G ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(nat,real,F,N_1))),hAPP(nat,real,G,N_1)))) -> (hBOOL(hAPP(fun(nat,real),bool,summable(real),G)) -> hBOOL(hAPP(fun(nat,real),bool,summable(real),F)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,real),real,suminf(real),F)),hAPP(fun(nat,real),real,suminf(real),G)))))) # label(fact_4097_summable__le2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1684 (all X_a (dense_linorder(X_a) -> (all Z_1 all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) -> ((all W_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),W_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),W_1),Y)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),W_1),Z_1))))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),Z_1))))))) # label(fact_2859_dense__le__bounded) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1685 (all X_a all X_b (t2_space(X_b) -> (all L_1 all F all L all A_2 (-hBOOL(hAPP(filter(X_a),bool,trivial_limit(X_a),A_2)) -> (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),A_2)) -> (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_1),A_2)) -> ti(X_b,L_1) = ti(X_b,L))))))) # label(fact_4792_tendsto__unique) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1686 (all N zero_zero(nat) != hAPP(nat,nat,fact_fact(nat),N)) # label(fact_2538_fact__nonzero__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1687 (all P all Q (-hBOOL(Q) | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fdisj,P),Q)))) # label(help_fdisj_2_1_U) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1688 (all Ya all Xa (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ya),zEven)) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xa),Ya)),zEven))))) # label(fact_1700_even__minus__even) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1689 (all X_a all F1 all Real1 all Real2 hAPP(complex,X_a,hAPP(fun(real,fun(real,X_a)),fun(complex,X_a),complex_case(X_a),F1),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Real1),Real2)) = hAPP(real,X_a,hAPP(real,fun(real,X_a),F1,Real1),Real2)) # label(fact_2789_complex_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1690 (all A_3 all Ma hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,A_3),Ma)))) # label(fact_3233_Bnor__fin) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1691 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx all Ry hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ly)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),Ry)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Rx)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ly),Ry))))) # label(fact_366_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1692 (all A_1 (hBOOL(hAPP(nat,bool,prime,A_1)) -> hBOOL(hAPP(int,bool,zprime,hAPP(nat,int,semiring_1_of_nat(int),A_1))))) # label(fact_3085_prime__impl__zprime__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1693 (all X_a all X_b (order(X_a) & order(X_b) -> (all C all A_3 all F all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),hAPP(X_b,X_a,F,B_2))) -> (hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),B_2),C)) -> ((all X_1 all Y_1 (hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Y_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,X_1)),hAPP(X_b,X_a,F,Y_1))))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),hAPP(X_b,X_a,F,C))))))))) # label(fact_2855_order__less__le__subst1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1694 (all X_a (archim1727834104eiling(X_a) -> (all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(X_a,int,archim2085082626_floor(X_a),Xa))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),Xa)))))) # label(fact_2907_zero__less__floor) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1695 (all X_a (linorder(X_a) -> (all A_1 all B all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),X)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)),X)))))) # label(fact_4825_min__max_Oless__infI2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1696 (all X_a (archim1727834104eiling(X_a) -> (all X all Y (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),hAPP(X_a,int,archim2085082626_floor(X_a),Y))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)))))) # label(fact_2890_floor__less__cancel) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1697 (all X_a all X_b all F all G all P_4 hAPP(product_prod(X_a,X_b),nat,hAPP(fun(X_b,nat),fun(product_prod(X_a,X_b),nat),hAPP(fun(X_a,nat),fun(fun(X_b,nat),fun(product_prod(X_a,X_b),nat)),product_prod_size(X_a,X_b),F),G),P_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(X_a,nat,F,hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),P_4))),hAPP(X_b,nat,G,hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),P_4)))),hAPP(nat,nat,suc,zero_zero(nat)))) # label(fact_3147_prod__size__simp) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1698 (all X_b all X_a (topological_space(X_a) & real_normed_algebra(X_b) -> (all Xa all G all L all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),hAPP(X_b,fun(X_b,X_b),times_times(X_b),Xa)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Xa),L)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4589_mult__right_OLIM) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1699 (all X_a (ord(X_a) -> ti(fun(X_a,fun(X_a,X_a)),ord_min(X_a)) = ord_min(X_a))) # label(tsy_c_Orderings_Oord__class_Omin_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1700 (all F (hBOOL(hAPP(fun(nat,real),bool,summable(real),F)) -> ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,F,N_1)))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(fun(nat,real),real,suminf(real),F)))))) # label(fact_4100_suminf__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1701 (all X_a (linordered_field(X_a) -> (all C_1 all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),C_1),B))))))))) # label(fact_1889_divide__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1702 (all Xa all Ya (hAPP(complex,real,re,Ya) = hAPP(complex,real,re,Xa) & hAPP(complex,real,im,Xa) = hAPP(complex,real,im,Ya) <-> Ya = Xa)) # label(fact_3022_complex__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1703 (all I_1 ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),zero_zero(int))) -> hAPP(int,int,abs_abs(int),I_1) = ti(int,I_1)) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),zero_zero(int))) -> hAPP(int,int,uminus_uminus(int),I_1) = hAPP(int,int,abs_abs(int),I_1)))) # label(fact_2088_zabs__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1704 (all X_a (group_add(X_a) -> (all A_3 all B_2 (ti(X_a,B_2) = hAPP(X_a,X_a,uminus_uminus(X_a),A_3) <-> hAPP(X_a,X_a,uminus_uminus(X_a),B_2) = ti(X_a,A_3))))) # label(fact_1922_equation__minus__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1705 (all Y all X all A_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A_1)) -> (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_eq(real),X),Y)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,Y),A_1))))))) # label(fact_3400_powr__mono2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1706 (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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Ma),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),K))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)))))) # label(fact_815_le__diff__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1707 (all X_a (real_normed_algebra(X_a) -> (all X all Y all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Y),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),A_1)),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Y),B)))))) # label(fact_1238_mult_Oprod__diff__prod) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1708 (all X_a (ab_sem1668676832m_mult(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)))) # label(fact_2975_mult__left__idem) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1709 (all Ma all Na ((exists J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J_1),Na)) & hAPP(nat,nat,suc,J_1) = Ma)) | zero_zero(nat) = Ma <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),hAPP(nat,nat,suc,Na))))) # label(fact_1374_less__Suc__eq__0__disj) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1710 (all X_a (ordered_ab_group_add(X_a) -> (all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2)),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2)))))) # label(fact_751_le__iff__diff__le__0) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1711 (all X_a (ordered_ring(X_a) -> (all A_3 all E_2 all C all B_2 all D_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),E_2)),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),E_2)),D_1))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2)),E_2)),C)),D_1)))))) # label(fact_773_le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1712 (all X_a all Q_1 all P_1 all A_3 (hBOOL(hAPP(X_a,bool,P_1,A_3)) -> ((all X_1 (hBOOL(hAPP(X_a,bool,P_1,X_1)) -> hBOOL(hAPP(X_a,bool,Q_1,X_1)))) -> hBOOL(hAPP(X_a,bool,Q_1,hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1)))))) # label(fact_4894_someI2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1713 (all X_a (zero(X_a) -> (all Xa (zero_zero(X_a) = ti(X_a,Xa) <-> zero_zero(X_a) = ti(X_a,Xa))))) # label(fact_281_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1714 (all A_3 all B_2 (ti(int,A_3) = ti(int,B_2) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),zero_zero(int))))) # label(fact_890_IntPrimes_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1715 (all A_3 all B_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_3),B_2)) <-> (all P_6 -(hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_6),A_3)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_6),B_2)) & hBOOL(hAPP(nat,bool,prime,P_6)))))) # label(fact_3088_coprime__prime__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1716 (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)))) # label(fact_958_dvd_Oless__le__not__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1717 (all X (hBOOL(hAPP(int,bool,nat_is_nat,X)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,fact_fact(int),X))))) # label(fact_4883_transfer__int__nat__factorial__closure) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1718 (all X all A_1 all B hAPP(real,real,hAPP(real,fun(real,real),powr,X),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A_1),B)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),B))) # label(fact_3383_powr__divide2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1719 (all A_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)),zero_zero(int))))) # label(fact_1421_neg__mod__sign) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1720 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),pi)),X)) -> hAPP(real,real,arccos,hAPP(real,real,cos,X)) = hAPP(real,real,uminus_uminus(real),X)))) # label(fact_2357_arccos__cos2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1721 (all X_a (semiring_div(X_a) -> (all A_1 all B all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C_1)),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C_1)))) # label(fact_1294_mod__add__left__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1722 (all X_a (comm_ring_1(X_a) -> (all X all N ((hBOOL(hAPP(nat,bool,even_odd_even(nat),N)) -> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),N) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N)) & (-hBOOL(hAPP(nat,bool,even_odd_even(nat),N)) -> hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),N)))))) # label(fact_2480_neg__power__if) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1723 (all X_a (real_normed_algebra(X_a) -> (all Xa all A_3 hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_a),fun(X_a,bool),isCont(X_a,X_a),hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa)),A_3))))) # label(fact_4914_mult__right_OisCont) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1724 (all N hAPP(real,nat,natceiling,hAPP(nat,real,real_1(nat),N)) = N) # label(fact_2370_natceiling__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1725 (all A_1 all B all C_1 hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),C_1))),C_1) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)),C_1)) # label(fact_1309_zmod__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1726 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),one_one(real))) -> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(real,fun(nat,real),power_power(real),Xa)),zero_zero(real)),sequentially))))) # label(fact_4520_LIMSEQ__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1727 (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),X),Y)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)))) # label(fact_954_dvd_Ole__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1728 (all X_a (field(X_a) -> (all Xa all Wa all Z all Ya (zero_zero(X_a) != ti(X_a,Ya) -> (zero_zero(X_a) != ti(X_a,Z) -> (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Wa),Ya) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),Z) <-> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Xa),Ya) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Wa),Z))))))) # label(fact_1852_frac__eq__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1729 (all Xa all Na (ti(int,Xa) != zero_zero(int) | zero_zero(nat) = Na <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,abs_abs(int),Xa)),Na))))) # label(fact_1035_zero__less__zpower__abs__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1730 (all Xa all Ya (zero_zero(real) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Ya),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) <-> zero_zero(real) = Ya & zero_zero(real) = Xa)) # label(fact_109_realpow__two__sum__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1731 (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_189_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1732 (all Na all F (hBOOL(hAPP(fun(nat,real),bool,summable(real),F)) -> ((all M_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),M_2)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,F,M_2))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(fun(nat,real),real,suminf(real),F)))))) # label(fact_4302_series__pos__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1733 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),K)),hAPP(int,int,number_number_of(int),L))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)))) # label(fact_16_less__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1734 (all F all A_3 all B_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2)) -> ((all X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),B_2)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),X_1)) -> (exists Y_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y_1),zero_zero(real))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),Y_1)))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,F,B_2)),hAPP(real,real,F,A_3)))))) # label(fact_3552_DERIV__nonpos__imp__nonincreasing) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1735 (all L all U hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),L),U) = hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,int),fun(fun(int,bool),fun(int,bool)),image(int,int),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),plus_plus(int)),L)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),U),L)))) # label(fact_4292_image__add__int__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1736 (all X_a (metric_space(X_a) -> (all Ra all X_2 all L_2 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),L_2),sequentially)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Ra)) -> (exists No all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),No),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),hAPP(nat,X_a,X_2,N_1)),L_2)),Ra))))))))) # label(fact_4546_metric__LIMSEQ__D) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1737 (all A_1 all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int)))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),inv,P_5),A_1)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int)))))))) # label(fact_1175_inv__less__p__minus__1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1738 (all Z_1 hAPP(complex,real,norm_norm(complex),Z_1) = hAPP(real,real,sqrt,hAPP(complex,real,re,hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),Z_1),hAPP(complex,complex,cnj,Z_1))))) # label(fact_3038_complex__mod__sqrt__Re__mult__cnj) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1739 (all A_3 all B_2 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),hAPP(int,int,uminus_uminus(int),Ma))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),Ma)))) # label(fact_2064_zcong__zminus) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1740 (all X_a all X_b (real_n1866405975lgebra(X_b) -> (all F all A_3 all A_2 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),A_3),A_2)) -> (ti(X_b,A_3) != zero_zero(X_b) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),inverse_inverse(X_b)),F)),hAPP(X_b,X_b,inverse_inverse(X_b),A_3)),A_2))))))) # label(fact_4476_tendsto__inverse) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1741 (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)) -> hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,ln,X)),hAPP(real,real,ln,Y))))) # label(fact_1800_ln__div) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1742 (all K_4 -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),K_4)),zero_zero(int)))) # label(fact_61_int__less__0__conv) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1743 (all X (hBOOL(hAPP(nat,bool,even_odd_even(nat),X)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,suc,X)),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),X),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))))) # label(fact_2501_even__nat__plus__one__div__two) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1744 (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),hAPP(nat,nat,suc,M)),N)))) # label(fact_1388_Suc__leI) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1745 (all X_a all C all P_1 ((exists X1 hBOOL(hAPP(X_a,bool,P_1,X1))) -> (ti(X_a,C) = hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1) -> hBOOL(hAPP(X_a,bool,P_1,C))))) # label(fact_4803_exE__some) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1746 (all X_a (ordere779506340up_add(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),B))))))) # label(fact_212_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1747 (all Xa (hAPP(real,real,cos,Xa) != zero_zero(real) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real)),sin)),cos)),Xa),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,cos,Xa)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) # label(fact_3943_lemma__DERIV__tan) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1748 (all P_1 all A_3 all B_2 hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),zero_zero(nat)) = hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_3),B_2)) # label(fact_3166_Bolzano__bisect_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1749 (all X_a all A_3 all F all Va hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),hAPP(bool,fun(X_a,fun(X_a,X_a)),if(X_a),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),A_3),hAPP(nat,X_a,F,hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va))))) = hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(X_a,fun(fun(nat,X_a),fun(nat,X_a)),nat_case(X_a),A_3),F),hAPP(int,nat,number_number_of(nat),Va))) # label(fact_4152_nat__case__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1750 (all N (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),N))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,N)))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,N)))))) # label(fact_339_number__of1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1751 (all Na all Ma (Na = Ma <-> hAPP(nat,real,real_1(nat),Ma) = hAPP(nat,real,real_1(nat),Na))) # label(fact_2371_real__of__nat__inject) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1752 (all Xa all A_3 (zero_zero(real) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa),hAPP(real,real,uminus_uminus(real),A_3)) <-> A_3 = Xa)) # label(fact_1990_real__add__minus__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1753 (all U_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),U_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),U_1),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),U_1)))) # label(fact_2171_lemma__real__divide__sqrt__less) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1754 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,Ma)),Na)) <-> hBOOL(hAPP(nat,bool,hAPP(fun(nat,bool),fun(nat,bool),hAPP(bool,fun(fun(nat,bool),fun(nat,bool)),nat_case(bool),fFalse),hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma)),Na)))) # label(fact_4147_less__eq__nat_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1755 (all Ra hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),plus_plus(real),Ra)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),uminus_uminus(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),suc))))),Ra),sequentially))) # label(fact_4456_LIMSEQ__inverse__real__of__nat__add__minus) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1756 (all X_a (real_normed_vector(X_a) & real_algebra_1(X_a) -> (all X all Y hAPP(real,X_a,of_real(X_a),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(real,X_a,of_real(X_a),X)),hAPP(real,X_a,of_real(X_a),Y))))) # label(fact_2803_of__real_Odiff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.68 1757 (all X_a (topological_space(X_a) -> (all G all L all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,complex),fun(X_a,real),hAPP(fun(complex,real),fun(fun(X_a,complex),fun(X_a,real)),combb(complex,real,X_a),im),G)),hAPP(complex,real,im,L)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4622_Im_OLIM) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1758 (all X_b (ring(X_b) & number(X_b) -> (all V all B all C_1 hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),B),C_1)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),B)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),C_1))))) # label(fact_756_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1759 (all A_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B))))) # label(fact_1359_neg__mod__bound) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1760 (all Xa all Ya (Xa = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Xa),Ya) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)))) # label(fact_3255_gcd__semilattice__nat_Ole__iff__inf) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1761 (all X_a (linordered_semidom(X_a) -> (all X all Y (hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,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,X) = ti(X_a,Y))))))) # label(fact_313_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1762 (all X_a (linord1117847801e_zero(X_a) -> (all A_3 all C all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C))) <-> (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),B_2),A_3))) & zero_zero(X_a) != ti(X_a,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(X_a),A_3),B_2))))))) # label(fact_1843_divide__less__cancel) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1763 (all Xa ((exists N_1 hAPP(int,real,real_1(int),N_1) = Xa) <-> hAPP(int,real,real_1(int),hAPP(real,int,archimedean_ceiling(real),Xa)) = Xa)) # label(fact_3365_real__of__int__ceiling__cancel) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1764 (all A_1 all B all C_1 all D_3 hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),C_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),D_3)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,C_1),D_3))) # label(fact_3778_mult__rat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1765 (all P_1 hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),hAPP(fun(int,nat),fun(int,bool),hAPP(fun(nat,bool),fun(fun(int,nat),fun(int,bool)),combb(nat,bool,int),P_1),nat_1)))) = hAPP(fun(nat,bool),fun(nat,bool),collect(nat),P_1)) # label(fact_3850_Nat__Transfer_Otransfer__nat__int__set__functions_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1766 (all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,exp(real),X)))) # label(fact_1769_exp__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1767 (all X_a all X_b (uminus(X_b) -> (all A_2 all X_1 hAPP(X_b,X_b,uminus_uminus(X_b),hAPP(X_a,X_b,A_2,X_1)) = hAPP(X_a,X_b,hAPP(fun(X_a,X_b),fun(X_a,X_b),uminus_uminus(fun(X_a,X_b)),A_2),X_1)))) # label(fact_2953_fun__Compl__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1768 (all G all Xa all Ma (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),G),Xa),Ma)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),cos),G)),Xa),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,hAPP(real,real,G,Xa)))),Ma))))) # label(fact_3900_DERIV__fun__cos) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1769 (all X all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5)) -> (hBOOL(hAPP(int,bool,zprime,P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),X))),one_one(int)),P_5)))))) # label(fact_1196_MultInv__prop2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1770 (all A_1 all B all C_1 hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),C_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),B),C_1))) # label(fact_3244_gcd__assoc__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1771 (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),one_one(real)),hAPP(real,real,exp(real),X))))) # label(fact_1788_exp__gt__one) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1772 (all S ((all M_2 exists N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_2),N_1)) & hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),N_1),S)))) <-> -hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),S)))) # label(fact_3680_infinite__nat__iff__unbounded__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1773 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),B)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)))) # label(fact_3133_div__neg__pos) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1774 (all X_a (ordered_ab_group_add(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_3)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)),zero_zero(X_a))))))) # label(fact_1954_neg__less__0__iff__less) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1775 (all Y all X (-hBOOL(hAPP(int,bool,even_odd_even(int),X)) -> (-hBOOL(hAPP(int,bool,even_odd_even(int),Y)) -> hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),Y)))))) # label(fact_1594_Parity_Oodd__plus__odd) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1776 (all Na all Va hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),hAPP(nat,nat,suc,Na)),hAPP(int,nat,number_number_of(nat),Va)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),hAPP(nat,nat,suc,Na)),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),Na),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va))))))) # label(fact_4969_max__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1777 (all X_a (number_semiring(X_a) -> hAPP(int,X_a,number_number_of(X_a),pls) = zero_zero(X_a))) # label(fact_84_semiring__numeral__0__eq__0) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1778 (all X_a (linord581940658strict(X_a) -> (all C_1 all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1)))))))) # label(fact_455_mult__strict__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1779 (all X_a (real_normed_vector(X_a) -> (all K all F all Sb (hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),Sb)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Sb),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K))))))))) # label(fact_4194_sums__split__initial__segment) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1780 (all X_a (semiring_div(X_a) -> (all A_1 all B (zero_zero(X_a) != ti(X_a,B) -> ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),A_1)),B))))) # label(fact_1087_div__mult__self1__is__id) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1781 (all X_a (banach(X_a) -> (all F all G ((exists N_2 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,F,N_1))),hAPP(nat,real,G,N_1))))) -> (hBOOL(hAPP(fun(nat,real),bool,summable(real),G)) -> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))))))) # label(fact_4019_summable__comparison__test) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1782 (all X_a (semiring_div(X_a) -> (all A_1 all C_1 all B (zero_zero(X_a) != ti(X_a,B) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1))),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B)))))) # label(fact_1115_div__mult__self2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1783 (all X_a (archim1804426504_field(X_a) -> (all X exists Z_2 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),Z_2)))))) # label(fact_3651_ex__le__of__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1784 (all X_a (ring_1(X_a) -> (all W all Z_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),W)),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)) = hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W),Z_1))))) # label(fact_3588_of__int__add) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1785 (all X all A_1 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(real,int,archimedean_ceiling(real),X)),A_1) = hAPP(real,int,archimedean_ceiling(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(int,real,real_1(int),A_1)))) # label(fact_3397_ceiling__add) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1786 (all M all K_4 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),K_4)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M))),K_4) = one_one(nat))) # label(fact_1554_Suc__times__mod__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1787 (all X_a ti(fun(X_a,fun(fun(code_code_numeral,X_a),fun(code_code_numeral,X_a))),code_c600343653l_case(X_a)) = code_c600343653l_case(X_a)) # label(tsy_c_Code__Numeral_Ocode__numeral_Ocode__numeral__case_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1788 (all X_a (number_semiring(X_a) -> (all X all Y hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X)),Y))))) # label(fact_575_power2__sum) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1789 (all X_a (linord1117847801e_zero(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_3)),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),zero_zero(X_a))))))) # label(fact_2584_inverse__nonpositive__iff__nonpositive) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1790 (all X_a (ordere453448008miring(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)))))))) # label(fact_426_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1791 (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),X),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X))),one_one(real)))))) # label(fact_2337_cos__double__less__one) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1792 (all X all Y hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y)),Y))) # label(fact_4373_gcd__dvd2__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1793 (all X_a (real_normed_vector(X_a) -> (all E_2 all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_2)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E_2)) -> (exists M_3 all M_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),M_2)) -> (all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,X_2,M_2)),hAPP(nat,X_a,X_2,N_1)))),E_2))))))))))) # label(fact_3341_CauchyD) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1794 (all K_4 all M all N hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),N)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,abs_abs(int),K_4)),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),N))) # label(fact_4384_gcd__mult__distrib__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1795 (all Xa all Ya all P_1 ((-hBOOL(P_1) -> hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),hAPP(bool,fun(int,fun(int,int)),if(int),P_1),Xa),Ya)) = hAPP(int,nat,nat_1,Ya)) & (hBOOL(P_1) -> hAPP(int,nat,nat_1,Xa) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),hAPP(bool,fun(int,fun(int,int)),if(int),P_1),Xa),Ya))))) # label(fact_964_nat__if__cong) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1796 (all B all Q_3 all R_3 all B_5 all Q_5 all R_4 (hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_3)),R_3) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_5),Q_5)),R_4) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_5),Q_5)),R_4)),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_3),B)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_5),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_5),Q_3))))))))) # label(fact_653_zdiv__mono2__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1797 (all X_a all F all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> (one_one(nat) = hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),A_2) <-> (exists X_1 ((all Xa_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa_1),A_2)) -> (ti(X_a,X_1) != ti(X_a,Xa_1) -> zero_zero(nat) = hAPP(X_a,nat,F,Xa_1)))) & one_one(nat) = hAPP(X_a,nat,F,X_1) & hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))))))) # label(fact_4284_setsum__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1798 (all X_a (semiring_div(X_a) -> (all A_1 (ti(X_a,A_1) != zero_zero(X_a) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),A_1) = one_one(X_a))))) # label(fact_1090_div__self) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1799 (all X_a (ring_div(X_a) -> (all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Y),X)) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),Y) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),X),Y)))))) # label(fact_1985_dvd__neg__div) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1800 (all X_a (real_vector(X_a) -> (all X hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),one_one(real)),X) = ti(X_a,X)))) # label(fact_4703_real__vector_Oscale__one) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1801 (all Xa hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),fact_fact(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),Xa)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat))))),hAPP(real,real,sin,Xa)))) # label(fact_4103_sin__paired) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1802 (all X_a all X_c all B_1_1 all B_2_1 hAPP(X_a,X_c,ti(fun(X_a,X_c),B_1_1),B_2_1) = hAPP(X_a,X_c,B_1_1,B_2_1)) # label(tsy_c_hAPP_arg1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1803 (all X_c all X_b all X_a all C all Xa all P_4 ((all A_4 all B_4 (ti(product_prod(X_a,X_b),P_4) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_4),B_4) -> hBOOL(hAPP(X_c,bool,hAPP(X_b,fun(X_c,bool),hAPP(X_a,fun(X_b,fun(X_c,bool)),C,A_4),B_4),Xa)))) -> hBOOL(hAPP(X_c,bool,hAPP(product_prod(X_a,X_b),fun(X_c,bool),hAPP(fun(X_a,fun(X_b,fun(X_c,bool))),fun(product_prod(X_a,X_b),fun(X_c,bool)),product_prod_case(X_a,X_b,fun(X_c,bool)),C),P_4),Xa)))) # label(fact_4086_splitI2_H) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1804 (all X_a (real_n1866405975lgebra(X_a) -> (all R_3 all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),R_3),hAPP(X_a,real,norm_norm(X_a),X))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_3)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),X))),hAPP(real,real,inverse_inverse(real),R_3)))))))) # label(fact_2582_Bseq__inverse__lemma) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1805 (all Xa all Ya (hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(product_prod(nat,nat),bool),accp(product_prod(nat,nat)),nat_gcd_rel),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),Ya))) -> (zero_zero(nat) = Ya -> Xa = hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,Xa),Ya)) & (zero_zero(nat) != Ya -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,Xa),Ya) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,Ya),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Xa),Ya))))) # label(fact_3751_nat__gcd_Opsimps) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1806 (all X_a (linordered_field(X_a) -> (all A_1 (zero_zero(X_a) != ti(X_a,A_1) -> hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)) = hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)))))) # label(fact_2601_nonzero__abs__inverse) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1807 (all A_2 all B_1 (B_1 = A_2 <-> hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B_1) = hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2))) # label(fact_3672_transfer__nat__int__set__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1808 (all X_a (linordered_idom(X_a) -> (all Xa all Ya (zero_zero(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Ya),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) <-> zero_zero(X_a) = ti(X_a,Xa) & ti(X_a,Ya) = zero_zero(X_a))))) # label(fact_2_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1809 (all X_a (linordered_field(X_a) -> (all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),zero_zero(X_a))) -> (ti(X_a,A_1) != zero_zero(X_a) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))))))) # label(fact_2587_inverse__negative__imp__negative) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1810 (all X_a (comm_monoid_add(X_a) -> (all F all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) -> hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Ma),Na)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,F,Ma)),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),hAPP(nat,nat,suc,Ma)),Na))))))) # label(fact_4232_setsum__head__upt__Suc) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1811 (all X X = hAPP(int,quickcheck_code_int,quickcheck_of_int,hAPP(quickcheck_code_int,int,quickcheck_int_of,X))) # label(fact_3562_int__of__inverse) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1812 (all X_a (order(X_a) -> (all Z_1 all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),Y)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),X))))))) # label(fact_517_xt1_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1813 (all X all M all Y hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),M)),Y)),M) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),Y)),M)) # label(fact_1318_zpower__zmod) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1814 (all X_a (linordered_idom(X_a) -> (all A_1 all N hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),N) = hAPP(X_a,X_a,abs_abs(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N))))) # label(fact_978_power__abs) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1815 (all X_a (real_normed_field(X_a) -> (all Ya all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2)) -> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),X_2)),Ya))))))) # label(fact_4036_divide_Osummable) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1816 (all Xa hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),cos),Xa))) # label(fact_4928_isCont__cos) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1817 (all Ya all Xa (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)) -> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Xa),Ya)),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Xa),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),Ya))) # label(fact_1736_even__prod__div__2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1818 (all X_a (linorder(X_a) -> (all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya)) <-> ti(X_a,Xa) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Xa),Ya))))) # label(fact_4845_min__max_Ole__iff__inf) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1819 (all X_2 (hBOOL(hAPP(fun(nat,complex),bool,cauchy(complex),X_2)) -> hBOOL(hAPP(fun(nat,real),bool,cauchy(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),re),X_2))))) # label(fact_3877_Re_OCauchy) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1820 (all X_a (linordered_semidom(X_a) -> (all N all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),N))))))))) # label(fact_310_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1821 (all X_a (real_normed_field(X_a) -> (all G all E_2 all F all Xa all D_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),Xa),E_2)) -> (zero_zero(X_a) != hAPP(X_a,X_a,G,Xa) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),inverse_divide(X_a)),F)),G)),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),D_1),hAPP(X_a,X_a,G,Xa))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),E_2),hAPP(X_a,X_a,F,Xa)))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,G,Xa)),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))))))))))) # label(fact_3928_DERIV__quotient) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1822 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(real,nat,natfloor,X)),one_one(nat)) = hAPP(real,nat,natfloor,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),one_one(real))))) # label(fact_1677_natfloor__add__one) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1823 (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_945_dvd_Oless__imp__neq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1824 (all N zero_zero(nat) = hAPP(int,nat,nat_1,hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),N)))) # label(fact_2086_nat__zminus__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1825 (all Xa all P_4 (hBOOL(hAPP(int,bool,zprime,P_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),sRStar,P_4))) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),standardRes,P_4),Xa)),hAPP(int,fun(int,bool),sRStar,P_4))))))) # label(fact_1564_StandardRes__SRStar__prop4) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1826 (all X_a (comm_semiring_1(X_a) -> (all A_3 all D_1 all Na hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a))),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3)),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),semiring_1_of_nat(X_a))),D_1))),hAPP(nat,fun(nat,bool),ord_lessThan(nat),Na))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)))),D_1))))))) # label(fact_4351_arith__series__general) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1827 (all X all A_1 all B (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_1),B)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),X)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),B)))))) # label(fact_3378_powr__less__mono) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1828 (all X_a (linordered_field(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)) -> (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),one_one(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)))))))) # label(fact_2628_one__le__inverse) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1829 (all M all K_4 all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)) # label(fact_814_diff__cancel2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1830 (all L all U hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),U),L) = hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),L),U))) # label(fact_4227_card__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1831 (all X_a all X_b (comm_monoid_mult(X_b) & dvd(X_b) -> (all F all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),dvd_dvd(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),B_1)))))))) # label(fact_3985_setprod__dvd__setprod__subset) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1832 (all U_1 all M all N all J_2 all I_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_2),I_1)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I_1),U_1)),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_2),U_1)),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I_1),J_2)),U_1)),M)),N))) # label(fact_868_nat__diff__add__eq1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1833 (all Xa all Ya 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),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Xa),Z)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Ya),Z))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya))))) # label(fact_634_real__mult__less__iff1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1834 (all X_a measure(X_a) = ti(fun(fun(X_a,nat),fun(product_prod(X_a,X_a),bool)),measure(X_a))) # label(tsy_c_Wellfounded_Omeasure_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1835 (all N all K_4 all M (hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K_4),M) = one_one(int) -> (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,N),M) -> one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),N)),M)))) # label(fact_3457_zgcd__zgcd__zmult) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1836 (all X_a (linord20386208strict(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),zero_zero(X_a)))))))) # label(fact_450_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1837 (all X_a ti(fun(bool,fun(X_a,fun(X_a,X_a))),if(X_a)) = if(X_a)) # label(tsy_c_If_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1838 (all Xa all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya)) & Xa != Ya <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya)))) # label(fact_351_real__less__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1839 (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)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,N)))))))) # label(fact_1430_power__gt1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1840 (all A_1 all B all C_1 all D_3 hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,C_1),D_3)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_1),C_1)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),D_3))) # label(fact_2763_complex__add) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1841 (all Wa all Z (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,nat_1,Wa)),hAPP(int,nat,nat_1,Z))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z)))) # label(fact_1019_zless__nat__conj) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1842 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,arccos,Y)),pi))))) # label(fact_2356_arccos__ubound) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1843 (all X_a (preorder(X_a) -> (all Z_1 all X all 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_eq(X_a),Y),Z_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Z_1))))))) # label(fact_555_order__less__le__trans) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1844 (all K_4 hAPP(product_prod(int,int),rat,frct,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,number_number_of(int),K_4)),one_one(int))) = hAPP(int,rat,number_number_of(rat),K_4)) # label(fact_3730_Frct__code__post_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1845 (all X_a (ring_1(X_a) -> (all F all Na all Ra hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),F)),Ra)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)),Ra))))) # label(fact_4219_sumr__diff__mult__const2) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1846 (all X all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> hAPP(real,real,hAPP(nat,fun(real,real),root,N),hAPP(real,real,abs_abs(real),X)) = hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)))) # label(fact_2693_real__root__abs) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1847 (all Y all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,exp(real),one_one(real))),X)) -> (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),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,ln,Y)),Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,ln,X)),X)))))) # label(fact_1762_ln__x__over__x__mono) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1848 (all Xa all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),root,Na)),Xa),hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),Na)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,zero_zero(nat))))))))))) # label(fact_2672_DERIV__real__root) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1849 (all Z_1 hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(complex,real,re,Z_1)),hAPP(complex,real,norm_norm(complex),Z_1)) = hAPP(complex,real,re,hAPP(complex,complex,sgn_sgn(complex),Z_1))) # label(fact_3034_Re__sgn) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1850 (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),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Y))),zero_zero(X_a)))))) # label(fact_488_not__sum__squares__lt__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1851 (all M all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),hAPP(nat,nat,suc,N)) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N))) # label(fact_1338_add__Suc__right) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1852 (all R_3 ((hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_3),zero_zero(real))) -> hAPP(real,real,abs_abs(real),R_3) = hAPP(real,real,uminus_uminus(real),R_3)) & (-hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_3),zero_zero(real))) -> hAPP(real,real,abs_abs(real),R_3) = R_3))) # label(fact_2025_real__abs__def) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1853 (all X_a (division_ring(X_a) -> (all B_2 all A_3 all C (zero_zero(X_a) != ti(X_a,C) -> (ti(X_a,A_3) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C) <-> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C) = ti(X_a,B_2)))))) # label(fact_1849_nonzero__divide__eq__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1854 (all M all N all Qr_1 (hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M),N),Qr_1)) -> hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M),N) = Qr_1)) # label(fact_3125_divmod__nat__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1855 (all X hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),hAPP(real,nat,natceiling,X)))) # label(fact_1668_zero__le__natceiling) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1856 (all X_a (ab_group_add(X_a) -> (all A_3 all B_2 all C all D_1 (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),C),D_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2) -> (ti(X_a,D_1) = ti(X_a,C) <-> ti(X_a,B_2) = ti(X_a,A_3)))))) # label(fact_715_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1857 (all M (hBOOL(hAPP(int,bool,zprime,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),M)),one_one(int)))) -> one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legendre,hAPP(int,int,number_number_of(int),min)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),M)),one_one(int))))) # label(fact_687_Legendre__1mod4) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1858 (all Xa all Ya (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) <-> Xa = Ya)) # label(fact_963_divides__antisym) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1859 (all Na hAPP(nat,fun(nat,bool),ord_atMost(nat),Na) = hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),Na)))) # label(fact_4959_SetInterval_Otransfer__nat__int__set__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1860 (all X_a ti(fun(fun(X_a,bool),nat),finite_card(X_a)) = finite_card(X_a)) # label(tsy_c_Finite__Set_Ocard_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1861 (all R_3 hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R_3),zero_zero(real)) = hAPP(real,complex,of_real(complex),R_3)) # label(fact_3190_rcis__zero__arg) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1862 (all X_a (real_normed_vector(X_a) -> (all X_2 all L_2 ((all R_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1)) -> (exists No all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),No),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,X_2,N_1)),L_2))),R_1)))))) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),L_2),sequentially)))))) # label(fact_4684_LIMSEQ__I) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1863 (all X all Y hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),X)),hAPP(int,real,real_1(int),Y)) = hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),Y))) # label(fact_3364_real__of__int__add) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1864 (all X_a all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) <-> hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,finite_finite_1(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),A_2))))) # label(fact_2849_finite__Pow__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1865 (all X_a (real_normed_field(X_a) -> (all Ya all A_3 hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_a),fun(X_a,bool),isCont(X_a,X_a),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),inverse_divide(X_a)),Ya)),A_3))))) # label(fact_4922_divide_OisCont) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1866 (all A_3 all D_1 all Na hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))),hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),hAPP(nat,fun(nat,nat),plus_plus(nat),A_3)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),D_1))),hAPP(nat,fun(nat,bool),ord_lessThan(nat),Na))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))),D_1))))) # label(fact_4349_arith__series__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1867 (all A_1 all B hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),B),A_1) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B)) # label(fact_4405_gcd__commute__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1868 (all X_a (monoid_mult(X_a) -> (all A_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),one_one(X_a)) = ti(X_a,A_1)))) # label(fact_390_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1869 (all K_4 all L_4 hAPP(int,int,bit0,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K_4),L_4)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,bit0,K_4)),hAPP(int,int,bit0,L_4))) # label(fact_743_diff__bin__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1870 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y)) -> (exists X_1 hAPP(real,real,exp(real),X_1) = Y))) # label(fact_2114_exp__total) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1871 (all X_a (archim1727834104eiling(X_a) -> (all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archim2085082626_floor(X_a),Xa)),zero_zero(int))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),one_one(X_a))))))) # label(fact_2908_floor__le__zero) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1872 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),pi)) -> hAPP(real,real,arccos,hAPP(real,real,cos,X)) = X))) # label(fact_2352_arccos__cos) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1873 (all Ma all Xa (hAPP(int,int,hAPP(int,fun(int,int),standardRes,Ma),Xa) = zero_zero(int) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Xa),zero_zero(int)),Ma)))) # label(fact_1417_StandardRes__eq__zcong) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1874 (all X_a (comm_monoid_add(X_a) -> (all F all P_4 all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)))) -> hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),P_4))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),Na))),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),P_4)))))))) # label(fact_4629_setsum__ub__add__nat) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1875 (all X all A_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1)) -> (A_1 != one_one(real) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) -> hAPP(real,real,hAPP(real,fun(real,real),log,A_1),hAPP(real,real,inverse_inverse(real),X)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),log,A_1),X)))))) # label(fact_2639_log__inverse) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1876 (all X_a (order(X_a) -> ti(fun(fun(nat,fun(nat,bool)),fun(fun(nat,X_a),bool)),strict_mono(nat,X_a)) = strict_mono(nat,X_a))) # label(tsy_c_Orderings_Oorder_Ostrict__mono_res) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1877 (all X all N_4 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),N),N_4)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N_4),X)))))))) # label(fact_2716_real__root__increasing) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1878 (all X hAPP(real,real,cos,X) = hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),X))) # label(fact_2300_cos__sin__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1879 (all V all W ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W))) -> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,V))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,W))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,int,number_number_of(int),W)))),one_one(int))) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W))) -> hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),V)),one_one(int))),hAPP(int,int,number_number_of(int),W)))),one_one(int)) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,V))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,W)))))) # label(fact_1262_zmod__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1880 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,K1)),hAPP(int,int,bit1,K2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)))) # label(fact_279_less__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1881 (all X_a all X_b all P_1 hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),the(product_prod(X_a,X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),hAPP(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,bool)),fun(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,bool))),combb(fun(X_b,product_prod(X_a,X_b)),fun(X_b,bool),X_a),hAPP(fun(product_prod(X_a,X_b),bool),fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,bool)),combb(product_prod(X_a,X_b),bool,X_b),P_1)),product_Pair(X_a,X_b)))) = hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),the(product_prod(X_a,X_b)),P_1)) # label(fact_4169_split__paired__The) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1882 (all I_1 all M all N (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),I_1),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),N))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),I_1),N)))) # label(fact_4401_dvd__gcd__D2__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1883 (all K 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_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),Na))))) # label(fact_412_nat__add__left__cancel__le) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1884 (all N -hBOOL(hAPP(int,bool,nat_neg,hAPP(nat,int,semiring_1_of_nat(int),N)))) # label(fact_1600_not__neg__int) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1885 (all X (X != zero_zero(real) -> hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sgn_sgn(real),X)),pi)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(real,real,arctan,X)) = hAPP(real,real,arctan,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),X)))) # label(fact_2248_arctan__inverse) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1886 (all X_a (power(X_a) & real_normed_algebra(X_a) -> (all Ma all X_2 all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(nat,fun(nat,X_a),hAPP(fun(nat,fun(nat,X_a)),fun(nat,fun(nat,X_a)),combc(nat,nat,X_a),hAPP(fun(nat,X_a),fun(nat,fun(nat,X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(fun(nat,X_a),fun(nat,fun(nat,X_a))),combb(X_a,fun(nat,X_a),nat),power_power(X_a)),X_2)),Ma)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_3),Ma)),sequentially)))))) # label(fact_4490_LIMSEQ__pow) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1887 (all Z_1 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_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Z_1))))) # label(fact_948_dvd_Oorder__trans) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1888 (all X_a (real_normed_field(X_a) -> (all C all F all Xa all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),hAPP(X_a,fun(X_a,X_a),times_times(X_a),C)),F)),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),D))))))) # label(fact_3887_DERIV__cmult) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1889 (all X_a (division_ring(X_a) -> (all A_1 all B (hAPP(X_a,X_a,inverse_inverse(X_a),B) = hAPP(X_a,X_a,inverse_inverse(X_a),A_1) -> (zero_zero(X_a) != ti(X_a,A_1) -> (zero_zero(X_a) != ti(X_a,B) -> ti(X_a,A_1) = ti(X_a,B))))))) # label(fact_2578_nonzero__inverse__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1890 (all X_a (number_ring(X_a) -> hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),pls))))) # label(fact_2156_iszero__Numeral0) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1891 (all X_b all X_a (order(X_b) & order(X_a) -> (all C all F all B_2 all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),A_3)) -> (hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),C),hAPP(X_a,X_b,F,B_2))) -> ((all X_1 all Y_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_1),X_1)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,Y_1)),hAPP(X_a,X_b,F,X_1))))) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),C),hAPP(X_a,X_b,F,A_3))))))))) # label(fact_2861_xt7) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1892 (all P_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4)) -> hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),sRStar,P_4))))) # label(fact_2733_SRStar__finite) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1893 (all K_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K_4)) -> -(all N_1 hAPP(nat,int,semiring_1_of_nat(int),N_1) != ti(int,K_4)))) # label(fact_344_nonneg__int__cases) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1894 (all K all P_1 all D_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_1)) -> ((all X_1 (hBOOL(hAPP(int,bool,P_1,X_1)) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),D_1))))) -> (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,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),D_1)))))))))) # label(fact_927_decr__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1895 (all Na all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),Na)),Ma)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),Na)),hAPP(int,real,real_1(int),Ma))))) # label(fact_3414_number__of__le__real__of__int__iff) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1896 (all F all J all Na all H (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H)) -> (exists B_6 hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),J)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),H))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B_6),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),H),Na)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na))))) = hAPP(real,real,F,H)))) # label(fact_4299_Maclaurin__lemma) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1897 (all X_a (real_normed_field(X_a) -> (all Y_2 all B_2 all X_2 all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially)) -> (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),Y_2),B_2),sequentially)) -> (ti(X_a,B_2) != zero_zero(X_a) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),X_2)),Y_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),B_2)),sequentially)))))))) # label(fact_4466_LIMSEQ__divide) # label(axiom) # label(non_clause). [assumption]. 7.62/7.69 1898 (all Ta all A_2 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Ta),one_one(int))),A_2)) -> (all X_1 ((all Xa_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D))) -> (all Xb (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2)) -> hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa_1) != ti(int,X_1))))) -> (ti(int,X_1) = ti(int,Ta) -> ti(int,Ta) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),D))))))) # label(fact_4661_aset_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.69 1899 (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,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_332_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.70/7.69 1900 (all X_a (number_ring(X_a) -> (all V all W hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),V),W)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),W))))) # label(fact_371_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 7.70/7.69 1901 (all X_a (ordere779506340up_add(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C_1))))))) # label(fact_213_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.70/7.69 1902 (all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Ya)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,sqrt,Ya))))) # label(fact_2140_real__sqrt__ge__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.69 1903 (all X_a (ordere142940540dd_abs(X_a) -> (all A_3 (zero_zero(X_a) = ti(X_a,A_3) <-> zero_zero(X_a) = hAPP(X_a,X_a,abs_abs(X_a),A_3))))) # label(fact_969_abs__eq__0) # label(axiom) # label(non_clause). [assumption]. 7.70/7.69 1904 (all X_b all X_a (real_normed_algebra(X_a) -> (all Xa all G all A_2 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),G),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa)),G)),A_2)))) # label(fact_4213_mult__right_Osetsum) # label(axiom) # label(non_clause). [assumption]. 7.70/7.69 1905 (all T_1 preorder(filter(T_1))) # label(arity_Limits_Ofilter___Orderings_Opreorder) # label(axiom) # label(non_clause). [assumption]. 7.70/7.69 1906 (all X_a all C all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2))) -> -hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),A_2)))) # label(fact_2236_ComplD) # label(axiom) # label(non_clause). [assumption]. 7.70/7.69 1907 (all X_a finite_psubset(X_a) = hAPP(fun(product_prod(fun(X_a,bool),fun(X_a,bool)),bool),fun(product_prod(fun(X_a,bool),fun(X_a,bool)),bool),collect(product_prod(fun(X_a,bool),fun(X_a,bool))),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(product_prod(fun(X_a,bool),fun(X_a,bool)),bool),product_prod_case(fun(X_a,bool),fun(X_a,bool),bool),hAPP(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),hAPP(fun(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool))),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(fun(X_a,bool),bool))),combc(fun(X_a,bool),fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool)),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),fun(bool,bool))),fun(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool))),hAPP(fun(fun(fun(X_a,bool),fun(bool,bool)),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool))),fun(fun(fun(X_a,bool),fun(fun(X_a,bool),fun(bool,bool))),fun(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool)))),combb(fun(fun(X_a,bool),fun(bool,bool)),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool)),fun(X_a,bool)),combs(fun(X_a,bool),bool,bool)),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(fun(X_a,bool),fun(bool,bool))),hAPP(fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(bool,bool))),fun(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(fun(X_a,bool),fun(bool,bool)))),combb(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(bool,bool)),fun(X_a,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(bool,bool))),combb(bool,fun(bool,bool),fun(X_a,bool)),fconj)),ord_less(fun(X_a,bool))))),finite_finite_1(X_a))))) # label(fact_4077_finite__psubset__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.69 1908 (all X_a (linordered_idom(X_a) -> (all A_1 all K_4 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),K_4))),zero_zero(X_a))) -> ti(X_a,A_1) = zero_zero(X_a))))) # label(fact_651_even__power__le__0__imp__0) # label(axiom) # label(non_clause). [assumption]. 7.70/7.69 1909 (all D_3 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,abs_abs(int),D_3)),D_3))) # label(fact_1227_zdvd__self__abs2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.69 1910 (all X_a (real_normed_vector(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,uminus_uminus(real),A_1)),B) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B))))) # label(fact_4741_scaleR_Ominus__left) # label(axiom) # label(non_clause). [assumption]. 7.70/7.69 1911 (all N hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,suc,N)))) # label(fact_1271_zero__less__Suc) # label(axiom) # label(non_clause). [assumption]. 7.70/7.69 1912 (all R_3 all A_1 hAPP(complex,complex,inverse_inverse(complex),hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R_3),A_1)) = hAPP(real,complex,hAPP(real,fun(real,complex),rcis,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),R_3)),hAPP(real,real,uminus_uminus(real),A_1))) # label(fact_3199_rcis__inverse) # label(axiom) # label(non_clause). [assumption]. 7.70/7.69 1913 (all Wa (hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,Wa)))) <-> hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),Wa))))) # label(fact_1604_neg__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.69 1914 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,sqrt,Xa)),zero_zero(real))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),zero_zero(real))))) # label(fact_2142_real__sqrt__lt__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.69 1915 (all A_3 all B_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_2),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_3),B_2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_3),zero_zero(int)))))) # label(fact_1106_neg__imp__zdiv__nonneg__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.69 1916 (all Xa (-hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa)) <-> hAPP(nat,nat,suc,zero_zero(nat)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Xa),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))))) # label(fact_2504_odd__nat__equiv__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.69 1917 (all X0 all F all R_2 ((all X_1 (hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_1),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_gr676655987ssThan(real),hAPP(real,real,uminus_uminus(real),R_2)),R_2))) -> hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),F)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),suc)))),hAPP(real,fun(nat,real),power_power(real),X_1)))))) -> (hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X0),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_gr676655987ssThan(real),hAPP(real,real,uminus_uminus(real),R_2)),R_2))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_2)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,fun(nat,real)),fun(real,real),hAPP(fun(fun(nat,real),real),fun(fun(real,fun(nat,real)),fun(real,real)),combb(fun(nat,real),real,real),suminf(real)),hAPP(fun(real,fun(nat,real)),fun(real,fun(nat,real)),hAPP(fun(fun(nat,real),fun(nat,real)),fun(fun(real,fun(nat,real)),fun(real,fun(nat,real))),combb(fun(nat,real),fun(nat,real),real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),F))),hAPP(fun(nat,nat),fun(real,fun(nat,real)),hAPP(fun(real,fun(fun(nat,nat),fun(nat,real))),fun(fun(nat,nat),fun(real,fun(nat,real))),combc(real,fun(nat,nat),fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(real,fun(fun(nat,nat),fun(nat,real))),hAPP(fun(fun(nat,real),fun(fun(nat,nat),fun(nat,real))),fun(fun(real,fun(nat,real)),fun(real,fun(fun(nat,nat),fun(nat,real)))),combb(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),real),combb(nat,real,nat)),power_power(real))),suc)))),X0),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),F)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),suc)))),hAPP(real,fun(nat,real),power_power(real),X0))))))))) # label(fact_4063_DERIV__power__series_H) # label(axiom) # label(non_clause). [assumption]. 7.70/7.69 1918 (all X_a all X_b (real_normed_field(X_b) -> (all Ya all G all A_3 all A_2 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),A_3),A_2)) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),inverse_divide(X_b)),G)),Ya)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),inverse_divide(X_b),A_3),Ya)),A_2)))))) # label(fact_4498_divide_Otendsto) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1919 (all X all Y hAPP(int,int,hAPP(int,fun(int,int),int_gcd,X),Y) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),X))),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),Y))))) # label(fact_3752_Nitpick_Oint__gcd__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1920 (all N all A_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1)) -> (exists R_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1)) & hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),R_1),hAPP(nat,nat,suc,N)) = A_1)))) # label(fact_1519_realpow__pos__nth2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1921 (all X_a all P_1 ti(fun(X_a,bool),P_1) = hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),P_1)) # label(fact_75_Collect__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1922 (all Ma hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),norRRset,Ma)) = hAPP(int,nat,phi,Ma)) # label(fact_3638_phi__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1923 (all X_a (ring_1(X_a) -> (all N all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M)) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)))))) # label(fact_720_of__nat__diff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1924 (all X_a (semiring_div(X_a) -> (all A_1 all B all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1))),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)))) # label(fact_1354_mod__mult__self2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1925 (all A_3 all P_4 (hBOOL(hAPP(int,bool,zprime,P_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),zero_zero(int)),P_4)) -> hAPP(int,fun(int,bool),sRStar,P_4) = hAPP(fun(fun(int,bool),bool),fun(int,bool),complete_Sup_Sup(fun(int,bool)),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,A_3),P_4)))))) # label(fact_1690_MultInvPair__prop2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1926 (all X_a (comm_monoid_add(X_a) -> (all F all Na all Ma ((hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),Ma)) -> zero_zero(X_a) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Ma),hAPP(nat,nat,suc,Na)))) & (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),Ma)) -> hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Ma),hAPP(nat,nat,suc,Na))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Ma),Na))),hAPP(nat,X_a,F,Na))))))) # label(fact_4231_setsum__op__ivl__Suc) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1927 (all M all N hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,real_1(nat),M)),hAPP(nat,real,real_1(nat),N)) = hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N))) # label(fact_2386_real__of__nat__add) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1928 (all X_a (comm_semiring_1(X_a) -> (all A_1 zero_zero(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),zero_zero(X_a))))) # label(fact_379_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1929 (all X_a (real_normed_field(X_a) & banach(X_a) -> (all X hAPP(X_a,X_a,exp(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)) = hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,exp(X_a),X))))) # label(fact_2606_exp__minus) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1930 (all X_a (real_normed_algebra(X_a) -> (all X all Y all Ya_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y)),Ya_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Ya_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Ya_1))))) # label(fact_1247_mult__left_Odiff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1931 (all X one_one(real) = hAPP(real,real,hAPP(real,fun(real,real),powr,X),zero_zero(real))) # label(fact_3376_powr__zero__eq__one) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1932 (all Ta all B_1 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ta),B_1)) -> (all X_1 ((all Xa_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D))) -> (all Xb (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1)) -> hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa_1) != ti(int,X_1))))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Ta),X_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Ta),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),D))))))))) # label(fact_4665_bset_I7_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1933 (all X_a (ring(X_a) -> (all X all Y all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Y),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),A_1)),B))))) # label(fact_755_mult__diff__mult) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1934 (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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Na)))))) # label(fact_657_nat__mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1935 (all Y all X (X = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y)) -> hAPP(real,real,sqrt,X) = Y))) # label(fact_2169_real__sqrt__unique) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1936 (all X_a (real_normed_algebra(X_a) -> (all Xa all X_2 all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),X_2),A_3)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa)),X_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),A_3))))))) # label(fact_4109_mult__right_Osums) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1937 (all X_a (semiring_char_0(X_a) -> (all N hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,fact_fact(nat),N)) != zero_zero(X_a)))) # label(fact_2541_of__nat__fact__not__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1938 (all X_a (number_ring(X_a) & real_algebra_1(X_a) -> (all W hAPP(real,X_a,of_real(X_a),hAPP(int,real,number_number_of(real),W)) = hAPP(int,X_a,number_number_of(X_a),W)))) # label(fact_2796_of__real__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1939 (all X_a (ord(X_a) -> max(X_a) = ti(fun(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,X_a))),max(X_a)))) # label(tsy_c_Orderings_Oord_Omax_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1940 (all Xa hAPP(rat,product_prod(int,int),quotient_of,Xa) = hAPP(fun(product_prod(int,int),bool),product_prod(int,int),the(product_prod(int,int)),hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool),hAPP(fun(product_prod(int,int),fun(bool,bool)),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool)),combs(product_prod(int,int),bool,bool),hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),fun(bool,bool))),combb(bool,fun(bool,bool),product_prod(int,int)),fconj),hAPP(fun(product_prod(int,int),rat),fun(product_prod(int,int),bool),hAPP(fun(rat,bool),fun(fun(product_prod(int,int),rat),fun(product_prod(int,int),bool)),combb(rat,bool,product_prod(int,int)),hAPP(rat,fun(rat,bool),fequal(rat),Xa)),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),rat),hAPP(fun(product_prod(int,int),fun(int,rat)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),rat)),combs(product_prod(int,int),int,rat),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,rat)),hAPP(fun(int,fun(int,rat)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,rat))),combb(int,fun(int,rat),product_prod(int,int)),fract),product_fst(int,int))),product_snd(int,int))))),hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool),hAPP(fun(product_prod(int,int),fun(bool,bool)),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool)),combs(product_prod(int,int),bool,bool),hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),fun(bool,bool))),combb(bool,fun(bool,bool),product_prod(int,int)),fconj),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),bool),hAPP(fun(int,bool),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),bool)),combb(int,bool,product_prod(int,int)),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int))),product_snd(int,int)))),hAPP(int,fun(product_prod(int,int),bool),hAPP(fun(product_prod(int,int),fun(int,bool)),fun(int,fun(product_prod(int,int),bool)),combc(product_prod(int,int),int,bool),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,bool)),hAPP(fun(int,fun(int,bool)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,bool))),combb(int,fun(int,bool),product_prod(int,int)),fequal(int)),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),int),hAPP(fun(product_prod(int,int),fun(int,int)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),int)),combs(product_prod(int,int),int,int),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,int))),combb(int,fun(int,int),product_prod(int,int)),gcd_gcd(int)),product_fst(int,int))),product_snd(int,int)))),one_one(int)))))) # label(fact_4436_quotient__of__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1941 (all X_c all X_b all X_a product_apsnd(X_c,X_b,X_a) = ti(fun(fun(X_c,X_b),fun(product_prod(X_a,X_c),product_prod(X_a,X_b))),product_apsnd(X_c,X_b,X_a))) # label(tsy_c_Product__Type_Oapsnd_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1942 (all X_a (order(X_a) -> (all C_1 all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1)) -> (ti(X_a,B) = ti(X_a,C_1) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),A_1))))))) # label(fact_505_xt1_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1943 (all W all Z_1 hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W),Z_1))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),W))),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),Z_1)))) # label(fact_966_nat__abs__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1944 (all X_a (banach(X_a) & real_n2089651433ebra_1(X_a) -> (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),Xa)),one_one(real))) -> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))))))) # label(fact_4053_complete__algebra__summable__geometric) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1945 (all X all N all Y (hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Y),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),N) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Y),X)) -> (exists Q_4 hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),Q_4)) = X)))) # label(fact_1568_nat__mod__eq__lemma) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1946 (all X all Y all Z_1 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Y),Z_1)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Y),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),Z_1))) # label(fact_46_zadd__left__commute) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1947 (all Ma all Na hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,Ma),Na) = hAPP(fun(product_prod(nat,nat),bool),product_prod(nat,nat),the(product_prod(nat,nat)),hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,Ma),Na))) # label(fact_4171_divmod__nat__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1948 (all X_a (linord1117847801e_zero(X_a) -> (all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_2)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),zero_zero(X_a))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),B_2))))))) # label(fact_1833_zero__le__divide__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1949 (all R_3 (hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R_3)) -> -(all S_1 (hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),S_1)) -> (all T_4 (hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),T_4)) -> R_3 != hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),S_1),T_4))))))) # label(fact_3736_obtain__pos__sum) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1950 (all A_1 all B all C_1 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),C_1))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),C_1))),C_1)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)),C_1)) # label(fact_1432_zdiv__zmult1__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1951 (all A_1 hAPP(real,real,cos,A_1) = hAPP(complex,real,re,hAPP(real,complex,cis,A_1))) # label(fact_3051_Re__cis) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1952 (all X_a (semiring_div(X_a) -> (all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B)) -> ti(X_a,B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),B),A_1)))))) # label(fact_1091_dvd__mult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1953 (all N one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,suc,N)),N)) # label(fact_3287_coprime__Suc__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1954 (all X_b all X_a (topological_space(X_a) & topological_space(X_b) -> (all L all Ma all F all G all A_3 all B_2 (ti(X_a,B_2) = ti(X_a,A_3) -> ((all X_1 (ti(X_a,X_1) != ti(X_a,B_2) -> hAPP(X_a,X_b,F,X_1) = hAPP(X_a,X_b,G,X_1))) -> (ti(X_b,Ma) = ti(X_b,L) -> (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) <-> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),Ma),hAPP(X_a,filter(X_a),at(X_a),B_2)))))))))) # label(fact_4787_LIM__cong) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1955 (all X_a (comm_monoid_add(X_a) -> (all F all P_4 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_eq(nat),Na),P_4)) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Ma),Na))),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Na),P_4))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Ma),P_4))))))) # label(fact_4228_setsum__add__nat__ivl) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1956 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),hAPP(int,int,bit1,K))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K)))) # label(fact_256_rel__simps_I22_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1957 (all Xa all Na all Diff all F (hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (zero_zero(real) != Xa -> ((all M_2 all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_2)),X_1),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),X_1)))) -> (exists T_4 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,abs_abs(real),T_4))) & hAPP(real,real,F,Xa) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,Na),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),Na))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),T_4)),hAPP(real,real,abs_abs(real),Xa)))))))))) # label(fact_4304_Maclaurin__all__lt) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1958 (all X_a (real_normed_field(X_a) -> (all G all E_3 all F all Xa all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),Xa),E_3)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),plus_plus(X_a)),F)),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),uminus_uminus(X_a)),G))),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),D),hAPP(X_a,X_a,uminus_uminus(X_a),E_3))))))))) # label(fact_3859_DERIV__add__minus) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1959 (all X_a (ring_div(X_a) -> (all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Y),X)) -> hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),X),hAPP(X_a,X_a,uminus_uminus(X_a),Y)))))) # label(fact_1986_dvd__div__neg) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1960 (all N hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,real_1(nat),N)),one_one(real)) = hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N))) # label(fact_2396_real__of__nat__Suc) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1961 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),hAPP(int,int,bit0,K))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K)))) # label(fact_260_rel__simps_I21_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1962 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C_1)))) # label(fact_384_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1963 (all L_4 hAPP(int,int,bit0,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),pls),L_4)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),pls),hAPP(int,int,bit0,L_4))) # label(fact_767_diff__bin__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1964 (all X_a all X_b (ordere779506340up_add(X_b) & semiring_1(X_b) -> (all F all K_3 all A_2 ((all I_2 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I_2),A_2)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,I_2)),K_3)))) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(nat,X_b,semiring_1_of_nat(X_b),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2))),K_3))))))) # label(fact_4323_setsum__bounded) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1965 (all Wa all Ma ((-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)) <-> Ma = hAPP(int,nat,nat_1,Wa))) # label(fact_1037_nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1966 (all P_1 all I all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I),K)) -> (hBOOL(hAPP(int,bool,P_1,K)) -> ((all I_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_2),K)) -> (hBOOL(hAPP(int,bool,P_1,I_2)) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),I_2),one_one(int))))))) -> hBOOL(hAPP(int,bool,P_1,I)))))) # label(fact_1143_int__le__induct) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1967 (all X_a (real_normed_field(X_a) -> (all C all Xa hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(X_a,fun(X_a,X_a),times_times(X_a),C)),Xa),C))))) # label(fact_2649_DERIV__cmult__Id) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1968 (all X_a all A_2 all B_1 (ti(fun(X_a,bool),B_1) = ti(fun(X_a,bool),A_2) | hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1)) <-> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)))) # label(fact_2198_subset__iff__psubset__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1969 (all X all N (-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),N))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(real,nat,natfloor,X)),hAPP(int,nat,number_number_of(nat),N)) = hAPP(real,nat,natfloor,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(int,real,number_number_of(real),N)))))) # label(fact_1658_natfloor__add__number__of) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1970 (all Y (Y != zero_zero(nat) -> -(all Nat_2 hAPP(nat,nat,suc,Nat_2) != Y))) # label(fact_2833_nat_Oexhaust) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1971 (all B all A_1 all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M)) -> (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),M) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M)) -> one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,B),M))))) # label(fact_3493_zgcd__zcong__zgcd) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1972 (all A_1 all B hAPP(complex,complex,cnj,hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),hAPP(real,real,uminus_uminus(real),B))) # label(fact_2973_complex__cnj) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1973 (all B_2 all D_1 all A_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_1),A_3)) & (all E (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),E),B_2)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),E),A_3)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),E),D_1)))) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_1),B_2)) <-> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_3),B_2) = D_1)) # label(fact_3258_gcd__unique__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1974 (all Ma all Na (Na != zero_zero(nat) & zero_zero(nat) = Ma <-> hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Ma),Na) = zero_zero(nat))) # label(fact_337_nat__power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1975 (all X_b (number(X_b) & ring(X_b) -> (all A_1 all B all V hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_1),hAPP(int,X_b,number_number_of(X_b),V))),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),B),hAPP(int,X_b,number_number_of(X_b),V))) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),A_1),B)),hAPP(int,X_b,number_number_of(X_b),V))))) # label(fact_757_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1976 (all X_b all X_a (ordere142940540dd_abs(X_a) -> (all F all A_2 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2))),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),abs_abs(X_a)),F)),A_2)))))) # label(fact_4192_setsum__abs) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1977 (all R_3 all A_1 hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,of_real(complex),R_3)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,cos,A_1)),hAPP(real,real,sin,A_1)))) = hAPP(real,real,abs_abs(real),R_3)) # label(fact_2824_cmod__complex__polar) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1978 (all A_1 all B all K_4 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),K_4) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),B)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B))),K_4)) # label(fact_1434_zdiv__zmod__equality2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1979 (all X_a (zero_neq_one(X_a) -> zero_zero(X_a) != one_one(X_a))) # label(fact_167_zero__neq__one) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1980 (all X_a (real_normed_vector(X_a) -> (all Ra all X_2 all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra)),X_2)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra),A_3)),sequentially)))))) # label(fact_4754_scaleR__right_OLIMSEQ) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1981 (all X_a (number_ring(X_a) -> hAPP(int,X_a,number_number_of(X_a),min) = hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)))) # label(fact_2017_number__of__Min) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1982 (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_944_dvd_Oless__not__sym) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1983 (all Xa all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),Xa)),Ya)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa),Ya))))) # label(fact_2022_real__0__less__add__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1984 (all X_a (number_ring(X_a) -> (all V all W all Z_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),W))),Z_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),W)),Z_1))))) # label(fact_90_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1985 (all V all W hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,int,number_number_of(int),W))) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,V))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,W)))) # label(fact_1471_zmod__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1986 (all X_a (semiring_div(X_a) -> (all B all A_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),A_1)),B)))) # label(fact_1296_mod__add__self1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1987 (all X hAPP(real,real,uminus_uminus(real),hAPP(real,real,sqrt,X)) = hAPP(real,real,sqrt,hAPP(real,real,uminus_uminus(real),X))) # label(fact_2132_real__sqrt__minus) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1988 (all X_a all X_b all A_2 all B_1 all P_1 (hBOOL(hAPP(fun(product_prod(fun(X_a,bool),fun(X_b,bool)),bool),bool,hAPP(product_prod(fun(X_a,bool),fun(X_b,bool)),fun(fun(product_prod(fun(X_a,bool),fun(X_b,bool)),bool),bool),member(product_prod(fun(X_a,bool),fun(X_b,bool))),hAPP(fun(X_b,bool),product_prod(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,bool),fun(fun(X_b,bool),product_prod(fun(X_a,bool),fun(X_b,bool))),product_Pair(fun(X_a,bool),fun(X_b,bool)),A_2),B_1)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(fun(X_a,bool),fun(X_b,bool)),bool),bijR(X_a,X_b),P_1))) -> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)))) # label(fact_4948_fin__bijRl) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1989 (all X_a (division_ring(X_a) -> (all A_1 (zero_zero(X_a) != ti(X_a,A_1) -> one_one(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),A_1))))) # label(fact_2616_left__inverse) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1990 (all X_b all X_a (topological_space(X_b) & topological_space(X_a) -> (all K all Xa hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),combk(X_b,X_a),K)),K),hAPP(X_a,filter(X_a),at(X_a),Xa)))))) # label(fact_4580_LIM__const) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1991 (all X_a (number_ring(X_a) -> (all V all W hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),W)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),W))))) # label(fact_92_number__of__add) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1992 (all X_a (real_normed_algebra(X_a) -> (all Y exists K_2 ((all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_1)),K_2)))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2)))))) # label(fact_2052_mult__left_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1993 -(all S_1 -(hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,s1),S_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int)))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),S_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int)))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),S_1)))) # label(fact_709__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]. 7.70/7.70 1994 (all A_1 hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,abs_abs(int),A_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(int,int,abs_abs(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) # label(fact_1054_abs__power2__distrib) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1995 (all X_a (linordered_field(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(X_a,X_a,inverse_inverse(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,fact_fact(nat),N)))))))) # label(fact_2632_inv__of__nat__fact__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1996 (all K_4 all L_4 all I_1 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_4),L_4)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),L_4)))))) # label(fact_138_add__less__mono) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1997 (all A_1 all B (ti(int,B) != zero_zero(int) -> (hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) != zero_zero(int) -> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,uminus_uminus(int),A_1)),B) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B))),one_one(int))) & (hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = zero_zero(int) -> hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,uminus_uminus(int),A_1)),B)))) # label(fact_2099_zdiv__zminus1__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1998 (all X all Y all N (N != zero_zero(nat) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),N)),Y)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))))) # label(fact_856_divides__exp2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 1999 (all V zero_zero(real) = hAPP(complex,real,im,hAPP(int,complex,number_number_of(complex),V))) # label(fact_3004_complex__Im__number__of) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2000 (all X_a (dvd(X_a) -> ti(fun(X_a,fun(X_a,X_a)),times_times(X_a)) = times_times(X_a))) # label(tsy_c_Groups_Otimes__class_Otimes_1_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2001 (all X_b all X_a (topological_space(X_b) & real_normed_vector(X_a) -> (all F all Xa (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_a),fun(X_a,X_b),hAPP(fun(X_a,X_b),fun(fun(X_a,X_a),fun(X_a,X_b)),combb(X_a,X_b,X_a),F),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Xa))),hAPP(X_a,X_b,F,Xa)),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a)))) <-> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),Xa)))))) # label(fact_4941_isCont__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2002 (all X_a (order(X_a) -> (all F ((all X_1 all Y_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,F,X_1)),hAPP(nat,X_a,F,Y_1))))) -> hBOOL(hAPP(fun(nat,X_a),bool,hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,X_a),bool),strict_mono(nat,X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(fun(nat,bool),fun(nat,bool))),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool))),combs(nat,fun(nat,bool),fun(nat,bool)),hAPP(fun(nat,fun(nat,fun(bool,bool))),fun(nat,fun(fun(nat,bool),fun(nat,bool))),hAPP(fun(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool))),fun(fun(nat,fun(nat,fun(bool,bool))),fun(nat,fun(fun(nat,bool),fun(nat,bool)))),combb(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),nat),combs(nat,bool,bool)),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,fun(bool,bool))),hAPP(fun(fun(nat,bool),fun(nat,fun(bool,bool))),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,fun(bool,bool)))),combb(fun(nat,bool),fun(nat,fun(bool,bool)),nat),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj)),dvd_dvd(nat)))),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),hAPP(fun(fun(nat,bool),fun(nat,bool)),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool))),combb(fun(nat,bool),fun(nat,bool),nat),hAPP(fun(bool,bool),fun(fun(nat,bool),fun(nat,bool)),combb(bool,bool,nat),fNot)),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),dvd_dvd(nat))))),F)))))) # label(fact_4816_dvd_Ostrict__monoI) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2003 (all M all K_4 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),J_2)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),K_4))),M) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,J_2)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_4),M)))) # label(fact_1452_diff__Suc__diff__eq2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2004 (all A_2 all B_1 all Ma (hBOOL(hAPP(fun(product_prod(fun(int,bool),fun(int,bool)),bool),bool,hAPP(product_prod(fun(int,bool),fun(int,bool)),fun(fun(product_prod(fun(int,bool),fun(int,bool)),bool),bool),member(product_prod(fun(int,bool),fun(int,bool))),hAPP(fun(int,bool),product_prod(fun(int,bool),fun(int,bool)),hAPP(fun(int,bool),fun(fun(int,bool),product_prod(fun(int,bool),fun(int,bool))),product_Pair(fun(int,bool),fun(int,bool)),A_2),B_1)),hAPP(fun(int,fun(int,bool)),fun(product_prod(fun(int,bool),fun(int,bool)),bool),bijR(int,int),hAPP(int,fun(int,fun(int,bool)),zcongm,Ma)))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),A_2)),hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),B_1)),Ma)))) # label(fact_4899_bijzcong__zcong__prod) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2005 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Xa),hAPP(int,int,bit1,pls))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,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]. 7.70/7.70 2006 (all X_a all F all A_2 ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2)) -> hBOOL(hAPP(int,bool,nat_is_nat,hAPP(X_a,int,F,X_1))))) -> hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co302546470etprod(X_a,int),F),A_2) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co302546470etprod(X_a,nat),hAPP(fun(X_a,int),fun(X_a,nat),hAPP(fun(int,nat),fun(fun(X_a,int),fun(X_a,nat)),combb(int,nat,X_a),nat_1),F)),A_2)))) # label(fact_4886_transfer__int__nat__sum__prod2_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2007 (all X_a (field(X_a) -> (all Z all Na all H (zero_zero(X_a) != ti(X_a,H) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),H),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(fun(nat,bool),X_a),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combb(fun(nat,bool),X_a,nat),hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z),H)))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Z)),hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat))),hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,zero_zero(nat))))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,zero_zero(nat)))))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z),H)),Na)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z),Na))),H)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,zero_zero(nat)))))))))) # label(fact_4265_lemma__termdiff2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2008 (all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,N)),hAPP(nat,nat,fact,N)) = hAPP(nat,nat,fact,hAPP(nat,nat,suc,N))) # label(fact_3571_fact_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2009 (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),hAPP(X_a,X_a,abs_abs(X_a),A_1)))))) # label(fact_971_abs__ge__self) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2010 (all R_3 all S_3 hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(rat,real,ratreal,hAPP(int,rat,number_number_of(rat),R_3))),hAPP(rat,real,ratreal,hAPP(int,rat,number_number_of(rat),S_3))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,number_number_of(real),R_3)),hAPP(int,real,number_number_of(real),S_3))) # label(fact_3802_Ratreal__number__of__quotient) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2011 (all R_3 all X hAPP(complex,real,im,hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),R_3),X)) = hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),R_3),hAPP(complex,real,im,X))) # label(fact_4707_Im_OscaleR) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2012 (all X X = hAPP(real,real,hAPP(nat,fun(real,real),root,hAPP(nat,nat,suc,zero_zero(nat))),X)) # label(fact_2675_real__root__Suc__0) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2013 (all X_a (real_normed_field(X_a) -> (all Ya all X_2 all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),X_2)),Ya)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),Ya)),sequentially)))))) # label(fact_4497_divide_OLIMSEQ) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2014 (all M all N (hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),N) = one_one(int) -> ti(int,M) = one_one(int) | hAPP(int,int,number_number_of(int),min) = ti(int,M))) # label(fact_690_pos__zmult__eq__1__iff__lemma) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2015 (all X_a (ord(X_a) -> (all X all Least ((all X_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Least),X_1))) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Least),X) = ti(X_a,Least))))) # label(fact_4861_min__leastL) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2016 (all C_1 all A_1 all B (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) -> (B = C_1 -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),C_1),A_1))))) # label(fact_938_dvd_Oord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2017 (all X_a all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) -> hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(fun(X_a,bool),bool),fun(fun(fun(X_a,bool),bool),bool),ord_less_eq(fun(fun(X_a,bool),bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),powp(X_a),A_2)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),powp(X_a),B_1))))) # label(fact_4895_Powp__mono) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2018 (all W (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),W)),zero_zero(int))) -> hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,hAPP(int,int,uminus_uminus(int),one_one(int))),hAPP(int,int,uminus_uminus(int),hAPP(int,int,number_number_of(int),W))))) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),one_one(int)),hAPP(int,int,number_number_of(int),W)))) # label(fact_3140_mod__pos__neg__1__number__of) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2019 (all S ((exists K_1 hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less_eq(fun(nat,bool)),S),hAPP(nat,fun(nat,bool),ord_lessThan(nat),K_1)))) <-> hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),S)))) # label(fact_4344_finite__nat__iff__bounded) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2020 (all B_2 all A_3 all D_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),D_1)) & (all E (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),E),A_3)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),E),B_2)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),E),D_1)))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),B_2)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),A_3)) <-> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_3),B_2) = ti(int,D_1))) # label(fact_4422_gcd__unique__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2021 (all X_a (ring(X_a) -> (all A_3 all E_2 all C all B_2 all D_1 (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),E_2)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),E_2)),D_1) <-> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B_2),A_3)),E_2)),D_1) = ti(X_a,C))))) # label(fact_754_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2022 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)))))))) # label(fact_230_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2023 (all X_a (metric_space(X_a) -> (all X_2 ((all E (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E)) -> (exists M_3 all M_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),M_2)) -> (all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),hAPP(nat,X_a,X_2,M_2)),hAPP(nat,X_a,X_2,N_1))),E)))))))) <-> hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_2)))))) # label(fact_3548_Cauchy__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2024 (all A_3 all B_2 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_3),B_2)),zOdd)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),zEven)) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),zOdd))))) # label(fact_1709_even__plus__odd__prop2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2025 (all X_a (division_ring(X_a) -> (all A_1 (ti(X_a,A_1) != zero_zero(X_a) -> one_one(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)))))) # label(fact_2618_right__inverse) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2026 (all A_1 all R_3 all B all M all C_1 all D_3 all N hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C_1),N))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),R_3),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),D_3),N)))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),R_3),B))),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),C_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),R_3),D_3))),N))) # label(fact_906_xzgcda__linear__aux1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2027 (all T_2 all T_1 (uminus(T_1) -> uminus(fun(T_2,T_1)))) # label(arity_fun___Groups_Ouminus) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2028 (all K_4 K_4 = hAPP(int,quickcheck_code_int,quickcheck_of_int,hAPP(quickcheck_code_int,int,quickcheck_int_of,K_4))) # label(fact_3561_of__int__int__of) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2029 (all X_c all X_a all X_b (comm_monoid_add(X_b) -> (all G all F all S (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S)) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),S) = hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co1399186613setsum(X_c,X_b),hAPP(fun(X_c,fun(X_a,bool)),fun(X_c,X_b),hAPP(fun(fun(X_a,bool),X_b),fun(fun(X_c,fun(X_a,bool)),fun(X_c,X_b)),combb(fun(X_a,bool),X_b,X_c),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G)),hAPP(fun(X_c,fun(X_a,bool)),fun(X_c,fun(X_a,bool)),hAPP(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_c,fun(X_a,bool)),fun(X_c,fun(X_a,bool))),combb(fun(X_a,bool),fun(X_a,bool),X_c),collect(X_a)),hAPP(fun(X_c,fun(X_a,bool)),fun(X_c,fun(X_a,bool)),hAPP(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_c,fun(X_a,bool)),fun(X_c,fun(X_a,bool))),combb(fun(X_a,bool),fun(X_a,bool),X_c),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),S)))),hAPP(fun(X_a,fun(X_c,bool)),fun(X_c,fun(X_a,bool)),combc(X_a,X_c,bool),hAPP(fun(X_a,X_c),fun(X_a,fun(X_c,bool)),hAPP(fun(X_c,fun(X_c,bool)),fun(fun(X_a,X_c),fun(X_a,fun(X_c,bool))),combb(X_c,fun(X_c,bool),X_a),fequal(X_c)),F)))))),hAPP(fun(X_a,bool),fun(X_c,bool),hAPP(fun(X_a,X_c),fun(fun(X_a,bool),fun(X_c,bool)),image(X_a,X_c),F),S)))))) # label(fact_4251_setsum__image__gen) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2030 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) & Ma != Na)) # label(fact_400_nat__less__le) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2031 (all A_1 all B hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)),ii) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,uminus_uminus(real),B)),A_1)) # label(fact_2784_Complex__mult__i) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2032 (all X_a (group_add(X_a) -> (all A_3 all B_2 (ti(X_a,B_2) = hAPP(X_a,X_a,uminus_uminus(X_a),A_3) <-> hAPP(X_a,X_a,uminus_uminus(X_a),B_2) = ti(X_a,A_3))))) # label(fact_1923_minus__equation__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2033 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C_1))))) # label(fact_298_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2034 (all N hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,fact_fact(nat),N)))) # label(fact_2544_fact__gt__zero__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2035 (all Xa hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),cos)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),Xa),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,Xa))),hAPP(real,real,cos,Xa))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,Xa))),hAPP(real,real,cos,Xa)))))) # label(fact_3942_DERIV__cos__realpow2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2036 (all X_a (ab_group_add(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(X_a,X_a,uminus_uminus(X_a),B)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B))))) # label(fact_1916_minus__add__distrib) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2037 (all X all Y hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),X),Y) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,re,X)),hAPP(complex,real,re,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,im,X)),hAPP(complex,real,im,Y)))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,re,X)),hAPP(complex,real,im,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,im,X)),hAPP(complex,real,re,Y))))) # label(fact_3041_complex__mult__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2038 (all X_a (linordered_field(X_a) -> (all Z_1 all W 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(X_a),X),Y)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),W)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),W),Z_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Z_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Y),W)))))))))) # label(fact_1863_frac__less) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2039 (all Na all Ma (hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),Na),Ma)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(quickcheck_code_int,int,quickcheck_int_of,Na)),hAPP(quickcheck_code_int,int,quickcheck_int_of,Ma))))) # label(fact_3186_less__code__int__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2040 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real))) -> (exists X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) & Y = hAPP(real,real,sin,X_1) & (all Y_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),Y_1)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) & Y = hAPP(real,real,sin,Y_1) -> Y_1 = X_1)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X_1))))))) # label(fact_2347_sin__total) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2041 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,ln,Xa))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),Xa))))) # label(fact_1728_ln__ge__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2042 (all X_a all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)),hAPP(fun(X_a,bool),nat,finite_card(X_a),B_1))),hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)))))) # label(fact_2735_diff__card__le__card__Diff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2043 (all Xa all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya)) -> (exists X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),X_1)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),Ya)) & hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_1),field_char_0_Rats(real))))))) # label(fact_3719_Rats__dense__in__real) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2044 (all A_1 all P_5 ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),P_5)) -> hAPP(int,int,hAPP(int,fun(int,int),legendre,A_1),P_5) = zero_zero(int)) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_5),A_1)) -> hAPP(int,int,hAPP(int,fun(int,int),legendre,A_1),P_5) = hAPP(int,int,number_number_of(int),min)) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_5),A_1)) -> one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legendre,A_1),P_5))))) # label(fact_794_Legendre__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2045 (all N all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,real_1(int),N)),hAPP(int,real,real_1(int),X))),hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),N),X)))),one_one(real)))) # label(fact_3435_real__of__int__div3) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2046 (all X_a (division_ring(X_a) -> (all A_1 (ti(X_a,A_1) != zero_zero(X_a) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),A_1) = one_one(X_a))))) # label(fact_1856_divide__self) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2047 (all X_b all X_a all B_1 all A_2 ((-(hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) & hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_1))) -> zero_zero(nat) = hAPP(fun(sum_sum(X_a,X_b),bool),nat,finite_card(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_2),B_1))) & (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) & hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_1)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)),hAPP(fun(X_b,bool),nat,finite_card(X_b),B_1)) = hAPP(fun(sum_sum(X_a,X_b),bool),nat,finite_card(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_2),B_1))))) # label(fact_3205_card__Plus__conv__if) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2048 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),zero_zero(real))) -> hAPP(int,real,number_number_of(real),min) = hAPP(real,real,sgn_sgn(real),X))) # label(fact_2311_real__sgn__neg) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2049 (all N all M all A_1 all B (hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B) = one_one(int) -> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),N)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),M)) = one_one(int))) # label(fact_4375_coprime__exp2__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2050 (all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z)))) # label(fact_282_int__one__le__iff__zero__less) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2051 (all X (-hBOOL(hAPP(nat,bool,even_odd_even(nat),X)) -> hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),X),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))))) = X)) # label(fact_2515_odd__nat__div__two__times__two__plus__one) # label(axiom) # label(non_clause). [assumption]. 7.70/7.70 2052 (all X_a (comm_ring(X_a) & monoid_mult(X_a) -> (all Z all H all Ma hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z),H))),hAPP(nat,fun(nat,nat),minus_minus(nat),Ma)))),hAPP(X_a,fun(nat,X_a),power_power(X_a),Z)))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z),Ma))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Ma)) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(X_a,fun(nat,X_a),power_power(X_a),Z))),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z),H))),hAPP(nat,fun(nat,nat),minus_minus(nat),Ma)))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Z)),hAPP(nat,fun(nat,nat),minus_minus(nat),Ma))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Ma))))) # label(fact_4253_lemma__termdiff1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2053 (all X_a all X_b (ord(X_b) -> (all Xa all F all G (hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,Xa)),hAPP(X_a,X_b,G,Xa))))))) # label(fact_500_le__funE) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2054 (all D_1 all K all P_4 (hBOOL(hAPP(nat,bool,prime,P_4)) -> ((exists I_2 (hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_4),I_2) = D_1 & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),K)))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_4),K)))))) # label(fact_3093_divides__primepow) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2055 (all Na all Va (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,Na)),hAPP(int,nat,number_number_of(nat),Va))) <-> (-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))))) & -hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))))) # label(fact_3922_less__Suc__number__of) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2056 (all X_a (real_normed_vector(X_a) -> (all A_1 all B hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B))))) # label(fact_4735_scaleR_Ominus__right) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2057 (all X_a (real_normed_vector(X_a) -> (all Ra all X_2 all L_2 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),L_2),sequentially)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Ra)) -> (exists No all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),No),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,X_2,N_1)),L_2))),Ra))))))))) # label(fact_4543_LIMSEQ__D) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2058 (all C_1 all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C_1),A_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C_1),B))))) # label(fact_826_divides__mul__l) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2059 (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),hAPP(int,int,bit1,K1)),hAPP(int,int,bit1,K2))))) # label(fact_30_less__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2060 (all Q all P (-hBOOL(P) | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fdisj,P),Q)))) # label(help_fdisj_1_1_U) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2061 (all R_3 all X hAPP(complex,real,im,hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),R_3),X)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),hAPP(complex,real,im,X))) # label(fact_4725_complex__Im__scaleR) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2062 (all X_b all X_a all R_2 all Xa all Ya (hBOOL(hAPP(fun(product_prod(X_a,X_b),bool),bool,hAPP(product_prod(X_a,X_b),fun(fun(product_prod(X_a,X_b),bool),bool),member(product_prod(X_a,X_b)),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),Xa),Ya)),R_2)) <-> hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),hAPP(fun(product_prod(X_a,X_b),bool),fun(X_a,fun(X_b,bool)),in_rel(X_a,X_b),R_2),Xa),Ya)))) # label(fact_3068_in__rel__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2063 (all X all Y hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,abs_abs(int),X)),hAPP(int,int,abs_abs(int),Y)) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y)) # label(fact_4391_gcd__abs__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2064 (all M all N ((zero_zero(nat) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)),hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)),one_one(nat)))) = hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N))) & (zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N) -> hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)) = one_one(nat)))) # label(fact_2536_fact__add__num__eq__if__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2065 (all X_b all X_a (topological_space(X_a) & topological_space(X_b) -> (all K_3 all F all L_2 all Xa (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),Xa))) -> (ti(X_b,K_3) = ti(X_b,L_2) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),K_3),hAPP(X_a,filter(X_a),at(X_a),Xa)))))))) # label(fact_4596_LIM__cong__limit) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2066 (all X_a (linordered_idom(X_a) -> (all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z),zero_zero(int))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)),zero_zero(X_a))))))) # label(fact_3626_of__int__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2067 (all N all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,real_1(nat),N)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,real_1(nat),N)),one_one(real)))) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) = hAPP(real,nat,natceiling,X)))) # label(fact_2439_natceiling__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2068 (all S (hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,S)) <-> (all X_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),S)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)))))) # label(fact_3684_nat__set__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2069 (all K_4 hAPP(int,int,bit1,K_4) = hAPP(int,int,succ,hAPP(int,int,bit0,K_4))) # label(fact_286_succ__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2070 (all Xa (one_one(real) = hAPP(real,real,sqrt,Xa) <-> Xa = one_one(real))) # label(fact_2129_real__sqrt__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2071 (all Xa all Ya all P_1 ((-hBOOL(P_1) -> hAPP(nat,int,semiring_1_of_nat(int),Ya) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),P_1),Xa),Ya))) & (hBOOL(P_1) -> hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),P_1),Xa),Ya)) = hAPP(nat,int,semiring_1_of_nat(int),Xa)))) # label(fact_163_int__if__cong) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2072 (all X_a all X_b all F all A_2 all B_1 hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2)),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),B_1))),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_2),B_1))))) # label(fact_3657_image__diff__subset) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2073 (all K_4 all M hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,K_4),K_4),M))) # label(fact_873_zcong__refl) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2074 (all W pls = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),pls),W)) # label(fact_398_mult__Pls) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2075 (all X_a (boolean_algebra(X_a) -> (all Xa all Ya (hAPP(X_a,X_a,uminus_uminus(X_a),Xa) = hAPP(X_a,X_a,uminus_uminus(X_a),Ya) <-> ti(X_a,Xa) = ti(X_a,Ya))))) # label(fact_2978_compl__eq__compl__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2076 (all X_a (banach(X_a) & real_normed_field(X_a) -> (all X_1 hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1))) = hAPP(X_a,X_a,exp(X_a),X_1)))) # label(fact_4775_lemma__exp__ext) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2077 (all X_a (idom(X_a) -> (all Xa all Ya (hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Ya),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))) <-> ti(X_a,Xa) = ti(X_a,Ya) | ti(X_a,Xa) = hAPP(X_a,X_a,uminus_uminus(X_a),Ya))))) # label(fact_2009_realpow__two__disj) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2078 (all X_a (real_normed_vector(X_a) & real_algebra_1(X_a) -> (all X_2 (hBOOL(hAPP(fun(nat,real),bool,cauchy(real),X_2)) -> hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),hAPP(fun(nat,real),fun(nat,X_a),hAPP(fun(real,X_a),fun(fun(nat,real),fun(nat,X_a)),combb(real,X_a,nat),of_real(X_a)),X_2))))))) # label(fact_4104_of__real_OCauchy) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2079 (all X_2 (hBOOL(hAPP(fun(nat,complex),bool,summable(complex),X_2)) -> hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),im),X_2))))) # label(fact_4074_Im_Osummable) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2080 (all X_a (linordered_field(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(X_a),Y),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),zero_zero(X_a)))))))) # label(fact_1861_divide__nonneg__neg) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2081 (all X_a all X_b all F all A_3 all B_2 (hBOOL(hAPP(product_prod(X_a,X_b),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),F),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_3),B_2))) -> hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),F,A_3),B_2)))) # label(fact_4069_splitD) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2082 (all X_a (linorder(X_a) -> (all X all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B))) -> -(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),X),B))))))) # label(fact_4854_min__max_Ole__infE) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2083 (all X_b all X_a (comm_monoid_add(X_a) -> (all A_2 zero_zero(X_a) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(X_a,fun(X_b,X_a),combk(X_a,X_b),zero_zero(X_a))),A_2)))) # label(fact_4239_setsum__0) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2084 (all Xa ((exists Y_1 ti(int,Xa) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Y_1)) <-> hBOOL(hAPP(int,bool,even_odd_even(int),Xa)))) # label(fact_1625_even__equiv__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2085 (all X_a (ab_group_add(X_a) -> (all A_1 all B hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),A_1)))) # label(fact_1918_minus__diff__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2086 (all A_3 all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),A_3)),Xa)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_3),hAPP(real,int,archim2085082626_floor(real),Xa))))) # label(fact_3389_le__floor__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2087 (all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),Ya)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),hAPP(real,real,sqrt,Ya))))) # label(fact_2151_real__sqrt__gt__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2088 (all I all M_1 (hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),zero_zero(nat)),M_1)) -> hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),combs(nat,bool,bool),hAPP(fun(nat,bool),fun(nat,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(fun(nat,bool),bool)),fun(fun(nat,bool),fun(nat,bool)),combc(nat,fun(nat,bool),bool),member(nat)),M_1))),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),hAPP(nat,nat,suc,I))))) = hAPP(nat,nat,suc,hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),combs(nat,bool,bool),hAPP(fun(nat,bool),fun(nat,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(fun(nat,bool),bool)),fun(fun(nat,bool),fun(nat,bool)),combc(nat,fun(nat,bool),bool),hAPP(fun(nat,nat),fun(nat,fun(fun(nat,bool),bool)),hAPP(fun(nat,fun(fun(nat,bool),bool)),fun(fun(nat,nat),fun(nat,fun(fun(nat,bool),bool))),combb(nat,fun(fun(nat,bool),bool),nat),member(nat)),suc)),M_1))),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),I))))))) # label(fact_3904_card__less__Suc) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2089 (all X_a (dist_norm(X_a) -> ti(fun(X_a,real),norm_norm(X_a)) = norm_norm(X_a))) # label(tsy_c_RealVector_Onorm__class_Onorm_1_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2090 (all X_a (linord581940658strict(X_a) -> (all Xa all Ya (zero_zero(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),Xa)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ya),Ya)) <-> zero_zero(X_a) = ti(X_a,Ya) & ti(X_a,Xa) = zero_zero(X_a))))) # label(fact_457_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2091 (all X all A_1 all B hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),B)) = hAPP(real,real,hAPP(real,fun(real,real),powr,X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_1),B))) # label(fact_3381_powr__add) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2092 (all X_a (archim1804426504_field(X_a) -> (all X exists Z_2 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),Z_2)))))) # label(fact_3650_ex__less__of__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2093 (all X_a ti(fun(int,fun(int,X_a)),frac(X_a)) = frac(X_a)) # label(tsy_c_Nitpick_Ofrac_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2094 (all N all K_4 all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),N)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N)))))) # label(fact_3463_zgcd__greatest) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2095 (all I all J (hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I),J) = zero_zero(int) <-> zero_zero(int) = ti(int,I) & zero_zero(int) = ti(int,J))) # label(fact_3460_zgcd0) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2096 (all X_a (comm_monoid_add(X_a) -> (all F all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,F,Ma)),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,suc,Ma)),Na))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),Na)))))) # label(fact_4623_setsum__head__Suc) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2097 (all M all N hAPP(nat,int,hAPP(int,fun(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,hAPP(nat,fun(nat,nat),power_power(nat),M),N))) # label(fact_20_zpower__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2098 (all X_a (real_normed_algebra(X_a) -> (all Y all S_3 all X all R_3 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),X)),R_3)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),Y)),S_3)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),S_3)))))))) # label(fact_2005_norm__mult__less) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2099 (all X_a (monoid_mult(X_a) -> (all Z_1 all W ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W))) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1),hAPP(int,nat,number_number_of(nat),W)))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1),hAPP(int,nat,number_number_of(nat),W))) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,W)))) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W))) -> one_one(X_a) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,W)))))))) # label(fact_3927_power__number__of__odd) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2100 (all A_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),one_one(int)))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B)))) # label(fact_931_norR__mem__unique__aux) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2101 (all X_a (zero_neq_one(X_a) & no_zero_divisors(X_a) & mult_zero(X_a) & power(X_a) -> (all A_3 all Na (ti(X_a,A_3) = zero_zero(X_a) & zero_zero(nat) != Na <-> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_3),Na) = zero_zero(X_a))))) # label(fact_188_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2102 (all X_b all X_a all A_2 all B_1 hAPP(fun(X_b,bool),fun(product_prod(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(product_prod(X_a,X_b),bool)),prod(X_a,X_b),A_2),B_1) = hAPP(fun(product_prod(X_a,X_b),bool),fun(product_prod(X_a,X_b),bool),collect(product_prod(X_a,X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),hAPP(fun(X_b,bool),fun(X_a,fun(X_b,bool)),hAPP(fun(X_a,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(X_a,fun(X_b,bool))),combc(X_a,fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(bool,bool)),fun(X_a,fun(fun(X_b,bool),fun(X_b,bool))),hAPP(fun(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_a,fun(bool,bool)),fun(X_a,fun(fun(X_b,bool),fun(X_b,bool)))),combb(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),X_a),combb(bool,bool,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),A_2)))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),B_1))))) # label(fact_4186_Nitpick_Oprod__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2103 (all X_a (power(X_a) -> power_power(X_a) = ti(fun(X_a,fun(nat,X_a)),power_power(X_a)))) # label(tsy_c_Power_Opower__class_Opower_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2104 (all N all M ((M = zero_zero(nat) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N) = N) & (M != zero_zero(nat) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N))))) # label(fact_1466_add__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2105 (all A_3 all A_2 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Ma)) -> (hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),Ma)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),A_2)) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(fun(int,bool),int,hilbert_Eps(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3)),Ma))),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(fun(int,bool),bool)),fun(fun(int,bool),fun(int,bool)),combc(int,fun(int,bool),bool),member(int)),hAPP(int,fun(int,bool),norRRset,Ma))))),hAPP(int,fun(int,bool),norRRset,Ma))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),hAPP(fun(int,bool),int,hilbert_Eps(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3)),Ma))),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(fun(int,bool),bool)),fun(fun(int,bool),fun(int,bool)),combc(int,fun(int,bool),bool),member(int)),hAPP(int,fun(int,bool),norRRset,Ma))))),Ma)))))) # label(fact_4183_aux__some) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2106 (all B all A_1 all A_6 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),A_6)) -> (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),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_6),B)))))) # label(fact_1102_zdiv__mono1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2107 (all X_a (semiring_div(X_a) -> (all A_1 zero_zero(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),zero_zero(X_a)),A_1)))) # label(fact_1282_mod__0) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2108 (all X_a all Ra all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,bool),accp(X_a),Ra),A_3)) <-> (exists X_1 ((all Xa_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),Ra,Xa_1),X_1)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,bool),accp(X_a),Ra),Xa_1)))) & ti(X_a,X_1) = ti(X_a,A_3))))) # label(fact_2944_accp_Osimps) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2109 (all N all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M)) -> hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,semiring_1_of_nat(int),M)),hAPP(nat,int,semiring_1_of_nat(int),N)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)))) # label(fact_870_zdiff__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2110 (all X_a (linordered_semidom(X_a) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a)))))) # label(fact_114_zero__less__two) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2111 (all T_2 all T_1 (boolean_algebra(T_1) -> boolean_algebra(fun(T_2,T_1)))) # label(arity_fun___Lattices_Oboolean__algebra) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2112 (all X_a (linordered_field(X_a) -> (all A_3 all B_2 all C (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_eq(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)))))))) # label(fact_1891_neg__le__divide__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2113 (all F all Xa all F_2 all Na all Ma ((all R_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),R_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),Na))) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),R_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),F,R_1)),Xa),hAPP(real,real,hAPP(nat,fun(real,real),F_2,R_1),Xa))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(nat,bool),fun(real,real),hAPP(fun(real,fun(fun(nat,bool),real)),fun(fun(nat,bool),fun(real,real)),combc(real,fun(nat,bool),real),hAPP(fun(real,fun(nat,real)),fun(real,fun(fun(nat,bool),real)),hAPP(fun(fun(nat,real),fun(fun(nat,bool),real)),fun(fun(real,fun(nat,real)),fun(real,fun(fun(nat,bool),real))),combb(fun(nat,real),fun(fun(nat,bool),real),real),big_co1399186613setsum(nat,real)),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),F))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Ma),Na))),Xa),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),F_2),Xa)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Ma),Na)))))) # label(fact_4316_DERIV__sumr) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2114 (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_eq(real),hAPP(real,real,ln,X)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),one_one(real)))))) # label(fact_1725_ln__le__minus__one) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2115 (all X_a (group_add(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B))))) # label(fact_1970_diff__minus__eq__add) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2116 (all Na all Ma (ti(int,Ma) != zero_zero(int) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Ma),Na)),Ma)) <-> one_one(int) = hAPP(int,int,abs_abs(int),Na)))) # label(fact_1044_zdvd__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2117 (all M all D_3 all R_3 (R_3 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),D_3) -> (exists Q_4 hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),R_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Q_4),D_3)) = M))) # label(fact_1574_mod__eqD) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2118 (all Xa (hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),Xa),field_char_0_Rats(real))) -> -(all M_2 all N_1 (N_1 != zero_zero(nat) -> (hAPP(real,real,abs_abs(real),Xa) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,real_1(nat),M_2)),hAPP(nat,real,real_1(nat),N_1)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M_2),N_1) != one_one(nat)))))) # label(fact_3697_Rats__abs__nat__div__natE) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2119 (all F (hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),abs_abs(real)),F))) -> hBOOL(hAPP(fun(nat,real),bool,summable(real),F)))) # label(fact_4037_summable__rabs__cancel) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2120 (all X_2 (hBOOL(hAPP(fun(nat,complex),bool,summable(complex),X_2)) -> hAPP(fun(nat,complex),complex,suminf(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(complex,complex),fun(fun(nat,complex),fun(nat,complex)),combb(complex,complex,nat),cnj),X_2)) = hAPP(complex,complex,cnj,hAPP(fun(nat,complex),complex,suminf(complex),X_2)))) # label(fact_4068_cnj_Osuminf) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2121 (all X_a (sgn_div_norm(X_a) -> ti(fun(real,fun(X_a,X_a)),scaleR_scaleR(X_a)) = scaleR_scaleR(X_a))) # label(tsy_c_RealVector_OscaleR__class_OscaleR_0_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2122 (all X_a (real_normed_vector(X_a) -> (all X_2 ((exists N_2 all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,X_2,N_1))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_2))))) <-> hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2)))))) # label(fact_3521_Bseq__iff1a) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2123 (all X_b all X_a (real_normed_vector(X_b) & metric_space(X_a) -> (all G all F all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,real),fun(X_a,fun(X_b,X_b)),hAPP(fun(real,fun(X_b,X_b)),fun(fun(X_a,real),fun(X_a,fun(X_b,X_b))),combb(real,fun(X_b,X_b),X_a),scaleR_scaleR(X_b)),F)),G)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3)))))))) # label(fact_4770_scaleR_OLIM__prod__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2124 (all X_a all P all X (-hBOOL(hAPP(X_a,bool,P,X)) | hBOOL(hAPP(fun(X_a,bool),bool,fEx(X_a),P)))) # label(help_fEx_1_1_U) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2125 (all Y all X (hBOOL(hAPP(int,bool,nat_is_nat,X)) -> (hBOOL(hAPP(int,bool,nat_is_nat,Y)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y)))))) # label(fact_4882_transfer__int__nat__gcd__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2126 (all P_1 all Na ((all M_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),Na)) -> hBOOL(hAPP(nat,bool,P_1,M_2)))) <-> (all X_1 (hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_1),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))) -> hBOOL(hAPP(nat,bool,P_1,X_1)))))) # label(fact_4226_all__nat__less__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2127 (all X_a (linorder(X_a) -> (all A_3 all B_2 all C all D_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),A_3),B_2)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),C),D_1))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),A_3)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),D_1)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),A_3)))))) # label(fact_4230_atLeastLessThan__subset__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2128 (all N all P_5 (hBOOL(hAPP(nat,bool,prime,P_5)) -> one_one(nat) = N | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,P_5),N)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),N)))) # label(fact_3096_prime__coprime) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2129 -(all X_1 all Y_1 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Y_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) != hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))) # label(fact_591__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]. 7.70/7.71 2130 (all K_4 all A_1 all B all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),A_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),B)),M)))) # label(fact_894_zcong__scalar2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2131 (all X_a (real_normed_vector(X_a) -> (all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2)) <-> (exists K_2 ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,X_2,N_1))),K_2))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2)))))))) # label(fact_3519_Bseq__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2132 (all K_4 ((K_4 = zero_zero(quickcheck_code_int) -> zero_zero(int) = hAPP(quickcheck_code_int,int,quickcheck_int_of,K_4)) & (zero_zero(quickcheck_code_int) != K_4 -> (zero_zero(quickcheck_code_int) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),K_4),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) -> hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(quickcheck_code_int,int,quickcheck_int_of,hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),K_4),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(quickcheck_code_int,int,quickcheck_int_of,K_4)) & (hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),K_4),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) != zero_zero(quickcheck_code_int) -> hAPP(quickcheck_code_int,int,quickcheck_int_of,K_4) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(quickcheck_code_int,int,quickcheck_int_of,hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),K_4),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),one_one(int)))))) # label(fact_3182_Quickcheck__Narrowing_Oint__of__code) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2133 (all Sb all Ta all A_3 all B_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A_3),B_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Sb),Ta)) -> hBOOL(hAPP(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),bool,hAPP(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),hAPP(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),hAPP(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A_3),Sb)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B_2),Ta))),pair_leq))))) # label(fact_3513_pair__leqI2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2134 (all X_a (division_ring(X_a) -> (all B all A_1 (ti(X_a,A_1) != zero_zero(X_a) -> (zero_zero(X_a) != ti(X_a,B) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B))),hAPP(X_a,X_a,inverse_inverse(X_a),B)))))))) # label(fact_2631_inverse__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2135 (all X_a (real_normed_vector(X_a) -> hAPP(X_a,real,norm_norm(X_a),zero_zero(X_a)) = zero_zero(real))) # label(fact_1962_norm__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2136 (all M all N all K_4 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N)),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M)),N)))) # label(fact_3455_zgcd__zdvd__zgcd__zmult) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2137 (all X_a (linord1117847801e_zero(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),A_3))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3)))))) # label(fact_2583_inverse__nonnegative__iff__nonnegative) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2138 (all X_a all X_b all X all Y all A_1 (hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X),Y)) = ti(X_a,A_1) -> ti(X_a,X) = ti(X_a,A_1))) # label(fact_3159_fst__eqD) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2139 (all X_a (comm_monoid_mult(X_a) & comm_monoid_add(X_a) -> (all F all A_2 hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co302546470etprod(nat,X_a),F),A_2) = hAPP(fun(int,bool),X_a,hAPP(fun(int,X_a),fun(fun(int,bool),X_a),big_co302546470etprod(int,X_a),hAPP(fun(int,nat),fun(int,X_a),hAPP(fun(nat,X_a),fun(fun(int,nat),fun(int,X_a)),combb(nat,X_a,int),F),nat_1)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2))))) # label(fact_3971_transfer__nat__int__sum__prod_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2140 (all X all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B))) -> -(hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),A_1)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),B))))) # label(fact_3269_gcd__semilattice__nat_Ole__infE) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2141 (all X_a (real_n1866405975lgebra(X_a) -> (all X all Y hAPP(X_a,X_a,sgn_sgn(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,sgn_sgn(X_a),X)),hAPP(X_a,X_a,sgn_sgn(X_a),Y))))) # label(fact_2265_sgn__mult) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2142 (all X_a (order(X_a) -> (all X_2 ((all N_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,X_2,hAPP(nat,nat,suc,N_1))),hAPP(nat,X_a,X_2,N_1)))) -> hBOOL(hAPP(fun(nat,X_a),bool,monoseq(X_a),X_2)))))) # label(fact_4168_mono__SucI2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2143 (all Xa all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (zero_zero(real) = hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa) <-> zero_zero(real) = Xa))) # label(fact_2683_real__root__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2144 (all X_a all Ra (hBOOL(hAPP(fun(product_prod(X_a,X_a),bool),bool,refl(X_a),Ra)) <-> (all X_1 hBOOL(hAPP(fun(product_prod(X_a,X_a),bool),bool,hAPP(product_prod(X_a,X_a),fun(fun(product_prod(X_a,X_a),bool),bool),member(product_prod(X_a,X_a)),hAPP(X_a,product_prod(X_a,X_a),hAPP(X_a,fun(X_a,product_prod(X_a,X_a)),product_Pair(X_a,X_a),X_1),X_1)),Ra))))) # label(fact_3579_Nitpick_Orefl_H__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2145 (all Ma all Xa hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,int),fun(fun(int,bool),fun(int,bool)),image(int,int),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),times_times(int)),Xa)),hAPP(int,fun(int,bool),norRRset,Ma)) = hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),noXRRset,Ma),Xa)) # label(fact_3908_noXRRset__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2146 (all X all Y all M (one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),X),Y)),M) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),M) = one_one(nat))) # label(fact_3317_invertible__coprime__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2147 (all X_a (linord1117847801e_zero(X_a) -> (all Xa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Xa)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),one_one(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),Xa))))))) # label(fact_2615_one__less__inverse__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2148 (all X_a all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) -> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))))) # label(fact_2741_rev__finite__subset) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2149 (all X hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),zero_zero(nat)),X) = X) # label(fact_3279_gcd__0__left__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2150 (all X_a all P_1 hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),P_1)) = hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(bool,bool),fun(fun(X_a,bool),fun(X_a,bool)),combb(bool,bool,X_a),fNot),P_1))) # label(fact_3890_Collect__neg__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2151 (all X hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,cos,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,sin,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X))) # label(fact_2305_cos__double) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2152 (all X_a (linorder(X_a) -> (all X all Y hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),X))))) # label(fact_4843_min__max_Oinf__le1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2153 (all X_a (linorder(X_a) -> (all B all A_1 all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),X)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)),X)))))) # label(fact_4826_min__max_Oless__infI1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2154 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) -> hAPP(real,real,arctan,hAPP(real,real,tan,X)) = X))) # label(fact_2453_arctan__tan) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2155 (all Ma all Na hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),Na) = hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),hAPP(nat,int,semiring_1_of_nat(int),Ma)),hAPP(nat,int,semiring_1_of_nat(int),Na)))) # label(fact_4567_SetInterval_Otransfer__nat__int__set__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2156 (all X_a (archim1727834104eiling(X_a) -> (all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(X_a,int,archimedean_ceiling(X_a),Xa))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),Xa)))))) # label(fact_3326_one__less__ceiling) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2157 (all L all U hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),L),U)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),U),L))) # label(fact_4286_card__atLeastLessThan__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2158 (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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),K))))) # label(fact_824_dvd__reduce) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2159 (all X_a (linordered_idom(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),A_3)),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a))))))) # label(fact_44_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2160 (all N -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,N)),N))) # label(fact_1339_Suc__n__not__le__n) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2161 (all X_a (real_vector(X_a) -> (all A_1 all X all Y hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),Y))))) # label(fact_4700_scaleR__right_Oadd) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2162 (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)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y)))))) # label(fact_633_real__mult__order) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2163 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B all C_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B),C_1)))))) # label(fact_722_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2164 (all X all Y (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),Y)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y),X)) | ti(int,Y) = ti(int,X))) # label(fact_15_zless__linear) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2165 (all X_a (real_normed_field(X_a) -> (all Y_2 all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2)) -> (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),Y_2)) -> hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),X_2)),Y_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),X_2)),hAPP(fun(nat,X_a),X_a,suminf(X_a),Y_2))))))) # label(fact_4010_suminf__diff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2166 (all Z_1 (-hBOOL(hAPP(int,bool,nat_neg,Z_1)) -> hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,nat_1,Z_1)) = ti(int,Z_1))) # label(fact_1611_not__neg__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2167 (all X_a (semiring_div(X_a) -> (all B all A_1 all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B))),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)))) # label(fact_1413_div__mod__equality2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2168 (all X_b all X_a (order(X_a) & order(X_b) -> (all C all F all B_2 all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),A_3)) -> (hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),C),hAPP(X_a,X_b,F,B_2))) -> ((all X_1 all Y_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_1),X_1)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,Y_1)),hAPP(X_a,X_b,F,X_1))))) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),C),hAPP(X_a,X_b,F,A_3))))))))) # label(fact_2862_xt5) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2169 (all X_a (linord1117847801e_zero(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_3)),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a))))))) # label(fact_2593_inverse__negative__iff__negative) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2170 (all X_a all N hAPP(int,X_a,number_of_frac(X_a),N) = hAPP(product_prod(int,int),X_a,abs_Frac(X_a),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),N),one_one(int)))) # label(fact_3820_Nitpick_Onumber__of__frac__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2171 (all X_a (mult_zero(X_a) -> (all A_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),zero_zero(X_a)),A_1) = zero_zero(X_a)))) # label(fact_373_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2172 (all I_1 all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,nat_1,I_1)),N) = hAPP(nat,nat,hAPP(int,fun(nat,nat),nat_aux,I_1),N)) # label(fact_1069_nat__aux__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2173 (all X_a (divisi14063676e_zero(X_a) -> (all A_1 ((zero_zero(X_a) != ti(X_a,A_1) -> one_one(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),A_1)) & (ti(X_a,A_1) = zero_zero(X_a) -> zero_zero(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),A_1)))))) # label(fact_1857_divide__self__if) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2174 (all Z_1 all W hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z_1),W) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),W),Z_1)) # label(fact_663_real__mult__commute) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2175 (all Ma all Na (hAPP(nat,nat,suc,zero_zero(nat)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ma),Na) <-> hAPP(nat,nat,suc,zero_zero(nat)) = Ma & hAPP(nat,nat,suc,zero_zero(nat)) = Na)) # label(fact_1381_mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2176 (all X_a (dense_linorder(X_a) -> (all A_3 all B_2 all C all D_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_gr676655987ssThan(X_a),A_3),B_2)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),C),D_1))) <-> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),D_1)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),A_3))))))) # label(fact_4261_greaterThanLessThan__subseteq__atLeastLessThan__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2177 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),X)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X))),X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) # label(fact_1742_abs__ln__one__plus__x__minus__x__bound) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2178 (all X_a (comm_monoid_add(X_a) -> (all F all K all Na hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),K))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(fun(nat,bool),X_a),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combb(fun(nat,bool),X_a,nat),hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F)),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,fun(nat,bool))),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combs(nat,nat,fun(nat,bool)),hAPP(fun(nat,nat),fun(nat,fun(nat,fun(nat,bool))),hAPP(fun(nat,fun(nat,fun(nat,bool))),fun(fun(nat,nat),fun(nat,fun(nat,fun(nat,bool)))),combb(nat,fun(nat,fun(nat,bool)),nat),ord_atLeastLessThan(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),K))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),K))),K)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))))) # label(fact_4198_sumr__group) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2179 (all X_a (linord1117847801e_zero(X_a) -> (all A_1 hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)) = hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1))))) # label(fact_2567_abs__inverse) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2180 (all X_a all P_1 all Xa (hBOOL(hAPP(X_a,bool,P_1,Xa)) -> hBOOL(hAPP(X_a,bool,P_1,hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1))))) # label(fact_4808_someI) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2181 (all X_a (field_inverse_zero(X_a) -> (all Z_1 all X all Y (zero_zero(X_a) != ti(X_a,Y) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),Y))),Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)))))) # label(fact_1881_add__num__frac) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2182 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),hAPP(nat,nat,suc,Na))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) | hAPP(nat,nat,suc,Na) = Ma)) # label(fact_1341_le__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2183 (all Ma all Na all Qr (hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,Ma),Na),Qr)) <-> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(product_prod(nat,nat),nat,product_fst(nat,nat),Qr)),Na)),hAPP(product_prod(nat,nat),nat,product_snd(nat,nat),Qr)) = Ma & (zero_zero(nat) != Na -> (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(product_prod(nat,nat),nat,product_snd(nat,nat),Qr)),zero_zero(nat))) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),hAPP(product_prod(nat,nat),nat,product_snd(nat,nat),Qr)))) & (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_eq(nat),zero_zero(nat)),hAPP(product_prod(nat,nat),nat,product_snd(nat,nat),Qr))) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(product_prod(nat,nat),nat,product_snd(nat,nat),Qr)),Na)))) & (Na = zero_zero(nat) -> zero_zero(nat) = hAPP(product_prod(nat,nat),nat,product_fst(nat,nat),Qr)))) # label(fact_3152_divmod__nat__rel__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2184 (all X_2 all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),X_2),A_3),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(complex,complex),fun(fun(nat,complex),fun(nat,complex)),combb(complex,complex,nat),cnj),X_2)),hAPP(complex,complex,cnj,A_3)),sequentially)))) # label(fact_4532_cnj_OLIMSEQ) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2185 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real))) -> hAPP(real,real,cos,hAPP(real,real,arccos,Y)) = Y))) # label(fact_2355_cos__arccos) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2186 (all Xa (hAPP(real,real,exp(real),Xa) = one_one(real) <-> zero_zero(real) = Xa)) # label(fact_1771_exp__eq__one__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2187 (all R_3 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),R_3),one_one(real))),hAPP(int,real,real_1(int),hAPP(real,int,archim2085082626_floor(real),R_3))))) # label(fact_3425_real__of__int__floor__ge__diff__one) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2188 (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_950_dvd__antisym) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2189 (all X_a (comm_monoid_add(X_a) -> (all F all Na all K hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),K))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K)))))) # label(fact_4199_sumr__offset3) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2190 (all X_a (banach(X_a) -> (all F ((all E (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E)) -> (exists N_2 all M_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),M_2)) -> (all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),M_2),N_1)))),E))))))) <-> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)))))) # label(fact_4314_summable__Cauchy) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2191 (all Na hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na) = hAPP(nat,fun(nat,bool),ord_lessThan(nat),Na)) # label(fact_4345_atLeast0LessThan) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2192 (all C_1 all A_1 all B all Q_3 all R_3 (hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,A_1),B),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Q_3),R_3))) -> (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)),C_1)) -> hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C_1)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),Q_3),C_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Q_3),C_1))),R_3)))))))) # label(fact_3120_divmod__nat__rel__mult2__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2193 (all X_a (archim1804426504_field(X_a) -> (all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X)) -> (exists N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1))),X)))))))) # label(fact_2644_ex__inverse__of__nat__less) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2194 (all X_a (real_normed_vector(X_a) -> (all Xa all X_2 (hBOOL(hAPP(fun(nat,real),bool,summable(real),X_2)) -> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),X_2)),Xa))))))) # label(fact_4742_scaleR__left_Osummable) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2195 (all X_a (linord1117847801e_zero(X_a) -> (all A_3 all B_2 ((hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_2)),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2))) & (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_2))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),A_3))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_3)),hAPP(X_a,X_a,inverse_inverse(X_a),B_2))))))) # label(fact_2627_inverse__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2196 (all X_a (real_field(X_a) -> (all A_1 all B all C_1 all D_3 all H_2 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),D_3))),H_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),D_3)),H_2))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),C_1)),H_2)),D_3))))) # label(fact_1780_DERIV__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2197 (all I_1 all K_4 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),J_2)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),J_2)),K_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),K_4)))) # label(fact_852_diff__add__assoc) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2198 (all M hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),hAPP(nat,nat,suc,zero_zero(nat))) = hAPP(nat,nat,suc,zero_zero(nat))) # label(fact_3283_gcd__Suc__0) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2199 (all X_a (comm_monoid_add(X_a) & topological_space(X_a) -> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(X_a,fun(nat,X_a),combk(X_a,nat),zero_zero(X_a))),zero_zero(X_a))))) # label(fact_4105_sums__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2200 (all W (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W))) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(int,int,number_number_of(int),W)))) -> hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,hAPP(int,int,number_number_of(int),W)),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,one_one(int)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,number_number_of(int),W)))) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,one_one(int)),hAPP(int,int,number_number_of(int),W))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(int,int,number_number_of(int),W)))) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,one_one(int)),hAPP(int,int,number_number_of(int),W)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,number_number_of(int),min)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(int,int,number_number_of(int),W)))))) # label(fact_1585_negDivAlg__eqn__1__number__of) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2201 (all A_1 all B (zero_zero(int) != hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B) -> (exists A_5 exists B_3 (ti(int,B) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_3),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B)) & hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_5),B_3) = one_one(int) & hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_5),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B)) = ti(int,A_1))))) # label(fact_4802_gcd__coprime__exists__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2202 (all K_4 all M all N hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K_4),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N)) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K_4),M)),N)) # label(fact_3467_zgcd__assoc) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2203 (all Xa hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),Xa))))) # label(fact_3896_bdd__nat__set__l__finite) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2204 (all N_3 (hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),N_3)) <-> (exists M_2 all X_1 (hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_1),N_3)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_1),M_2)))))) # label(fact_3692_finite__nat__set__iff__bounded) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2205 (all X_a (metric_space(X_a) -> ti(fun(fun(X_a,bool),X_a),the(X_a)) = the(X_a))) # label(tsy_c_HOL_OThe_0_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2206 (all X all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),zero_zero(real))) -> (exists R_1 exists A_4 (Y = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_1),hAPP(real,real,sin,A_4)) & X = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_1),hAPP(real,real,cos,A_4)))))) # label(fact_2559_polar__ex2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2207 (all X_a (topological_space(X_a) & comm_monoid_add(X_a) -> suminf(X_a) = ti(fun(fun(nat,X_a),X_a),suminf(X_a)))) # label(tsy_c_Series_Osuminf_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2208 (all P_4 hAPP(product_prod(int,int),product_prod(int,int),hAPP(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_prod_case(int,int,product_prod(int,int)),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int)),abs_abs(int))),hAPP(rat,product_prod(int,int),quotient_of,P_4)) = hAPP(rat,product_prod(int,int),quotient_of,hAPP(rat,rat,abs_abs(rat),P_4))) # label(fact_4057_rat__abs__code) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2209 (all Y all X (hAPP(real,real,exp(real),Y) = X -> hAPP(real,real,ln,X) = Y)) # label(fact_1761_ln__unique) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2210 (all X all Y hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),X)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Y),Y)))))) # label(fact_2159_real__sqrt__mult__self__sum__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2211 (all X_a (ring_1(X_a) -> (all W all Z_1 hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),W),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),W)),hAPP(int,X_a,ring_1_of_int(X_a),Z_1))))) # label(fact_3586_of__int__diff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2212 (all X_a (metric_space(X_a) -> (all Xa all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Xa),Ya))) <-> ti(X_a,Ya) != ti(X_a,Xa))))) # label(fact_3538_dist__nz) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2213 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,arctan,Y))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,arctan,Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) # label(fact_2227_arctan__bounded) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2214 (all M all K_4 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K_4)),N)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),N)))) # label(fact_417_add__leD2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2215 (all P_1 all I hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),combs(nat,bool,bool),hAPP(fun(nat,bool),fun(nat,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj),P_1)),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),I)))))) # label(fact_3901_finite__M__bounded__by__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2216 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,arcsin,Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,arcsin,Y)))))) # label(fact_2230_arcsin__bounded) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2217 (all X all N all P_5 (hBOOL(hAPP(nat,bool,prime,P_5)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),N))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),N)))))) # label(fact_3082_prime__divexp__n) # label(axiom) # label(non_clause). [assumption]. 7.70/7.71 2218 (all Z_4 all Z_1 all W_2 all W (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_2),W)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_4),Z_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W_2),Z_4)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W),Z_1)))))) # label(fact_263_zadd__zless__mono) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2219 (all Xa all A_2 (hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),A_2)) -> hBOOL(hAPP(int,bool,nat_is_nat,Xa))))) # label(fact_4868_Nat__Transfer_Otransfer__int__nat__set__function__closures_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2220 (all A_9 all B_8 all Q_6 all Y (hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_8),Q_6)),Y) = ti(int,A_9) -> ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_8)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Y),zero_zero(int))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_8),Y))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_8)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y),B_8)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))) -> (ti(int,B_8) != zero_zero(int) -> ti(int,Y) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_9),B_8))))) # label(fact_1478_divmod__int__rel__mod__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2221 (all X_a all X_b (real_n2089651433ebra_1(X_a) & metric_space(X_b) -> (all M_1 all F all L_2 all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),M_1),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> ti(X_b,M_1) = ti(X_b,L_2)))))) # label(fact_4595_LIM__unique) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2222 (all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(complex,real,im,X))),hAPP(complex,real,norm_norm(complex),X)))) # label(fact_3008_abs__Im__le__cmod) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2223 (all X all Y hAPP(complex,real,norm_norm(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,X),Y)) = hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) # label(fact_2755_complex__norm) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2224 (all Na all Ma (zero_zero(nat) = Na & zero_zero(nat) = Ma <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),Na)),hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),Ma)))))) # label(fact_2084_int__zle__neg) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2225 (all B_2 all A_3 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),zEven)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),zOdd)) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_3),B_2)),zOdd))))) # label(fact_1711_IntNatAux_Oeven__plus__odd) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2226 (all X_a (linorder(X_a) -> (all Y all Z_1 all X hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Z_1),X)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z_1)),X)))) # label(fact_4978_min__max_Osup__inf__distrib2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2227 (all X_a (linordered_semidom(X_a) -> (all K all F all K_3 all Na ((all P_6 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),P_6),Na)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,F,P_6)),K_3)))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),K_3)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),K)))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)),K_3)))))))) # label(fact_4300_real__setsum__nat__ivl__bounded2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2228 (all Xa (-hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa)) <-> (exists Y_1 hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))),Y_1)) = Xa))) # label(fact_2497_odd__nat__equiv__def2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2229 (all X_a (semiring_div(X_a) -> (all C all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_3),B_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_3),C)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B_2),C)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),B_2),A_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),C),A_3))))))))) # label(fact_1076_div__dvd__div) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2230 (all X_a (real_normed_field(X_a) -> (all Xa hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),combi(X_a)),Xa),one_one(X_a)))))) # label(fact_3871_DERIV__ident) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2231 (all M all K_4 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)),K_4)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),M)) -> (K_4 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),K_4))))))) # label(fact_1442_prod__mn__less__k) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2232 (all X_b all X_a image(X_b,X_a) = ti(fun(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool))),image(X_b,X_a))) # label(tsy_c_Set_Oimage_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2233 (all X_a (number_ring(X_a) & field(X_a) -> hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls))) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls)))) # label(fact_2621_inverse__numeral__1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2234 (all N all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),N)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,real_1(nat),N)),one_one(real)))) -> N = hAPP(real,nat,natfloor,X)))) # label(fact_2428_natfloor__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2235 (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,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_314_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2236 (all X_a (real_n2089651433ebra_1(X_a) -> (all N hAPP(nat,real,semiring_1_of_nat(real),N) = hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N))))) # label(fact_1938_norm__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2237 (all M all N all P_5 (hBOOL(hAPP(nat,bool,prime,P_5)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),M)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),N))))) # label(fact_3079_prime__dvd__mult) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2238 (all J_2 all I_1 -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),I_1)),I_1))) # label(fact_122_not__add__less2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2239 (all X_a all X_b (comm_monoid_add(X_b) & ordere779506340up_add(X_b) -> (all F all G all K_3 ((all I_2 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I_2),K_3)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,I_2)),hAPP(X_a,X_b,G,I_2))))) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),K_3)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),K_3))))))) # label(fact_4370_setsum__mono) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2240 (all X all A_1 all B hAPP(real,real,hAPP(real,fun(real,real),powr,hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),B) = hAPP(real,real,hAPP(real,fun(real,real),powr,X),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_1),B))) # label(fact_3355_powr__powr) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2241 (all X_a (linordered_field(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)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1))) -> (ti(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)),A_1))))))) # label(fact_2591_inverse__positive__imp__positive) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2242 (all X_b all X_a all X_c all F all Xa hAPP(product_prod(X_b,X_a),X_a,product_snd(X_b,X_a),hAPP(product_prod(X_b,X_c),product_prod(X_b,X_a),hAPP(fun(X_c,X_a),fun(product_prod(X_b,X_c),product_prod(X_b,X_a)),product_apsnd(X_c,X_a,X_b),F),Xa)) = hAPP(X_c,X_a,F,hAPP(product_prod(X_b,X_c),X_c,product_snd(X_b,X_c),Xa))) # label(fact_4092_snd__apsnd) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2243 (all X_a (ring_1(X_a) -> (all Q_2 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Q_2),ring_1_Ints(X_a))) -> -(all Z_2 ti(X_a,Q_2) != hAPP(int,X_a,ring_1_of_int(X_a),Z_2)))))) # label(fact_3839_Ints__cases) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2244 (all X_a (finite_finite(X_a) -> (all A_2 hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))))) # label(fact_2738_finite) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2245 (all X_a (linord581940658strict(X_a) -> (all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),zero_zero(X_a))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),zero_zero(X_a))) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_2)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_2))))))) # label(fact_424_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2246 (all M M = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),zero_zero(nat))) # label(fact_806_minus__nat_Odiff__0) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2247 (all G all F all Xa all D_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_1)) -> ((all Z_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),Xa))),D_1)) -> hAPP(real,real,G,hAPP(real,real,F,Z_2)) = Z_2)) -> ((all Z_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),Xa))),D_1)) -> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),F),Z_2)))) -> (exists E (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E)) & (all Y_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Y_1),hAPP(real,real,F,Xa))))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Y_1),hAPP(real,real,F,Xa)))),E)) -> hAPP(real,real,F,hAPP(real,real,G,Y_1)) = Y_1)))))))) # label(fact_4954_isCont__inv__fun__inv) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2248 (all X_a (real_normed_vector(X_a) -> (all A_1 all B hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(X_a,real,norm_norm(X_a),A_1)),hAPP(X_a,real,norm_norm(X_a),B))),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B))))))) # label(fact_2012_norm__diff__ineq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2249 (all Code_numeral all Code_numeral_5 (Code_numeral_5 = Code_numeral <-> hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,Code_numeral) = hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,Code_numeral_5))) # label(fact_2961_code__numeral_Oinject) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2250 (all Xa all Ya (-hBOOL(hAPP(int,bool,even_odd_even(int),Ya)) & -hBOOL(hAPP(int,bool,even_odd_even(int),Xa)) | hBOOL(hAPP(int,bool,even_odd_even(int),Ya)) & hBOOL(hAPP(int,bool,even_odd_even(int),Xa)) <-> hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xa),Ya))))) # label(fact_1598_even__sum) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2251 (all W hAPP(int,int,ring_1_of_int(int),W) = hAPP(int,int,number_number_of(int),W)) # label(fact_3593_int__number__of__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2252 (all Xa hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),Xa))) = Xa) # label(fact_3893_card__bdd__nat__set__l) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2253 (all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_1),B)) -> (one_one(nat) != B -> (exists X_1 exists Y_1 hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),Y_1)),one_one(nat)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),X_1))))) # label(fact_2922_coprime__bezout__strong) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2254 (all X_a (field(X_a) -> (all A_1 (ti(X_a,A_1) != zero_zero(X_a) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),A_1) = one_one(X_a))))) # label(fact_2617_field__inverse) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2255 (all X_a (ordere216010020id_add(X_a) -> (all B all C_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),B),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)))))))) # label(fact_244_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2256 (all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) <-> (exists M_2 hAPP(nat,nat,suc,M_2) = Na))) # label(fact_1376_gr0__conv__Suc) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2257 (all A_3 all B_2 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_3),B_2)),zOdd)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),zOdd)) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),zEven))))) # label(fact_1710_even__plus__odd__prop1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2258 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),zero_zero(X_a)))))))) # label(fact_200_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2259 (all M hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),one_one(int)) = one_one(int)) # label(fact_3458_zgcd__1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2260 (all X_a (linordered_idom(X_a) -> (all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),zero_zero(int))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)),zero_zero(X_a))))))) # label(fact_3628_of__int__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2261 (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_264_zero__zle__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2262 (all X_a (ring_char_0(X_a) -> (all Wa all Z (ti(int,Wa) = ti(int,Z) <-> hAPP(int,X_a,ring_1_of_int(X_a),Z) = hAPP(int,X_a,ring_1_of_int(X_a),Wa))))) # label(fact_3600_of__int__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2263 (all X_a (semiring_div(X_a) -> (all C all B_2 all A_3 (zero_zero(X_a) != ti(X_a,A_3) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_3),B_2)) -> (ti(X_a,C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),B_2),A_3) <-> ti(X_a,B_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_3))))))) # label(fact_1119_dvd__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2264 (all X all Y hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,re,X)),hAPP(complex,real,im,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,im,X)),hAPP(complex,real,re,Y))) = hAPP(complex,real,im,hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),X),Y))) # label(fact_3035_complex__Im__mult) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2265 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)))))))) # label(fact_246_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2266 (all X_a (real_normed_vector(X_a) -> (all Xa all X_2 (hBOOL(hAPP(fun(nat,real),bool,summable(real),X_2)) -> hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),X_2)),Xa)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(fun(nat,real),real,suminf(real),X_2)),Xa))))) # label(fact_4766_scaleR__left_Osuminf) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2267 (all Xa all Na (Xa = one_one(nat) | Na = zero_zero(nat) <-> hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Xa),Na) = one_one(nat))) # label(fact_195_exp__eq__1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2268 (all A_3 all B_2 all A_2 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Ma)) -> (hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),Ma)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),Ma)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),A_2)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),A_2)) -> ti(int,A_3) = ti(int,B_2))))))) # label(fact_3646_RRset__zcong__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2269 (all X_a (linordered_field(X_a) -> (all B_2 all A_3 all C (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)),B_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)),A_3))))))) # label(fact_1874_neg__divide__less__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2270 (all A_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B)),B)))) # label(fact_4424_gcd__le2__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2271 (all A_1 all B ti(int,A_1) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B))) # label(fact_1431_zmod__zdiv__equality) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2272 (all Xa (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> hAPP(int,int,number_number_of(int),min) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),hAPP(int,nat,nat_1,Xa))))) # label(fact_1229_EvenOdd_Oneg__one__odd__power) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2273 (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),hAPP(real,real,arctan,X)),hAPP(real,real,arctan,Y))))) # label(fact_2133_arctan__monotone_H) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2274 (all X all A_1 all B (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),B))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),X)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_1),B))))) # label(fact_3377_powr__less__cancel) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2275 (all M all N (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) -> M = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)))) # label(fact_838_add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2276 (all X_a all X_b (topological_space(X_b) -> ti(fun(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool))),tendsto(X_a,X_b)) = tendsto(X_a,X_b))) # label(tsy_c_Limits_Otendsto_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2277 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) -> hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,inverse_inverse(real),hAPP(real,real,sqrt,X))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(real,real,inverse_inverse(real),X))) # label(fact_2640_real__inv__sqrt__pow2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2278 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),Y)),one_one(real))) -> hAPP(real,real,cos,hAPP(real,real,arccos,Y)) = Y)) # label(fact_2353_cos__arccos__abs) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2279 (all X_2 all A_3 (hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),X_2),A_3)) -> hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),re),X_2)),hAPP(complex,real,re,A_3))))) # label(fact_4141_Re_Osums) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2280 (all N (N != one_one(nat) -> (exists P_6 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_6),N)) & hBOOL(hAPP(nat,bool,prime,P_6)))))) # label(fact_3113_prime__factor) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2281 (all X_a (linord20386208strict(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(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_1),B))))))) # label(fact_476_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2282 (all F all G ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,F,N_1)),hAPP(nat,real,G,N_1)))) -> (hBOOL(hAPP(fun(nat,real),bool,summable(real),F)) -> (hBOOL(hAPP(fun(nat,real),bool,summable(real),G)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,real),real,suminf(real),F)),hAPP(fun(nat,real),real,suminf(real),G))))))) # label(fact_4156_summable__le) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2283 (all X_a (number(X_a) & linord1117847801e_zero(X_a) -> (all B_2 all Wa all A_3 ((hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Wa))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),hAPP(int,X_a,number_number_of(X_a),Wa))))) & (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Wa))) -> (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3))) & (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),hAPP(int,X_a,number_number_of(X_a),Wa))),B_2)))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),hAPP(int,X_a,number_number_of(X_a),Wa))),A_3)))))) # label(fact_1801_divide__le__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2284 (all Ma all K all Na (hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),K),Na) = one_one(nat) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ma),Na))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),Ma))))) # label(fact_3314_coprime__dvd__mult__iff__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2285 (all X_a (linorder(X_a) -> (all Z_1 all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Z_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z_1)))))))) # label(fact_4852_min__max_Oinf__greatest) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2286 (all N hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),pi)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),N)) # label(fact_2433_cos__npi) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2287 (all K_4 all L_4 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit0,K_4)),hAPP(int,int,bit1,L_4)) = hAPP(int,int,bit1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),L_4))) # label(fact_94_add__Bit0__Bit1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2288 (all B_2 all A_3 all P_4 (hBOOL(hAPP(int,bool,zprime,P_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),P_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int)))) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,A_3),P_4))) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),inv,P_4),B_2)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,A_3),P_4)))))))) # label(fact_1198_wset__mem__inv__mem) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2289 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),B)))))) # label(fact_1653_posDivAlg__correct) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2290 (all X_a (archim1727834104eiling(X_a) -> (all Xa all Z (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)),one_one(X_a)))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archimedean_ceiling(X_a),Xa)),Z)))))) # label(fact_3636_ceiling__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2291 (all X_a (real_normed_field(X_a) -> (all X_2 all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),X_2),A_3)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),X_2)),hAPP(X_a,X_a,uminus_uminus(X_a),A_3))))))) # label(fact_4106_sums__minus) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2292 (all Z_1 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,uminus_uminus(int),Z_1)),Z_1) = zero_zero(int)) # label(fact_2073_zadd__zminus__inverse2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2293 (all X_a (linorder(X_a) -> (all X all Y all Z_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),Z_1))))) # label(fact_4977_min__max_Oinf__sup__distrib1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2294 (all X_a (real_algebra_1(X_a) & real_normed_vector(X_a) -> (exists K_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2)) & (all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(real,X_a,of_real(X_a),X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),X_1)),K_2)))))))) # label(fact_2825_of__real_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2295 (all X_2 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),resSet,Ma),X_2)) -> hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),X_2))))) # label(fact_2940_ResSet__finite) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2296 (all N (-hBOOL(hAPP(nat,bool,even_odd_even(nat),N)) -> (exists X_1 hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),X_1)),one_one(nat)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) # label(fact_2532_odd__square) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2297 (all K_4 hAPP(int,int,succ,K_4) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K_4),min)) # label(fact_770_diff__bin__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2298 (all X_a (linorder(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),B),A_1)))) # label(fact_4831_min__max_Oinf_Ocommute) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2299 (all X_a (order(X_a) -> (all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) -> (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)))))) # label(fact_502_xt1_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2300 (all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,cos,X)),one_one(real)))) # label(fact_2274_cos__le__one) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2301 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))) -> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B))))))) # label(fact_3145_mod__pos__neg) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2302 (all X_a (real_normed_field(X_a) -> (all G all E_3 all F all Xa all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),Xa),E_3)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),times_times(X_a)),F)),G)),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,F,Xa)),E_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),D),hAPP(X_a,X_a,G,Xa)))))))))) # label(fact_3857_DERIV__mult_H) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2303 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),hAPP(int,int,bit1,K))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),K)))) # label(fact_680_rel__simps_I26_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2304 (all X_a (comm_semiring_1(X_a) -> (all X all Q_3 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),Q_3)),X) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,suc,Q_3))))) # label(fact_1369_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2305 (all X_a (linordered_field(X_a) -> (all A_3 all B_2 all 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_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)),B_2))))))) # label(fact_1823_pos__le__divide__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2306 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ly)),Rx) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Rx)),Ly)))) # label(fact_363_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2307 (all Wa all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Wa),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z),one_one(int)))))) # label(fact_779_zle__diff1__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2308 (all X_a (real_normed_field(X_a) -> (all G all E_3 all F all Xa all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),Xa),E_3)) -> (hAPP(X_a,X_a,G,Xa) != zero_zero(X_a) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),inverse_divide(X_a)),F)),G)),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),D),hAPP(X_a,X_a,G,Xa))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,F,Xa)),E_3))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,G,Xa)),hAPP(X_a,X_a,G,Xa))))))))))) # label(fact_3846_DERIV__divide) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2309 (all X hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,sin,X))),hAPP(real,real,cos,X)) = hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X))) # label(fact_2278_sin__double) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2310 (all X all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) -> X = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y))) # label(fact_3261_gcd__proj1__if__dvd__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2311 (all X_a (ab_sem1668676832m_mult(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),A_1)))) # label(fact_2977_times_Oidem) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2312 (all A_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)),zero_zero(int))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B))))) # label(fact_1422_neg__mod__conj) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2313 (all Z_1 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_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z_1),Y)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Z_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z_1),X))))) # label(fact_946_dvd_Ole__less__trans) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2314 (all Na all Ma (hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),Na)) = hAPP(nat,int,semiring_1_of_nat(int),Ma) <-> Na = zero_zero(nat) & Ma = zero_zero(nat))) # label(fact_2074_negative__eq__positive) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2315 (all X_a (archim1727834104eiling(X_a) -> (all X all V hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),hAPP(int,int,number_number_of(int),V)) = hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),hAPP(int,X_a,number_number_of(X_a),V)))))) # label(fact_2916_floor__diff__number__of) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2316 (all X_a (real_normed_algebra(X_a) -> (all Xa_2 all R_3 all X hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_2),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),X)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_2),X))))) # label(fact_4720_mult__right_OscaleR) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2317 (all Na (hBOOL(hAPP(nat,bool,even_odd_even(nat),Na)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Na)))) # label(fact_2521_even__dvd) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2318 (all Na all Ma (hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),Ma) = one_one(nat) <-> one_one(nat) = Na & Ma = one_one(nat))) # label(fact_618_nat__mult__eq__one) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2319 (all Na (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Na)) -> hAPP(int,int,fact_fact(int),Na) = hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),Na)))) # label(fact_4639_fact__altdef__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2320 (all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),P_5)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int)))),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))))))) # label(fact_1267_inv__inv__aux) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2321 (all X_a (real_normed_vector(X_a) -> (all Xa all A_3 hBOOL(hAPP(filter(real),bool,hAPP(X_a,fun(filter(real),bool),hAPP(fun(real,X_a),fun(X_a,fun(filter(real),bool)),tendsto(real,X_a),hAPP(X_a,fun(real,X_a),hAPP(fun(real,fun(X_a,X_a)),fun(X_a,fun(real,X_a)),combc(real,X_a,X_a),scaleR_scaleR(X_a)),Xa)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_3),Xa)),hAPP(real,filter(real),at(real),A_3)))))) # label(fact_4758_scaleR__left_Ocont) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2322 (all X_a (linord1117847801e_zero(X_a) -> (all A_3 all B_2 ((hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_2))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),A_3))) & (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_2)),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_3)),hAPP(X_a,X_a,inverse_inverse(X_a),B_2))))))) # label(fact_2626_inverse__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2323 (all A_3 all B_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2)) -> (hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_3))))) # label(fact_3812_zero__le__Fract__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2324 (all X_a (metric_space(X_a) -> (all X_2 all L_2 all A_3 ((all S_2 ((all N_1 A_3 != hAPP(nat,real,S_2,N_1)) & hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),S_2),A_3),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,real),fun(nat,X_a),hAPP(fun(real,X_a),fun(fun(nat,real),fun(nat,X_a)),combb(real,X_a,nat),X_2),S_2)),L_2),sequentially)))) -> hBOOL(hAPP(filter(real),bool,hAPP(X_a,fun(filter(real),bool),hAPP(fun(real,X_a),fun(X_a,fun(filter(real),bool)),tendsto(real,X_a),X_2),L_2),hAPP(real,filter(real),at(real),A_3))))))) # label(fact_4686_LIMSEQ__SEQ__conv2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2325 (all Na all F (hBOOL(hAPP(fun(nat,real),bool,summable(real),F)) -> ((all M_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),M_2)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,F,M_2))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(fun(nat,real),real,suminf(real),F)))))) # label(fact_4301_series__pos__less) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2326 (all X_a all X_b (comm_monoid_add(X_b) -> (all F all S (hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,finite_finite_1(fun(X_a,bool)),S)) -> ((all X_1 (hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),X_1),S)) -> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),X_1)))) -> ((all T1 (hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),T1),S)) -> (all T2 (hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),T2),S)) -> (ti(fun(X_a,bool),T1) != ti(fun(X_a,bool),T2) -> (all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),T1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),T2)) -> hAPP(X_a,X_b,F,X_1) = zero_zero(X_b))))))))) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),S)) = hAPP(fun(fun(X_a,bool),bool),X_b,hAPP(fun(fun(X_a,bool),X_b),fun(fun(fun(X_a,bool),bool),X_b),big_co1399186613setsum(fun(X_a,bool),X_b),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F)),S))))))) # label(fact_4337_setsum__UNION__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2327 (all Z_1 all W hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z_1),W)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,uminus_uminus(int),Z_1)),hAPP(int,int,uminus_uminus(int),W))) # label(fact_2062_zminus__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2328 (all X_a all F all A_2 hAPP(fun(X_a,bool),real,hAPP(fun(X_a,real),fun(fun(X_a,bool),real),big_co1399186613setsum(X_a,real),hAPP(fun(X_a,nat),fun(X_a,real),hAPP(fun(nat,real),fun(fun(X_a,nat),fun(X_a,real)),combb(nat,real,X_a),real_1(nat)),F)),A_2) = hAPP(nat,real,real_1(nat),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),A_2))) # label(fact_4271_real__of__nat__setsum) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2329 (all X_a (linorder(X_a) -> (all Y all Z_1 all X hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),Z_1)),X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Z_1),X))))) # label(fact_4979_min__max_Oinf__sup__distrib2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2330 (all X_a all X_b (comm_semiring_1(X_b) -> (all D_1 all F all A_2 ((all I_2 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I_2),A_2)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),dvd_dvd(X_b),D_1),hAPP(X_a,X_b,F,I_2))))) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),dvd_dvd(X_b),D_1),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2))))))) # label(fact_4798_dvd__setsum) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2331 (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),hAPP(real,real,sqrt,Xa)),hAPP(real,real,sqrt,Ya))))) # label(fact_2122_real__sqrt__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2332 (all X_a all X_b (comm_monoid_add(X_b) -> (all B_2 all A_3 all S (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),S)) -> hAPP(X_a,X_b,B_2,A_3) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),A_3))),B_2)),zero_zero(X_b))),S)) & (-hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),S)) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),A_3))),B_2)),zero_zero(X_b))),S) = zero_zero(X_b)))))) # label(fact_4236_setsum__delta) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2333 (all Ma (Ma = hAPP(nat,nat,suc,zero_zero(nat)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),hAPP(nat,nat,suc,zero_zero(nat)))))) # label(fact_1377_dvd__1__iff__1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2334 (all X_a (real_normed_vector(X_a) -> (all R_3 exists K_2 ((all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_1)),K_2)))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2)))))) # label(fact_4782_scaleR__right_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2335 (all X_a (linorder(X_a) -> (all X all Y all Z_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),Z_1)))) # label(fact_4838_min__max_Oinf__assoc) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2336 (all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),N),zero_zero(nat)) = N) # label(fact_4989_max__0R) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2337 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,K)),min)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),min)))) # label(fact_675_rel__simps_I13_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2338 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 all B hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B))),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),A_1))))))) # label(fact_994_abs__triangle__ineq2__sym) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2339 (all X_a (linord219039673up_add(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),A_3))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_3)))))) # label(fact_197_zero__less__double__add__iff__zero__less__single__add) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2340 (all X_a (semiring_div(X_a) -> (all K_4 all M all N (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_4),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),M),N))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_4),N)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_4),M))))))) # label(fact_1302_dvd__mod__imp__dvd) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2341 (all X_a (ordered_ab_group_add(X_a) -> (all X all Y all Z_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Z_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Y),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),Z_1)))) # label(fact_4840_min__diff__distrib__left) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2342 (all M all V (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),V)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M)),hAPP(int,nat,number_number_of(nat),V)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),hAPP(int,nat,number_number_of(nat),hAPP(int,int,pred,V))))) # label(fact_2937_Suc__diff__number__of) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2343 (all C all Xa all Ta all A_3 all D_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_3),D_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xa),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),D_1))),Ta))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xa),Ta)))))) # label(fact_764_zdvd__period) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2344 (all Xa ((exists Y_1 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Y_1)),one_one(int)) = ti(int,Xa)) <-> -hBOOL(hAPP(int,bool,even_odd_even(int),Xa)))) # label(fact_1639_odd__equiv__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2345 (all X_a (ordere142940540dd_abs(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,abs_abs(X_a),A_3))) <-> zero_zero(X_a) != ti(X_a,A_3))))) # label(fact_990_zero__less__abs__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2346 (all M all N hAPP(product_prod(nat,nat),nat,product_snd(nat,nat),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)) # label(fact_3151_mod__nat__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2347 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 all B hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)))),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B))))))) # label(fact_996_abs__triangle__ineq3) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2348 (all X_a (linord219039673up_add(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),A_3)),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),zero_zero(X_a))))))) # label(fact_231_double__add__le__zero__iff__single__add__le__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2349 (all X all Y (hAPP(nat,nat,suc,Y) = hAPP(nat,nat,suc,X) -> X = Y)) # label(fact_1275_Suc__inject) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2350 (all Y all X exists R_1 exists A_4 (X = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_1),hAPP(real,real,cos,A_4)) & Y = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_1),hAPP(real,real,sin,A_4)))) # label(fact_3517_polar__Ex) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2351 (all M all N all Z_1 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N))),Z_1) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,semiring_1_of_nat(int),M)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,semiring_1_of_nat(int),N)),Z_1))) # label(fact_22_zadd__int__left) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2352 (all X_a (ring_1(X_a) -> (all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1)) -> hAPP(int,X_a,ring_1_of_int(X_a),Z_1) = hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,nat_1,Z_1)))))) # label(fact_3629_of__nat__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2353 (all K_4 all I_1 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_4)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_2),K_4)))))) # label(fact_627_mult__less__mono1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2354 (all Ta all A_2 all D_1 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),D)) -> (all X_1 ((all Xa_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D))) -> (all Xb (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2)) -> ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa_1))))) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),Ta))) -> -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),D)),Ta)))))))) # label(fact_4672_aset_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2355 (all X all Y hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,real_1(int),X)),hAPP(int,real,real_1(int),Y)) = hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y))) # label(fact_3359_real__of__int__mult) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2356 (all C_1 all D_3 all A_1 all B all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,C_1),D_3),M)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),C_1)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),D_3)),M))))) # label(fact_901_zcong__zdiff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2357 (all K_4 hAPP(quickcheck_code_int,int,quickcheck_int_of,hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),K_4)) = hAPP(int,int,number_number_of(int),K_4)) # label(fact_3185_int__of__number) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2358 (all N all K_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K_4)) -> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),N)) = ti(int,K_4))) # label(fact_3481_zgcd__zmult__eq__self) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2359 (all X_a all A_2 hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),A_2) = hAPP(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool),collect(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),combc(fun(X_a,bool),fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool))),A_2))) # label(fact_2845_Pow__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2360 (all X all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),P_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),standardRes,P_5),X)),P_5)))) # label(fact_1415_StandardRes__ubound) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2361 (all X_a (topological_space(X_a) -> (all G all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),zero_zero(complex)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,complex),fun(X_a,real),hAPP(fun(complex,real),fun(fun(X_a,complex),fun(X_a,real)),combb(complex,real,X_a),re),G)),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4640_Re_OLIM__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2362 (all Va all K all A_3 all B_2 (B_2 != A_3 -> ((all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),Va),X_1),K))) -> hAPP(real,real,Va,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_3),B_2)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,Va,A_3)),hAPP(real,real,Va,B_2))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) # label(fact_2658_DERIV__const__average) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2363 (all X_a (real_normed_field(X_a) & banach(X_a) -> (all Xa all C all K_3 (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),C)),hAPP(X_a,fun(nat,X_a),power_power(X_a),K_3)))) -> (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),C))),hAPP(X_a,fun(nat,X_a),power_power(X_a),K_3)))) -> (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),C)))),hAPP(X_a,fun(nat,X_a),power_power(X_a),K_3)))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),Xa)),hAPP(X_a,real,norm_norm(X_a),K_3))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,fun(nat,X_a)),fun(X_a,X_a),hAPP(fun(fun(nat,X_a),X_a),fun(fun(X_a,fun(nat,X_a)),fun(X_a,X_a)),combb(fun(nat,X_a),X_a,X_a),suminf(X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,X_a)),hAPP(fun(fun(nat,X_a),fun(nat,X_a)),fun(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,X_a))),combb(fun(nat,X_a),fun(nat,X_a),X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),C))),power_power(X_a)))),Xa),hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),C))),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa)))))))))))) # label(fact_4151_termdiffs) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2364 (all Z_1 Z_1 = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(complex,real,re,Z_1)),hAPP(complex,real,im,Z_1))) # label(fact_3025_complex__surj) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2365 (all X_a (divisi14063676e_zero(X_a) -> (all A_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),zero_zero(X_a)) = zero_zero(X_a)))) # label(fact_1825_divide__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2366 (all A_1 all B all C_1 hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),C_1))),C_1) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)),C_1)) # label(fact_1308_zmod__zmult1__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2367 (all X_a (real_normed_vector(X_a) -> (all Xa (hAPP(X_a,real,norm_norm(X_a),Xa) = zero_zero(real) <-> ti(X_a,Xa) = zero_zero(X_a))))) # label(fact_1963_norm__eq__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2368 (all X_a (semiring_div(X_a) -> (all A_1 all B (zero_zero(X_a) != ti(X_a,B) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),A_1)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B)),one_one(X_a)))))) # label(fact_1118_div__add__self1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2369 (all X_a (zero_neq_one(X_a) -> one_one(X_a) != zero_zero(X_a))) # label(fact_166_one__neq__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2370 (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),X),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,sin,X)))))) # label(fact_2321_sin__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2371 (all K_4 all M all N hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),N)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,abs_abs(int),K_4)),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N))) # label(fact_3476_zgcd__zmult__distrib2__abs) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2372 (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)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))))) # label(fact_334_sum__power2__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.72 2373 (all X all Y all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),Y),P_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),X)),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),Y)),P_5))))) # label(fact_1180_MultInv__prop1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2374 (all X hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,X)) = hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),pi),X))) # label(fact_2291_sin__periodic__pi2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2375 (all X_b all X_a (real_normed_algebra(X_b) & topological_space(X_a) -> (all Ya all G all L all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),G)),Ya)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),L),Ya)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4590_mult__left_OLIM) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2376 (all X_a (semiri456707255roduct(X_a) -> (all C all D_1 all A_3 all B_2 (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),D_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),C)) != hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),D_1)) <-> ti(X_a,D_1) != ti(X_a,C) & ti(X_a,B_2) != ti(X_a,A_3))))) # label(fact_383_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2377 (all N -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat)))) # label(fact_145_less__nat__zero__code) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2378 (all A_1 all B hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),B))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),A_1)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))))) # label(fact_348_zadd__power3) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2379 (all Z_1 (hBOOL(hAPP(int,bool,nat_neg,Z_1)) -> zero_zero(nat) = hAPP(int,nat,nat_1,Z_1))) # label(fact_1608_neg__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2380 (all X_a (topological_space(X_a) & comm_monoid_add(X_a) -> sums(X_a) = ti(fun(fun(nat,X_a),fun(X_a,bool)),sums(X_a)))) # label(tsy_c_Series_Osums_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2381 (all Z (zero_zero(complex) = hAPP(complex,complex,cnj,Z) <-> Z = zero_zero(complex))) # label(fact_2967_complex__cnj__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2382 (all E_1 all B all D_3 all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_3),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),E_1),B)) -> (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B) -> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_3),E_1) = one_one(int))))) # label(fact_4397_coprime__divisors__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2383 (all X_a (comm_monoid_add(X_a) -> (all F all Ma all Na hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),suc)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Ma),Na)) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),hAPP(nat,nat,suc,Ma)),hAPP(nat,nat,suc,Na)))))) # label(fact_4209_setsum__shift__bounds__Suc__ivl) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2384 (all X_a all F all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> ((exists X_1 (hAPP(X_a,nat,F,X_1) = hAPP(nat,nat,suc,zero_zero(nat)) & (all Xa_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa_1),A_2)) -> (ti(X_a,X_1) != ti(X_a,Xa_1) -> hAPP(X_a,nat,F,Xa_1) = zero_zero(nat)))) & hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2)))) <-> hAPP(nat,nat,suc,zero_zero(nat)) = hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),A_2)))) # label(fact_4283_setsum__eq__Suc0__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2385 (all P_1 all K (hBOOL(hAPP(nat,bool,P_1,K)) -> ((all N_1 (hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,suc,N_1))) -> hBOOL(hAPP(nat,bool,P_1,N_1)))) -> hBOOL(hAPP(nat,bool,P_1,zero_zero(nat)))))) # label(fact_2834_zero__induct) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2386 (all X_a (real_n1866405975lgebra(X_a) -> (all A_3 (ti(X_a,A_3) != zero_zero(X_a) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),inverse_inverse(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),A_3)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4594_LIM__inverse__fun) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2387 (all X_b all X_a all B_1 all F all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B_1),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),F),A_2))) -> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_1))))) # label(fact_3656_finite__surj) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2388 (all Xa (hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa)) <-> hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(nat,int,semiring_1_of_nat(int),Xa))))) # label(fact_2483_even__nat__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2389 (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),hAPP(nat,nat,fact_fact(nat),N))))))) # label(fact_2549_of__nat__fact__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2390 (all Xa all Ya (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Xa),Ya)),zOdd)) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd)))) # label(fact_1220_odd__mult__odd__prop) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2391 (all X_a (linordered_semiring(X_a) -> (all A_1 all C_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(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_1),B))))))) # label(fact_479_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2392 (all X all Y hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),X)),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) # label(fact_2179_real__sqrt__ge__abs1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2393 (all X_a (number_semiring(X_a) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) # label(fact_9_semiring__one__add__one__is__two) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2394 (all X_a (ordere453448008miring(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),A_1)),zero_zero(X_a)))))))) # label(fact_428_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2395 (all F all Na all Ra hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),Na)),Ra)) = hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),minus_minus(real)),F)),Ra)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))) # label(fact_4205_sumr__diff__mult__const) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2396 (all Xa all Ya (hAPP(real,real,uminus_uminus(real),Xa) = Ya <-> zero_zero(real) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa),Ya))) # label(fact_1989_real__add__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2397 (all K all N_1 all F_3 hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),F_3),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N_1))),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F_3),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K))) = hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F_3),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),K)))) # label(fact_4200_sumr__offset4) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2398 (all X_a (real_normed_vector(X_a) & real_algebra_1(X_a) -> zero_zero(X_a) = hAPP(real,X_a,of_real(X_a),zero_zero(real)))) # label(fact_2793_of__real_Ozero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2399 (all B_1 all K all Diff all H all Na ((all M_2 all T_4 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),Na)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T_4)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_4),H)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_2)),T_4),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),T_4))))) -> (hAPP(nat,nat,suc,K) = Na -> (all M_2 all T_4 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),Na)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_4),H)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T_4)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),minus_minus(real)),hAPP(nat,fun(real,real),Diff,M_2))),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),plus_plus(real)),hAPP(fun(nat,bool),fun(real,real),hAPP(fun(real,fun(fun(nat,bool),real)),fun(fun(nat,bool),fun(real,real)),combc(real,fun(nat,bool),real),hAPP(fun(real,fun(nat,real)),fun(real,fun(fun(nat,bool),real)),hAPP(fun(fun(nat,real),fun(fun(nat,bool),real)),fun(fun(real,fun(nat,real)),fun(real,fun(fun(nat,bool),real))),combb(fun(nat,real),fun(fun(nat,bool),real),real),big_co1399186613setsum(nat,real)),hAPP(fun(real,fun(nat,real)),fun(real,fun(nat,real)),hAPP(fun(fun(nat,real),fun(nat,real)),fun(fun(real,fun(nat,real)),fun(real,fun(nat,real))),combb(fun(nat,real),fun(nat,real),real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,nat),fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,real)),fun(fun(nat,nat),fun(nat,fun(real,real))),combb(nat,fun(real,real),nat),Diff),hAPP(nat,fun(nat,nat),plus_plus(nat),M_2))),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat)))))),power_power(real)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),M_2))))),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),hAPP(real,fun(real,real),times_times(real),B_1)),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real)),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),power_power(real)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),M_2)))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),M_2)))))))),T_4),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),T_4)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,nat),fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,real)),fun(fun(nat,nat),fun(nat,fun(real,real))),combb(nat,fun(real,real),nat),Diff),hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,suc,M_2)))),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),T_4))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,M_2))))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),T_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,M_2)))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,M_2))))))))))))))) # label(fact_4295_Maclaurin__lemma2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2400 (all X_a all Xa all A_2 (hBOOL(hAPP(X_a,bool,A_2,Xa)) <-> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa),A_2)))) # label(fact_74_mem__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2401 (all P_1 all I ((all N_1 (hAPP(nat,int,semiring_1_of_nat(int),N_1) = ti(int,I) -> hBOOL(hAPP(nat,bool,P_1,N_1)))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I),zero_zero(int))) -> hBOOL(hAPP(nat,bool,P_1,zero_zero(nat)))) <-> hBOOL(hAPP(nat,bool,P_1,hAPP(int,nat,nat_1,I))))) # label(fact_1036_split__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2402 (all Ra all X_2 (hBOOL(hAPP(fun(nat,rat),bool,vanishes,X_2)) -> (hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),Ra)) -> (exists K_1 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N_1)) -> hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,abs_abs(rat),hAPP(nat,rat,X_2,N_1))),Ra))))))) # label(fact_4888_vanishesD) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2403 (all P_1 all K all I (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),I)) -> (hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K),one_one(int)))) -> ((all I_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),I_2)) -> (hBOOL(hAPP(int,bool,P_1,I_2)) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),I_2),one_one(int))))))) -> hBOOL(hAPP(int,bool,P_1,I)))))) # label(fact_115_int__gr__induct) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2404 (all X_a (real_algebra_1(X_a) -> (all X all Y hAPP(real,X_a,of_real(X_a),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(real,X_a,of_real(X_a),X)),hAPP(real,X_a,of_real(X_a),Y))))) # label(fact_2795_of__real__mult) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2405 (all X 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_eq(real),zero_zero(real)),X)) -> X = hAPP(real,real,hAPP(nat,fun(real,real),root,N),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N))))) # label(fact_2712_real__root__pos2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2406 (all X hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),pi)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,X))) # label(fact_2292_sin__periodic__pi) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2407 (all X_2 (hBOOL(hAPP(fun(nat,complex),bool,summable(complex),X_2)) -> hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),re),X_2))))) # label(fact_4073_Re_Osummable) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2408 (all Na all Xa all Diff all F (hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) & zero_zero(real) != Xa & (all M_2 all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_2)),X_1),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),X_1)))) -> (exists T_4 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,abs_abs(real),T_4))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),T_4)),hAPP(real,real,abs_abs(real),Xa))) & hAPP(real,real,F,Xa) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,Na),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),Na))))))) # label(fact_4303_Maclaurin__all__lt__objl) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2409 (all W hAPP(int,rat,hAPP(int,fun(int,rat),fract,W),one_one(int)) = hAPP(int,rat,number_number_of(rat),W)) # label(fact_3777_rat__number__of__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2410 (all X_a (field_char_0(X_a) & field_inverse_zero(X_a) -> (all B_2 all A_3 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),field_char_0_Rats(X_a))) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),field_char_0_Rats(X_a))) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),B_2)),field_char_0_Rats(X_a)))))))) # label(fact_3715_Rats__divide) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2411 (all A_3 all B_2 all Ma ((exists K_1 ti(int,B_2) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Ma),K_1))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),Ma)))) # label(fact_908_zcong__iff__lin) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2412 (all Wa all Z ((exists N_1 ti(int,Z) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Wa),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N_1)))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z)))) # label(fact_1451_zless__iff__Suc__zadd) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2413 (all X_a (semiring_div(X_a) -> (all B all A_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),A_1)),B) = zero_zero(X_a)))) # label(fact_1352_mod__mult__self1__is__0) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2414 (all X_a (real_normed_field(X_a) -> (all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2)) -> hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),X_2)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),X_2)))))) # label(fact_4017_suminf__minus) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2415 (all X_a (linorder(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)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)))))) # label(fact_499_linorder__le__cases) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2416 (all X_a (number(X_a) -> (all Wa all Xa (hAPP(int,X_a,number_number_of(X_a),Wa) = ti(X_a,Xa) <-> hAPP(int,X_a,number_number_of(X_a),Wa) = ti(X_a,Xa))))) # label(fact_41_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2417 (all Na all N_6 all F ((all N_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,N_1)),hAPP(nat,nat,F,hAPP(nat,nat,suc,N_1))))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),N_6)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,Na)),hAPP(nat,nat,F,N_6)))))) # label(fact_1522_dvd_Olift__Suc__mono__le) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2418 (all Q_3 all B all R_3 all C_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_3),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Q_3),C_1))),R_3))))))) # label(fact_1474_zmult2__lemma__aux3) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2419 (all Xa all Ya (ti(int,Ya) = ti(int,Xa) <-> hAPP(int,real,real_1(int),Xa) = hAPP(int,real,real_1(int),Ya))) # label(fact_3353_real__of__int__inject) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2420 (all X_a (order(X_a) -> (all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) -> ti(X_a,Y) != ti(X_a,X))))) # label(fact_526_order__less__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2421 (all M all I_1 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_2)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),M))))) # label(fact_135_trans__less__add1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2422 (all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A_3),zero_zero(real)),sequentially)) -> (hBOOL(hAPP(fun(nat,real),bool,monoseq(real),A_3)) -> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,fun(nat,bool)),fun(nat,real),hAPP(fun(fun(nat,bool),real),fun(fun(nat,fun(nat,bool)),fun(nat,real)),combb(fun(nat,bool),real,nat),hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat))))),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))),sequentially))))) # label(fact_4451_summable__Leibniz_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2423 (all D_1 all A_3 all B_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_3),B_2))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_1),B_2)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_1),A_3)))) # label(fact_2881_coprime__mul__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2424 (all X all N hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),N)),pi))) = hAPP(real,real,tan,X)) # label(fact_2416_tan__periodic__n) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2425 (all X_a (real_normed_vector(X_a) -> (all Y all S_3 all X all R_3 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),X)),R_3)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),Y)),S_3)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),R_3),S_3)))))))) # label(fact_2008_norm__add__less) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2426 (all B all A_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),B),A_1))),hAPP(complex,real,norm_norm(complex),B))),hAPP(complex,real,norm_norm(complex),A_1)))) # label(fact_1987_complex__mod__triangle__ineq2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2427 (all Xa all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),Xa)),hAPP(int,real,real_1(int),Ya))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),Ya)))) # label(fact_3357_real__of__int__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2428 (all N exists P_6 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),P_6)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),P_6),hAPP(nat,nat,suc,hAPP(nat,nat,fact,N)))) & hBOOL(hAPP(nat,bool,prime,P_6)))) # label(fact_3576_euclid__bound) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2429 (all X hAPP(int,nat,nat_1,hAPP(real,int,archimedean_ceiling(real),X)) = hAPP(real,nat,natceiling,X)) # label(fact_3309_natceiling__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2430 (all K all Xa hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),cos),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),plus_plus(real)),K))),Xa),hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa),K)))))) # label(fact_3848_DERIV__cos__add) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2431 (all X_a (number_ring(X_a) -> (all Ya (hAPP(int,X_a,number_number_of(X_a),Ya) = one_one(X_a) <-> hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),hAPP(int,int,uminus_uminus(int),Ya))))))))) # label(fact_2119_eq__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2432 (all D_3 all A_1 all B (one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_3),B) = one_one(nat))) # label(fact_3297_coprime__rmult__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2433 (all K_4 hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit0,K_4)) = hAPP(int,int,bit0,hAPP(int,int,uminus_uminus(int),K_4))) # label(fact_2059_minus__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2434 (all X all Y hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y)),X))) # label(fact_3253_gcd__semilattice__nat_Oinf__le1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2435 (all X_a (number_ring(X_a) -> (all Wa hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),ring_1_Ints(X_a)))))) # label(fact_3829_Ints__number__of) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2436 (all X_a (archim1727834104eiling(X_a) -> (all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archimedean_ceiling(X_a),Y)),hAPP(X_a,int,archimedean_ceiling(X_a),X))))))) # label(fact_3306_ceiling__mono) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2437 (all M all N hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M),N) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),M),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),M),N)),N))) # label(fact_1436_zmod__zdiv__equality_H) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2438 (all F all Xa all L (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),Xa),L)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),L),zero_zero(real))) -> (exists D_2 ((all H_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H_1)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),H_1),D_2)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa),H_1))),hAPP(real,real,F,Xa)))))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_2))))))) # label(fact_2665_DERIV__neg__dec__right) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2439 (all X_a (linordered_idom(X_a) -> (all A_1 all N hAPP(X_a,X_a,abs_abs(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)) = hAPP(X_a,X_a,abs_abs(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),N))))) # label(fact_1983_abs__power__minus) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2440 (all X_a all A_2 all B_1 (ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1) <-> hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),B_1) = hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2))) # label(fact_2237_Compl__eq__Compl__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2441 (all X_a (real_algebra_1(X_a) & real_normed_vector(X_a) -> (all X_2 all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),X_2),A_3),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,real),fun(nat,X_a),hAPP(fun(real,X_a),fun(fun(nat,real),fun(nat,X_a)),combb(real,X_a,nat),of_real(X_a)),X_2)),hAPP(real,X_a,of_real(X_a),A_3)),sequentially)))))) # label(fact_4489_of__real_OLIMSEQ) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2442 (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),M),N)) -> zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N)) & (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),N))))) # label(fact_1468_div__if) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2443 (all Na hAPP(nat,real,hAPP(fun(nat,real),fun(nat,real),diffs(real),sin_coeff),Na) = hAPP(nat,real,cos_coeff,Na)) # label(fact_4143_sin__fdiffs2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2444 (all X_a (real_normed_field(X_a) -> (all G all Db all F all Xa all Da (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),Da)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),Xa),Db)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),times_times(X_a)),F)),G)),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Da),hAPP(X_a,X_a,G,Xa))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Db),hAPP(X_a,X_a,F,Xa)))))))))) # label(fact_3858_DERIV__mult) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2445 (all N hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),N))),zero_zero(int)))) # label(fact_2085_negative__zle__0) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2446 (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),hAPP(int,nat,nat_1,Xa)),hAPP(int,nat,nat_1,Ya))))))) # label(fact_1033_transfer__nat__int__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2447 (all Xa (hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(nat,int,semiring_1_of_nat(int),Xa))) <-> hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa)))) # label(fact_2484_Parity_Otransfer__int__nat__relations) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2448 (all X_a (ab_group_add(X_a) -> (all A_1 all B hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(X_a,X_a,uminus_uminus(X_a),B))))) # label(fact_1917_minus__diff__minus) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2449 (all X_a (ordered_ab_group_add(X_a) -> (all A_3 (hBOOL(hAPP(fun(nat,X_a),bool,monoseq(X_a),A_3)) -> hBOOL(hAPP(fun(nat,X_a),bool,monoseq(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),A_3))))))) # label(fact_3948_monoseq__minus) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2450 (all X all Y hAPP(complex,real,im,hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,X),Y)) = Y) # label(fact_2997_Im) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2451 (all A_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),zero_zero(nat)))) # label(fact_1254_gcd__lcm__complete__lattice__nat_Otop__greatest) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2452 (all K_4 all I_1 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_2),K_4))))) # label(fact_615_mult__le__mono1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2453 (all X 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),zero_zero(nat)),N)) -> hAPP(real,real,hAPP(nat,fun(real,real),root,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)),X) = hAPP(real,real,hAPP(nat,fun(real,real),root,M),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X))))) # label(fact_2687_real__root__mult__exp) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2454 (all X_a (number_ring(X_a) -> (all K_4 hAPP(int,X_a,number_number_of(X_a),K_4) = hAPP(int,X_a,ring_1_of_int(X_a),K_4)))) # label(fact_3594_number__of__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2455 (all Ma all K all Na (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Na)) -> ((exists S_1 exists T_4 hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),xzgcd,Ma),Na) = hAPP(product_prod(int,int),product_prod(int,product_prod(int,int)),hAPP(int,fun(product_prod(int,int),product_prod(int,product_prod(int,int))),product_Pair(int,product_prod(int,int)),K),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),S_1),T_4))) <-> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,Ma),Na) = ti(int,K)))) # label(fact_3500_xzgcd__correct) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2456 (all A_1 all B all C_1 all Q_3 all R_3 (hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,B),C_1),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_3),R_3))) -> (zero_zero(int) != ti(int,C_1) -> hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)),C_1),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),Q_3)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),R_3)),C_1))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),R_3)),C_1))))))) # label(fact_1643_zmult1__lemma) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2457 (all N all M hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),N))),hAPP(nat,int,semiring_1_of_nat(int),M)))) # label(fact_2054_negative__zle) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2458 (all Ma all K all Na (zero_zero(nat) = K | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),Na)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ma),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),K))))) # label(fact_843_nat__mult__dvd__cancel__disj_H) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2459 (all T_2 all T_1 (finite_finite(T_2) & finite_finite(T_1) -> finite_finite(product_prod(T_2,T_1)))) # label(arity_prod___Finite__Set_Ofinite) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2460 (all P_4 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),P_4)) & (all M_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M_2),P_4)) -> one_one(nat) = M_2 | P_4 = M_2)) <-> hBOOL(hAPP(nat,bool,prime,P_4)))) # label(fact_3092_prime__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2461 (all X_a (field(X_a) -> (all X all Y all Z_1 (zero_zero(X_a) != ti(X_a,Z_1) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Y),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),X)),Y)),Z_1))))) # label(fact_1882_diff__divide__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2462 (all X_a (archim1727834104eiling(X_a) -> (all X hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)) = hAPP(int,int,uminus_uminus(int),hAPP(X_a,int,archim2085082626_floor(X_a),X))))) # label(fact_3319_ceiling__minus) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2463 (all Z_1 all W (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Z_1),W)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),W),Z_1)))) # label(fact_598_real__le__linear) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2464 (all X_a (ordered_ab_group_add(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,uminus_uminus(X_a),A_3))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a))))))) # label(fact_1953_neg__0__less__iff__less) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2465 (exists K_2 all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(complex,real,im,X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_1)),K_2)))) # label(fact_3014_Im_Obounded) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2466 (all X_a (banach(X_a) & real_normed_field(X_a) -> (all Xa all C all K_3 (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),C)))),hAPP(X_a,fun(nat,X_a),power_power(X_a),K_3)))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),Xa)),hAPP(X_a,real,norm_norm(X_a),K_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(fun(X_a,fun(nat,X_a)),fun(X_a,X_a),hAPP(fun(fun(nat,X_a),X_a),fun(fun(X_a,fun(nat,X_a)),fun(X_a,X_a)),combb(fun(nat,X_a),X_a,X_a),suminf(X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,X_a)),hAPP(fun(fun(nat,X_a),fun(nat,X_a)),fun(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,X_a))),combb(fun(nat,X_a),fun(nat,X_a),X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),C))),hAPP(fun(nat,X_a),fun(X_a,fun(nat,X_a)),hAPP(fun(X_a,fun(fun(nat,X_a),fun(nat,X_a))),fun(fun(nat,X_a),fun(X_a,fun(nat,X_a))),combc(X_a,fun(nat,X_a),fun(nat,X_a)),hAPP(fun(X_a,fun(nat,fun(X_a,X_a))),fun(X_a,fun(fun(nat,X_a),fun(nat,X_a))),hAPP(fun(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a))),fun(fun(X_a,fun(nat,fun(X_a,X_a))),fun(X_a,fun(fun(nat,X_a),fun(nat,X_a)))),combb(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),X_a),combs(nat,X_a,X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,fun(X_a,X_a))),hAPP(fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),fun(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,fun(X_a,X_a)))),combb(fun(nat,X_a),fun(nat,fun(X_a,X_a)),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a))),hAPP(fun(X_a,X_a),fun(X_a,fun(nat,X_a)),hAPP(fun(X_a,fun(X_a,fun(nat,X_a))),fun(fun(X_a,X_a),fun(X_a,fun(nat,X_a))),combs(X_a,X_a,fun(nat,X_a)),hAPP(fun(X_a,fun(nat,fun(X_a,X_a))),fun(X_a,fun(X_a,fun(nat,X_a))),hAPP(fun(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a))),fun(fun(X_a,fun(nat,fun(X_a,X_a))),fun(X_a,fun(X_a,fun(nat,X_a)))),combb(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),X_a),combc(nat,X_a,X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,fun(X_a,X_a))),hAPP(fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),fun(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,fun(X_a,X_a)))),combb(fun(nat,X_a),fun(nat,fun(X_a,X_a)),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a))),hAPP(fun(nat,X_a),fun(X_a,fun(nat,X_a)),hAPP(fun(X_a,fun(fun(nat,X_a),fun(nat,X_a))),fun(fun(nat,X_a),fun(X_a,fun(nat,X_a))),combc(X_a,fun(nat,X_a),fun(nat,X_a)),hAPP(fun(X_a,fun(nat,fun(X_a,X_a))),fun(X_a,fun(fun(nat,X_a),fun(nat,X_a))),hAPP(fun(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a))),fun(fun(X_a,fun(nat,fun(X_a,X_a))),fun(X_a,fun(fun(nat,X_a),fun(nat,X_a)))),combb(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),X_a),combs(nat,X_a,X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,fun(X_a,X_a))),hAPP(fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),fun(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,fun(X_a,X_a)))),combb(fun(nat,X_a),fun(nat,fun(X_a,X_a)),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a))),hAPP(fun(X_a,X_a),fun(X_a,fun(nat,X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(nat,X_a))),combb(X_a,fun(nat,X_a),X_a),power_power(X_a)),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Xa))))),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))))),combi(X_a))))),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),semiring_1_of_nat(X_a))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),minus_minus(nat)),hAPP(nat,nat,suc,zero_zero(nat))))))))),zero_zero(X_a)),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a))))))))) # label(fact_4557_termdiffs__aux) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2467 (all X_a (archim1727834104eiling(X_a) -> (all Z_1 ti(int,Z_1) = hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1))))) # label(fact_3613_ceiling__of__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2468 (all X_a (monoid_mult(X_a) & linordered_ring(X_a) -> (all X all N (hBOOL(hAPP(nat,bool,even_odd_even(nat),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,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N))))))) # label(fact_2489_zero__le__even__power) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2469 (all X_a (field_char_0(X_a) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),one_one(X_a)),field_char_0_Rats(X_a))))) # label(fact_3711_Rats__1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2470 (all A_3 all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Xa)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A_3),hAPP(real,nat,natfloor,Xa))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),A_3)),Xa))))) # label(fact_2410_le__natfloor__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2471 (all X all K_4 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> hAPP(real,real,hAPP(nat,fun(real,real),root,N),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),K_4)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)),K_4))) # label(fact_2688_real__root__power) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2472 (all X_a (comm_semiring_1(X_a) -> (all A_1 all C_1 all D_3 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),D_3)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)),D_3)))) # label(fact_299_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2473 (all X_a (number(X_a) & linord1117847801e_zero(X_a) -> (all A_3 all B_2 all Wa ((hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Wa))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),hAPP(int,X_a,number_number_of(X_a),Wa))),B_2))) & (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Wa))) -> (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a)))) & (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),hAPP(int,X_a,number_number_of(X_a),Wa)))))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),hAPP(int,X_a,number_number_of(X_a),Wa)))))))) # label(fact_1792_less__divide__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2474 (all Xa hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) # label(fact_4130_Ln_Oaux2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2475 (all L_4 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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),L_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),L_4))))) # label(fact_819_diff__le__mono) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2476 (all X_a (linordered_field(X_a) -> (all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),zero_zero(X_a)))))))) # label(fact_1840_divide__neg__pos) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2477 (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)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)))))))) # label(fact_492_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2478 (all X_a (ring(X_a) -> (all A_1 all B hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B))))) # label(fact_1908_minus__mult__right) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2479 (all N all X (X = hAPP(nat,real,real_1(nat),hAPP(real,nat,natfloor,X)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(real,nat,natfloor,X)),N) = hAPP(real,nat,natfloor,hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N)))) # label(fact_2403_natfloor__power) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2480 (all Xa all A_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(real,int,archimedean_ceiling(real),Xa)),A_3)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,real_1(int),A_3)),one_one(real)))))) # label(fact_3441_ceiling__less__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2481 (all X_a (linorder(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),A_1)))) # label(fact_4830_min__max_Oinf_Oidem) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2482 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),zero_zero(real))) -> hAPP(real,real,cos,hAPP(complex,real,arg,hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,zero_zero(real)),Y))) = zero_zero(real))) # label(fact_2771_cos__arg__i__mult__zero__neg) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2483 (all P_4 all Na (hBOOL(hAPP(nat,bool,prime,P_4)) & one_one(nat) = Na <-> hBOOL(hAPP(nat,bool,prime,hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_4),Na))))) # label(fact_3084_prime__exp) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2484 (all X_a (linordered_field(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)) -> (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),one_one(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)))))))) # label(fact_2614_one__less__inverse) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2485 (all N all K_4 all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),M)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),N)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N)))))) # label(fact_3263_gcd__greatest__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2486 (all L (hAPP(int,int,bit1,L) = min <-> ti(int,L) = min)) # label(fact_666_rel__simps_I43_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2487 (all P_5 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_5),Y)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Y),N)))))) # label(fact_917_zpower__zdvd__prop1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2488 (all X_a (linord219039673up_add(X_a) -> (all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),hAPP(X_a,X_a,uminus_uminus(X_a),Y)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y))))) # label(fact_4973_minus__min__eq__max) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2489 (all X_a (comm_monoid_add(X_a) -> (all F all Na all Ma ((-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,Na)),Ma)) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,F,Ma)),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),suc)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),Na))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),hAPP(nat,nat,suc,Na)))) & (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,Na)),Ma)) -> hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),hAPP(nat,nat,suc,Na))) = zero_zero(X_a)))))) # label(fact_4631_setsum__cl__ivl__Suc2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2490 (all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),plus_plus(real)),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),sin)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),cos)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X_1),zero_zero(real)))) # label(fact_3936_DERIV__sin__circle__all__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2491 (all X_a nat_case(X_a) = ti(fun(X_a,fun(fun(nat,X_a),fun(nat,X_a))),nat_case(X_a))) # label(tsy_c_Nat_Onat_Onat__case_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2492 (all Real1 all Real2 all Real1_2 all Real2_2 (hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Real1),Real2) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Real1_2),Real2_2) <-> Real2 = Real2_2 & Real1_2 = Real1)) # label(fact_2759_complex_Oinject) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2493 (all P (fFalse = ti(bool,P) | ti(bool,P) = fTrue)) # label(help_fFalse_1_1_T) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2494 (all Y (Y != zero_zero(real) -> hAPP(real,real,cos,hAPP(complex,real,arg,hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,zero_zero(real)),Y))) = zero_zero(real))) # label(fact_2773_cos__arg__i__mult__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2495 (all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A_3),zero_zero(real)),sequentially)) -> ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,A_3,N_1)))) -> ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,A_3,hAPP(nat,nat,suc,N_1))),hAPP(nat,real,A_3,N_1)))) -> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,fun(nat,bool)),fun(nat,real),hAPP(fun(fun(nat,bool),real),fun(fun(nat,fun(nat,bool)),fun(nat,real)),combb(fun(nat,bool),real,nat),hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat))),hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))),sequentially)))))) # label(fact_4540_summable__Leibniz_H_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2496 (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_170_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2497 (all X_a (real_normed_vector(X_a) -> (all X ((ti(X_a,X) != zero_zero(X_a) -> one_one(real) = hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,sgn_sgn(X_a),X))) & (zero_zero(X_a) = ti(X_a,X) -> zero_zero(real) = hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,sgn_sgn(X_a),X))))))) # label(fact_2319_norm__sgn) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2498 (all X_a (number_ring(X_a) -> (all K_4 hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,succ,K_4)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),hAPP(int,X_a,number_number_of(X_a),K_4))))) # label(fact_308_number__of__succ) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2499 (all X_a (field(X_a) -> (all X all Y all Z_1 (ti(X_a,Z_1) != zero_zero(X_a) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),Y))),Z_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Z_1)),Y))))) # label(fact_1883_divide__diff__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2500 (all B all A_1 (zero_zero(nat) != A_1 -> (exists X_1 exists Y_1 hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),X_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),Y_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B))))) # label(fact_3338_bezout__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2501 (all K all Ma all Na (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,Ma),Na))) <-> 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),Ma)))) # label(fact_3464_zgcd__greatest__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2502 (all M all N hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),M),N)) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(nat,int,semiring_1_of_nat(int),M)),N)) # label(fact_21_int__power) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2503 (all X_a (preorder(X_a) -> (all P_1 all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ya),Xa)) -> hBOOL(P_1)))))) # label(fact_525_order__less__imp__triv) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2504 (all P_5 (hBOOL(hAPP(nat,bool,prime,P_5)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),P_5)))) # label(fact_3089_prime__g__one) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2505 (all X_a (linord1117847801e_zero(X_a) -> (all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y)) -> hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,abs_abs(X_a),X)),Y))))) # label(fact_1887_abs__div__pos) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2506 (all A_3 hBOOL(hAPP(complex,bool,hAPP(fun(complex,complex),fun(complex,bool),isCont(complex,complex),cnj),A_3))) # label(fact_4932_cnj_OisCont) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2507 (all X_a (ring(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(X_a,X_a,uminus_uminus(X_a),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)))) # label(fact_1905_minus__mult__minus) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2508 (all M all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N) = hAPP(product_prod(nat,nat),nat,product_fst(nat,nat),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M),N))) # label(fact_3150_div__nat__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2509 (all A_1 hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),A_1) = A_1) # label(fact_3252_gcd__semilattice__nat_Oinf_Oidem) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2510 (all X_a (linord1117847801e_zero(X_a) -> (all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_2)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),zero_zero(X_a))) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),zero_zero(X_a))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),B_2)),zero_zero(X_a))))))) # label(fact_1834_divide__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2511 (all P (fTrue = ti(bool,P) | fFalse = ti(bool,P))) # label(help_fTrue_1_1_T) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2512 (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) = hAPP(X_a,X_a,abs_abs(X_a),A_1))))) # label(fact_988_abs__of__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2513 (all X_a (comm_semiring_1(X_a) -> (all X all Y all Q_3 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),Q_3)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),Q_3)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)),Q_3)))) # label(fact_395_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2514 (all X_4 all Y_3 hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X_4),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_4),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y_3),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(real,real,sin,hAPP(real,real,arccos,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X_4),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_4),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y_3),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))))) # label(fact_2363_sin__arccos__lemma1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2515 (all X_a (linordered_field(X_a) -> (all B_2 all A_3 all 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),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)),A_3)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)))))))) # label(fact_1824_pos__divide__le__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.73 2516 (all B all M all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M)) -> ti(int,B) = ti(int,A_1))))))) # label(fact_916_zcong__zless__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2517 (all X_a (ordered_semiring(X_a) -> (all C_1 all D_3 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),D_3)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D_3)))))))))) # label(fact_437_mult__mono) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2518 (all Xa hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),cos_coeff)),hAPP(real,fun(nat,real),power_power(real),Xa))) = hAPP(real,real,cos,Xa)) # label(fact_3998_cos__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2519 (all X_a (linordered_field(X_a) -> (all N hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,fact_fact(nat),N)))))))) # label(fact_2633_inv__of__nat__fact__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2520 (all X_a (division_ring(X_a) -> ti(fun(X_a,fun(X_a,X_a)),inverse_divide(X_a)) = inverse_divide(X_a))) # label(tsy_c_Fields_Oinverse__class_Odivide_1_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2521 (all Ma all K all Na (hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ma),K) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),K) <-> zero_zero(nat) = K | Na = Ma)) # label(fact_606_mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2522 (all X all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5)) -> (hBOOL(hAPP(int,bool,zprime,P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_5),X)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),one_one(int)),P_5))))))) # label(fact_1136_Euler__part3) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2523 (all N all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),X)),one_one(real))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),one_one(real))),N))))) # label(fact_2431_LIMSEQ__inverse__realpow__zero__lemma) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2524 (all X_a (group_add(X_a) -> (all A_3 all B_2 (ti(X_a,A_3) = ti(X_a,B_2) <-> hAPP(X_a,X_a,uminus_uminus(X_a),B_2) = hAPP(X_a,X_a,uminus_uminus(X_a),A_3))))) # label(fact_1924_neg__equal__iff__equal) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2525 (all U all Ma all Na all I all J (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),I)),U)),Na))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I),U)),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J),U)),Na)))))) # label(fact_878_nat__less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2526 (all X_a (linorder(X_a) -> (all Xa all Ya all Z (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ya),Z)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Z)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Xa),Ya)),Z)))))) # label(fact_4855_min__le__iff__disj) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2527 (all Y_2 all X_2 ((exists A_4 (hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A_4)) & (all N_1 hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,abs_abs(rat),hAPP(nat,rat,X_2,N_1))),A_4))))) -> (hBOOL(hAPP(fun(nat,rat),bool,vanishes,Y_2)) -> hBOOL(hAPP(fun(nat,rat),bool,vanishes,hAPP(fun(nat,rat),fun(nat,rat),hAPP(fun(nat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,rat)),combs(nat,rat,rat),hAPP(fun(nat,rat),fun(nat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,fun(rat,rat))),combb(rat,fun(rat,rat),nat),times_times(rat)),X_2)),Y_2)))))) # label(fact_4863_vanishes__mult__bounded) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2528 (all X_a (number_ring(X_a) -> (all W hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,W)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),W))),hAPP(int,X_a,number_number_of(X_a),W))))) # label(fact_102_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2529 (all X_a (field_inverse_zero(X_a) -> (all A_3 all B_2 all C ((zero_zero(X_a) = ti(X_a,C) -> ti(X_a,A_3) = zero_zero(X_a)) & (ti(X_a,C) != zero_zero(X_a) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C) = ti(X_a,B_2)) <-> ti(X_a,A_3) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C))))) # label(fact_1846_eq__divide__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2530 (all X ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) -> hAPP(int,int,fact_fact(int),X) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,fact_fact(nat),hAPP(int,nat,nat_1,X)))) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) -> zero_zero(int) = hAPP(int,int,fact_fact(int),X)))) # label(fact_3203_fact__int__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2531 (all I_1 all U_1 all J_2 all K_4 hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),J_2)),U_1)),K_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I_1),U_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_2),U_1)),K_4))) # label(fact_694_left__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2532 (all X_a (linordered_field(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(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)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),C_1),B))))))))) # label(fact_1876_divide__strict__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2533 (all X_a all X_b ti(fun(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool))),sum_Plus(X_a,X_b)) = sum_Plus(X_a,X_b)) # label(tsy_c_Sum__Type_OPlus_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2534 (all Ma all Na all P_1 ((all M_2 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,M_2),zero_zero(nat)))) -> ((all M_2 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,N_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_2),N_1))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,M_2),N_1))))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,Ma),Na))))) # label(fact_3555_gcd__nat__induct) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2535 (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)) -> (hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),one_one(int)) != ti(int,X) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),one_one(int)))))))) # label(fact_913_Euler_Oaux1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2536 (all X_a (comm_monoid_add(X_a) & topological_space(X_a) -> (all F (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) <-> (exists X1 hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),X1))))))) # label(fact_4117_summable__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2537 (all X_a (ord(X_a) -> hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,X_a)),min_1(X_a),ord_less_eq(X_a)) = ord_min(X_a))) # label(fact_4862_min__ord__min) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2538 (all X_a (real_normed_vector(X_a) -> (exists K_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2)) & (all A_4 all B_4 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_4),B_4))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),A_4)),hAPP(X_a,real,norm_norm(X_a),B_4))),K_2)))))))) # label(fact_4780_scaleR_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2539 (all Xa all Ya (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Xa),Ya)),zEven)) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ya),zEven)) | hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)))) # label(fact_1697_EvenOdd_Oeven__product) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2540 (all A_1 all B all X (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),B)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),X)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B))) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),X)))) # label(fact_3271_gcd__semilattice__nat_Oless__infI2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2541 (all X_a (semiring_div(X_a) -> (all A_1 all B all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C_1))),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C_1)))) # label(fact_1292_zmod__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2542 (all A_1 all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> (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_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),A_1)),one_one(int)),P_5)) -> one_one(int) = ti(int,A_1) | hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int)) = ti(int,A_1)))))) # label(fact_796_zcong__square__zless) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2543 (all X X = hAPP(real,real,ln,hAPP(real,real,exp(real),X))) # label(fact_1759_ln__exp) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2544 (all X all N_4 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),N),N_4)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),X)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N_4),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X))))))) # label(fact_2706_real__root__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2545 (all X all Y ((zero_zero(nat) = Y -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,X),Y) = X) & (Y != zero_zero(nat) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,X),Y)))) # label(fact_3747_nat__gcd_Osimps) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2546 (all X_a all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),A_2))) -> hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,finite_finite_1(fun(X_a,bool)),A_2)))) # label(fact_2744_finite__UnionD) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2547 (all X_2 (hBOOL(hAPP(fun(nat,complex),bool,summable(complex),X_2)) -> hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),im),X_2)) = hAPP(complex,real,im,hAPP(fun(nat,complex),complex,suminf(complex),X_2)))) # label(fact_4076_Im_Osuminf) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2548 (all X_a (archim1727834104eiling(X_a) -> (all X all Z_1 hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),Z_1))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),Z_1)))) # label(fact_3623_floor__diff__of__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2549 (all K all I all J hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,nat),fun(fun(nat,bool),fun(nat,bool)),image(nat,nat),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),I),J)) = hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),K))) # label(fact_4566_image__add__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2550 (all N (-hBOOL(hAPP(nat,bool,even_odd_even(nat),N)) -> hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) # label(fact_2527_lemma__not__even__div2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2551 (all N all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,real_1(nat),N)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N)))) -> N = hAPP(int,nat,nat_1,hAPP(real,int,archim2085082626_floor(real),X))))) # label(fact_2903_floor__eq3) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2552 (all Xa all Ya (hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),Xa),Ya)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(rat,real,ratreal,Xa)),hAPP(rat,real,ratreal,Ya))))) # label(fact_3781_real__less__code) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2553 (all X_a (real_normed_vector(X_a) -> (all Ra all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_2)) -> hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra)),X_2))))))) # label(fact_4745_scaleR__right_OCauchy) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2554 (all I_1 ((ti(int,I_1) = zero_zero(int) -> zero_zero(int) = hAPP(int,int,sgn_sgn(int),I_1)) & (zero_zero(int) != ti(int,I_1) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),I_1)) -> one_one(int) = hAPP(int,int,sgn_sgn(int),I_1)) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),I_1)) -> hAPP(int,int,sgn_sgn(int),I_1) = hAPP(int,int,uminus_uminus(int),one_one(int)))))) # label(fact_2351_zsgn__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2555 (all X_a (linordered_semidom(X_a) -> (all Ma all Na (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Ma)),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)))))) # label(fact_208_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2556 (all X_a (real_normed_field(X_a) -> (all K all F (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(fun(nat,bool),X_a),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combb(fun(nat,bool),X_a,nat),hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F)),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,fun(nat,bool))),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combs(nat,nat,fun(nat,bool)),hAPP(fun(nat,nat),fun(nat,fun(nat,fun(nat,bool))),hAPP(fun(nat,fun(nat,fun(nat,bool))),fun(fun(nat,nat),fun(nat,fun(nat,fun(nat,bool)))),combb(nat,fun(nat,fun(nat,bool)),nat),ord_atLeastLessThan(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),K))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),K))),K)))),hAPP(fun(nat,X_a),X_a,suminf(X_a),F)))))))) # label(fact_4258_sums__group) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2557 (all Xa all P_4 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),sRStar,P_4))) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),sr,P_4))))) # label(fact_1542_SRStar__SR__prop) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2558 (all X_a (linord893533164strict(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B)))))))) # label(fact_454_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2559 (all X_a (ord(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (ti(X_a,C_1) = ti(X_a,B) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),C_1))))))) # label(fact_520_ord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2560 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit0,K)),hAPP(int,int,bit0,L))))) # label(fact_221_rel__simps_I31_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2561 (all X_a (monoid_mult(X_a) -> (all A_1 all M all N hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N))))) # label(fact_465_power__add) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2562 (all X all Y hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,cnj,X)),hAPP(complex,complex,cnj,Y)) = hAPP(complex,complex,cnj,hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),X),Y))) # label(fact_2957_cnj_Oadd) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2563 (all A_1 all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),P_5)) -> (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_5)) -> hAPP(int,int,hAPP(int,fun(int,int),inv,P_5),hAPP(int,int,hAPP(int,fun(int,int),inv,P_5),A_1)) = ti(int,A_1)))))) # label(fact_1141_inv__inv) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2564 (all N hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N)))))) # label(fact_2092_neg__zminus__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2565 (all X_a all F all A_2 hAPP(int,real,real_1(int),hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),F),A_2)) = hAPP(fun(X_a,bool),real,hAPP(fun(X_a,real),fun(fun(X_a,bool),real),big_co1399186613setsum(X_a,real),hAPP(fun(X_a,int),fun(X_a,real),hAPP(fun(int,real),fun(fun(X_a,int),fun(X_a,real)),combb(int,real,X_a),real_1(int)),F)),A_2)) # label(fact_4272_real__of__int__setsum) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2566 (all L all U hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),L),U)))) # label(fact_4559_finite__atLeastAtMost__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2567 (all X_b all X_a all P all Q hAPP(X_b,X_a,hAPP(X_a,fun(X_b,X_a),combk(X_a,X_b),P),Q) = ti(X_a,P)) # label(help_COMBK_1_1_U) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2568 (all N hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),hAPP(nat,nat,fact_fact(nat),N)))) # label(fact_2548_fact__ge__one__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2569 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,ln,X))))) # label(fact_1718_ln__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2570 (all X all Y (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,X),Y)) -> (exists P_6 (hBOOL(hAPP(nat,bool,prime,P_6)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_6),Y)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_6),X)))))) # label(fact_3111_coprime__prime__dvd__ex) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2571 (all X_a (linordered_idom(X_a) -> (all K_4 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),K_4),hAPP(X_a,X_a,sgn_sgn(X_a),K_4)) = hAPP(X_a,X_a,abs_abs(X_a),K_4)))) # label(fact_2283_abs__sgn) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2572 (all X_b (number(X_b) & semiring(X_b) -> (all V all B all C_1 hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),B),C_1)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),B)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),C_1))))) # label(fact_460_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2573 (all X_a all C all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),A_2)) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),B_1))))) # label(fact_2189_subsetD) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2574 (all N (zero_zero(nat) != N -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),N)))) # label(fact_2902_coprime__minus1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2575 (all N all I_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,zero_zero(nat))),I_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),I_1),N))))) # label(fact_1454_nat__one__le__power) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2576 (all X all Y hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sqrt,X)),hAPP(real,real,sqrt,Y)) = hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y))) # label(fact_2131_real__sqrt__mult) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2577 (all X all Y (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)))) # label(fact_943_dvd_Oless__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2578 (all B_2 all A_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_3),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),B_2),A_3))))) # label(fact_1110_nonneg1__imp__zdiv__pos__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2579 (all X_a (real_normed_vector(X_a) -> ti(fun(fun(nat,X_a),bool),bseq(X_a)) = bseq(X_a))) # label(tsy_c_SEQ_OBseq_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2580 (all X_a all X_b (comm_monoid_mult(X_b) -> (all B_2 all C all A_3 all S (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S)) -> (-hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),S)) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),A_3))),B_2)),C)),S) = hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),C),hAPP(fun(X_a,bool),nat,finite_card(X_a),S))) & (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),S)) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),A_3))),B_2)),C)),S) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_a,X_b,B_2,A_3)),hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),S)),one_one(nat))))))))) # label(fact_3963_setprod__gen__delta) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2581 (all X_a (real_normed_vector(X_a) -> (all R_3 exists K_2 all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_1)),K_2)))))) # label(fact_4786_scaleR__right_Obounded) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2582 (all A_1 -(-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),zero_zero(nat))) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A_1)))) # label(fact_1255_gcd__lcm__complete__lattice__nat_Onot__top__less) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2583 (all K_4 all M all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N))) # label(fact_612_add__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2584 (all X_a all X_b (field_inverse_zero(X_b) -> (all F all G all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),inverse_divide(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),A_2)) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),inverse_divide(X_b)),F)),G)),A_2))))) # label(fact_3984_setprod__dividef) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2585 (all X_a (comm_ring(X_a) & dvd(X_a) -> (all Ta all D_1 all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),D)) -> (all X_1 all K_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),Ta))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),K_1),D))),Ta))))))))) # label(fact_719_inf__period_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2586 (all X_a (linorder(X_a) -> (all X all Y all Z_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Z_1))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),Z_1))))))) # label(fact_4981_min__max_Odistrib__inf__le) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2587 (all X_b all X_a (topological_space(X_b) & metric_space(X_a) -> (all L all F all G all A_3 all R_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_2)) -> ((all X_1 (ti(X_a,A_3) != ti(X_a,X_1) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X_1),A_3)),R_2)) -> hAPP(X_a,X_b,G,X_1) = hAPP(X_a,X_b,F,X_1)))) -> (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3))))))))) # label(fact_4677_metric__LIM__equal2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2588 (all X_a (linorder(X_a) -> (all A_1 all B all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),X)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)),X)))))) # label(fact_4848_min__max_Ole__infI2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2589 (all K all Na all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Na),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),Ma)))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),Na)))) # label(fact_763_zdvd__reduce) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2590 (all X_a (comm_semiring_1(X_a) -> (all C_1 all D_3 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),C_1),D_3)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D_3)))))))) # label(fact_725_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2591 (all S_3 all T_5 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),S_3),T_5)) -> S_3 != T_5)) # label(fact_134_less__not__refl3) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2592 (all Y Y = hAPP(real,real,tan,hAPP(real,real,arctan,Y))) # label(fact_2375_tan__arctan) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2593 (all N all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N),M)) -> zero_zero(nat) = M | M = N | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)),M)))) # label(fact_887_divides__cases) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2594 (all X_a (real_normed_vector(X_a) -> (all X exists K_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2)) & (all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),X_1),X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),X_1)),K_2)))))))) # label(fact_4781_scaleR__left_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2595 (all X_a all X_b all P_1 ((all A_4 all B_4 hBOOL(hAPP(product_prod(X_a,X_b),bool,P_1,hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_4),B_4)))) <-> (all X1 hBOOL(hAPP(product_prod(X_a,X_b),bool,P_1,X1))))) # label(fact_3153_split__paired__All) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2596 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) -> hAPP(real,real,ln,Xa) = hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),inverse_divide(real),one_one(real))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),one_one(nat))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Xa),one_one(real)))),suc)))))) # label(fact_4004_ln__series) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2597 (all A_1 all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(real,nat,natfloor,X)),A_1) = hAPP(real,nat,natfloor,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(nat,real,real_1(nat),A_1))))) # label(fact_2425_natfloor__add) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2598 (all X_a (linorder(X_a) -> (all Xa all Ya all Z (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Ya),Z))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Z)))))) # label(fact_4846_min__max_Ole__inf__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2599 (all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X)),hAPP(real,real,exp(real),X)))) # label(fact_1789_exp__ge__add__one__self) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2600 (all M all N hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),N) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),M),N)),N)) # label(fact_4385_gcd__add1__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2601 (all Q_2 all P_1 ((all A_4 all B_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_4)) -> hBOOL(hAPP(rat,bool,P_1,hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_4),B_4))))) -> hBOOL(hAPP(rat,bool,P_1,Q_2)))) # label(fact_3817_Rat__induct__pos) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2602 (all X_a (semiring_div(X_a) -> (all A_1 all C_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),C_1),B)) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),B),C_1)))))) # label(fact_1092_div__mult__swap) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2603 (all A_3 all B_2 (one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_3),B_2) <-> (all D_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_2),A_3)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_2),B_2)) <-> D_2 = hAPP(nat,nat,suc,zero_zero(nat)))))) # label(fact_3321_coprime__Suc__0__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2604 (all Z1 all Z2 all W hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z1),W)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z2),W)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Z1),Z2)),W)) # label(fact_624_real__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2605 (all M all K_4 all N (hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),K_4),N) = one_one(nat) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),M))))) # label(fact_3315_coprime__dvd__mult__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2606 (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N))))))) # label(fact_253_one__le__power) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2607 (all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),N),zero_zero(nat)) = zero_zero(nat)) # label(fact_4827_min__0R) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2608 (all X_a all X_b (comm_monoid_mult(X_b) -> (all F all A_2 (-hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2) = one_one(X_b))))) # label(fact_3978_setprod__infinite) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2609 (all D_1 hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool),collect(product_prod(int,int)),hAPP(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_prod_case(int,int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),hAPP(fun(fun(int,bool),fun(int,bool)),fun(fun(int,fun(int,bool)),fun(int,fun(int,bool))),combb(fun(int,bool),fun(int,bool),int),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),D_1)))),ord_less(int)))) = hAPP(int,fun(product_prod(int,int),bool),int_ge_less_than2,D_1)) # label(fact_4101_int__ge__less__than2__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2610 (all F (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),abs_abs(real)),F)),zero_zero(real)),sequentially)) <-> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),F),zero_zero(real)),sequentially)))) # label(fact_4512_LIMSEQ__rabs__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2611 (all X all Y hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,cos,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,X)),hAPP(real,real,sin,Y)))) # label(fact_2270_cos__diff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2612 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),B))))))) # label(fact_1978_abs__leI) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2613 (all A_3 all B_2 (A_3 = B_2 <-> hAPP(rat,product_prod(int,int),quotient_of,A_3) = hAPP(rat,product_prod(int,int),quotient_of,B_2))) # label(fact_3738_quotient__of__inject__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2614 (all A_1 all B all C_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C_1)) -> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C_1)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),C_1))) # label(fact_1100_zdiv__zmult2__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2615 (all X_a all X all Y ti(X_a,Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),hAPP(bool,fun(X_a,fun(X_a,X_a)),if(X_a),fFalse),X),Y)) # label(help_If_2_1_T) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2616 (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M)),hAPP(nat,X_a,hAPP(X_a,fun(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_186_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2617 (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_713_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2618 (all X_a (ord(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]. 7.70/7.74 2619 (all P_1 all A0 all A1 (hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(product_prod(nat,nat),bool),accp(product_prod(nat,nat)),nat_gcd_rel),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A0),A1))) -> ((all X_1 all Y_1 (hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(product_prod(nat,nat),bool),accp(product_prod(nat,nat)),nat_gcd_rel),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X_1),Y_1))) -> ((Y_1 != zero_zero(nat) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,Y_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X_1),Y_1)))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,X_1),Y_1))))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,A0),A1))))) # label(fact_3753_nat__gcd_Opinduct) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2620 (all X_a (dist_norm(X_a) -> dist_dist(X_a) = ti(fun(X_a,fun(X_a,real)),dist_dist(X_a)))) # label(tsy_c_RealVector_Odist__class_Odist_0_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2621 (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),A_1),zero_zero(X_a))) -> hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) = hAPP(X_a,X_a,sgn_sgn(X_a),A_1))))) # label(fact_2316_sgn__neg) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2622 (all X_a (semiring_div(X_a) -> (all A_1 zero_zero(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),zero_zero(X_a))))) # label(fact_1074_div__by__0) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2623 (all A_1 all Q_3 all R_3 (hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),A_1),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_3),R_3))) -> (zero_zero(int) != ti(int,A_1) -> ti(int,Q_3) = one_one(int)))) # label(fact_1617_self__quotient) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2624 (all X_a (semiring_1(X_a) -> (all M all N hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),N) = hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),M),N))))) # label(fact_175_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2625 (all X_b all X_a all X_c (real_normed_vector(X_c) -> (all F all A_3 all A_2 all S ((all I_2 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I_2),S)) -> hBOOL(hAPP(filter(X_b),bool,hAPP(X_c,fun(filter(X_b),bool),hAPP(fun(X_b,X_c),fun(X_c,fun(filter(X_b),bool)),tendsto(X_b,X_c),hAPP(X_a,fun(X_b,X_c),F,I_2)),hAPP(X_a,X_c,A_3,I_2)),A_2)))) -> hBOOL(hAPP(filter(X_b),bool,hAPP(X_c,fun(filter(X_b),bool),hAPP(fun(X_b,X_c),fun(X_c,fun(filter(X_b),bool)),tendsto(X_b,X_c),hAPP(fun(X_a,bool),fun(X_b,X_c),hAPP(fun(X_b,fun(fun(X_a,bool),X_c)),fun(fun(X_a,bool),fun(X_b,X_c)),combc(X_b,fun(X_a,bool),X_c),hAPP(fun(X_b,fun(X_a,X_c)),fun(X_b,fun(fun(X_a,bool),X_c)),hAPP(fun(fun(X_a,X_c),fun(fun(X_a,bool),X_c)),fun(fun(X_b,fun(X_a,X_c)),fun(X_b,fun(fun(X_a,bool),X_c))),combb(fun(X_a,X_c),fun(fun(X_a,bool),X_c),X_b),big_co1399186613setsum(X_a,X_c)),hAPP(fun(X_a,fun(X_b,X_c)),fun(X_b,fun(X_a,X_c)),combc(X_a,X_b,X_c),F))),S)),hAPP(fun(X_a,bool),X_c,hAPP(fun(X_a,X_c),fun(fun(X_a,bool),X_c),big_co1399186613setsum(X_a,X_c),A_3),S)),A_2)))))) # label(fact_4788_tendsto__setsum) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2626 (all Ma all Na (zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Ma),Na) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)))) # label(fact_835_diff__is__0__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2627 (all X all Y (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),hAPP(real,real,sqrt,X)),hAPP(real,real,sqrt,Y))))) # label(fact_2124_real__sqrt__less__mono) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2628 (all K_4 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),one_one(int)),K_4)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),K_4)),X))))) # label(fact_1098_int__div__less__self) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2629 (all A_3 all B_2 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),Ma)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,B_2),A_3),Ma)))) # label(fact_874_zcong__sym) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2630 (all I_1 all K_4 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),J_2)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),I_1)),K_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),K_4)),I_1))) # label(fact_853_add__diff__assoc2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2631 (all X all M all Y hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),M)),Y)),M) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),Y)),M)) # label(fact_1310_zdiff__zmod__left) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2632 (all Ma all Na all K (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K)) -> (hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Ma) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Na) <-> Na = Ma))) # label(fact_696_nat__mult__eq__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2633 (all X_a (ring_1(X_a) -> (all X hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),one_one(X_a))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),one_one(X_a))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X)),one_one(X_a))))) # label(fact_776_real__squared__diff__one__factored) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2634 (all N all M hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),N),M) = hAPP(int,quickcheck_code_int,quickcheck_of_int,hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(quickcheck_code_int,int,quickcheck_int_of,N)),hAPP(quickcheck_code_int,int,quickcheck_int_of,M)))) # label(fact_3558_mod__code__int__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2635 (all D_1 (D_1 = one_one(nat) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_1),zero_zero(nat))))) # label(fact_2892_coprime__0) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2636 (all X_b all X_a all H all F_1 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),F_1)) -> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),H),F_1))))) # label(fact_3655_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2637 (all X_a (metric_space(X_a) -> (all B_2 all X_2 all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially)) -> (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),B_2),sequentially)) -> ti(X_a,A_3) = ti(X_a,B_2)))))) # label(fact_4464_LIMSEQ__unique) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2638 (all N all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_1),B)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A_1),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),N))))) # label(fact_2875_coprime__exp__imp) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2639 (all X_a all Y_2 hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,X_a),fun(fun(X_a,bool),fun(X_a,bool)),image(X_a,X_a),combi(X_a)),Y_2) = ti(fun(X_a,bool),Y_2)) # label(fact_3874_image__ident) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2640 (all X_a (linord20386208strict(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),zero_zero(X_a)))))))) # label(fact_445_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2641 (all Xa all Ya (hBOOL(hAPP(int,bool,even_odd_even(int),Xa)) & hBOOL(hAPP(int,bool,even_odd_even(int),Ya)) | -hBOOL(hAPP(int,bool,even_odd_even(int),Ya)) & -hBOOL(hAPP(int,bool,even_odd_even(int),Xa)) <-> hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xa),Ya))))) # label(fact_1601_even__difference) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2642 (all Na all Ma all F ((all N_1 (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,hAPP(nat,nat,suc,N_1))),hAPP(nat,nat,F,N_1))) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,N_1)),hAPP(nat,nat,F,hAPP(nat,nat,suc,N_1)))))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),Ma)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,Na)),hAPP(nat,nat,F,Ma))) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,Ma)),hAPP(nat,nat,F,Na)))))) # label(fact_1569_dvd_Olift__Suc__mono__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2643 (all X_a (cancel_semigroup_add(X_a) -> ti(fun(X_a,fun(X_a,X_a)),plus_plus(X_a)) = plus_plus(X_a))) # label(tsy_c_Groups_Oplus__class_Oplus_3_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2644 (all M all N all K_4 hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)),K_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),K_4))) # label(fact_127_nat__add__assoc) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2645 (all X_a (inverse(X_a) & real_normed_vector(X_a) -> (all F all A_3 all D (hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),inverse_divide(X_a)),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),minus_minus(X_a)),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),F),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3)))),hAPP(X_a,X_a,F,A_3)))),combi(X_a))),D),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a)))) <-> hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),inverse_divide(X_a)),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),minus_minus(X_a)),F)),hAPP(X_a,X_a,F,A_3)))),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),minus_minus(X_a)),A_3))),D),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4632_DERIV__LIM__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2646 (all X_a (ordere142940540dd_abs(X_a) -> hAPP(X_a,X_a,abs_abs(X_a),zero_zero(X_a)) = zero_zero(X_a))) # label(fact_970_abs__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2647 (all X_a (linorder(X_a) -> (all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ya),Xa)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya)) <-> ti(X_a,Ya) != ti(X_a,Xa))))) # label(fact_535_linorder__neq__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2648 (all X hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = X) # label(fact_1806_real__sum__of__halves) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2649 (all K 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_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K)),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K)),Na))))) # label(fact_1397_Suc__mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2650 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) -> zero_zero(real) != hAPP(real,real,sqrt,X))) # label(fact_2143_real__sqrt__not__eq__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2651 (all X_a all P_3 all X_1 (hBOOL(hAPP(X_a,bool,P_3,X_1)) -> hBOOL(hAPP(X_a,bool,P_3,hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_3))))) # label(fact_4809_tfl__some) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2652 (all N all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),N))) # label(fact_3299_gcd__diff1__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2653 (all X_a all A_2 all B_1 (hBOOL(hAPP(filter(X_a),bool,hAPP(filter(X_a),fun(filter(X_a),bool),ord_less_eq(filter(X_a)),A_2),B_1)) & -hBOOL(hAPP(filter(X_a),bool,hAPP(filter(X_a),fun(filter(X_a),bool),ord_less_eq(filter(X_a)),B_1),A_2)) <-> hBOOL(hAPP(filter(X_a),bool,hAPP(filter(X_a),fun(filter(X_a),bool),ord_less(filter(X_a)),A_2),B_1)))) # label(fact_4535_less__filter__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2654 (all X_a (division_ring(X_a) -> (all B all A_1 (ti(X_a,A_1) != zero_zero(X_a) -> (zero_zero(X_a) != ti(X_a,B) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),A_1))),hAPP(X_a,X_a,inverse_inverse(X_a),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B))))))) # label(fact_2619_division__ring__inverse__diff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2655 (all B hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,hAPP(int,int,number_number_of(int),min)),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,number_number_of(int),min)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),one_one(int)))) # label(fact_1636_negDivAlg__minus1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2656 (all B all A_1 (zero_zero(int) != ti(int,A_1) | ti(int,B) != zero_zero(int) -> (exists C_4 exists D_2 (hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,C_4),D_2) = one_one(int) & ti(int,B) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B)),D_2) & hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B)),C_4) = ti(int,A_1))))) # label(fact_3509_make__zrelprime) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2657 (all Xa hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),fact_fact(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),Xa)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) # label(fact_4061_exp__tail__after__first__two__terms__summable) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2658 (all Q all P (hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fimplies,P),Q)) | hBOOL(P))) # label(help_fimplies_1_1_U) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2659 (all X all Y all Z_1 hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),X),Z_1))) # label(fact_126_nat__add__left__commute) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2660 (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)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Ya),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) <-> ti(X_a,Ya) != zero_zero(X_a) | zero_zero(X_a) != ti(X_a,Xa))))) # label(fact_100_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2661 (all X_a ti(fun(fun(X_a,bool),bool),fEx(X_a)) = fEx(X_a)) # label(tsy_c_fEx_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2662 (all M hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),zero_zero(nat)) = zero_zero(nat)) # label(fact_603_mult__0__right) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2663 (all X_a (order(X_a) -> monoseq(X_a) = ti(fun(fun(nat,X_a),bool),monoseq(X_a)))) # label(tsy_c_SEQ_Omonoseq_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2664 (all X_a (linordered_field(X_a) -> (all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a))))))))) # label(fact_1886_less__half__sum) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2665 (all Y all X 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_eq(real),zero_zero(real)),X)) -> (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),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),Y))))))) # label(fact_2709_real__root__less__mono__lemma) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2666 (all Na all Xa all Diff all F (hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F & (all M_2 all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_2)),X_1),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),X_1)))) -> (exists T_4 (hAPP(real,real,F,Xa) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,Na),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),Na))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),T_4)),hAPP(real,real,abs_abs(real),Xa))))))) # label(fact_4313_Maclaurin__all__le__objl) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2667 (all X_a (real_algebra(X_a) -> (all A_1 all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)),Y) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y))))) # label(fact_4721_mult__scaleR__left) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2668 (all X all Y hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(rat,real,ratreal,X)),hAPP(rat,real,ratreal,Y)) = hAPP(rat,real,ratreal,hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),X),Y))) # label(fact_3789_real__plus__code) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2669 (all X all N hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,real_1(int),X)),N) = hAPP(int,real,real_1(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),N))) # label(fact_3362_power__real__of__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2670 (all F ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,F,N_1)))) -> (hBOOL(hAPP(fun(nat,real),bool,summable(real),F)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(fun(nat,real),real,suminf(real),F)))))) # label(fact_4099_suminf__0__le) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2671 (all X_a (comm_ring_1(X_a) -> (all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Xa),Ya)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Xa)),Ya)))))) # label(fact_1919_minus__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2672 (all X_b all X_a (comm_monoid_add(X_a) -> (all H all G all S all R_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_2,zero_zero(X_a)),zero_zero(X_a))) -> ((all X1 all Y1 all X2 all Y2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_2,X1),X2)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_2,Y1),Y2)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_2,hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X1),Y1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X2),Y2))))) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),S)) -> ((all X_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),S)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_2,hAPP(X_b,X_a,H,X_1)),hAPP(X_b,X_a,G,X_1))))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_2,hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),H),S)),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),G),S)))))))))) # label(fact_4820_setsum_Orelated) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2673 (all X all Y (X = Y -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)))) # label(fact_956_dvd_Oeq__refl) # label(axiom) # label(non_clause). [assumption]. 7.70/7.74 2674 (all Z hAPP(complex,real,arg,Z) = hAPP(fun(real,bool),real,hilbert_Eps(real),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(fun(real,real),fun(real,bool),hAPP(fun(real,bool),fun(fun(real,real),fun(real,bool)),combb(real,bool,real),hAPP(real,fun(real,bool),fequal(real),hAPP(complex,real,re,hAPP(complex,complex,sgn_sgn(complex),Z)))),cos))),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(fun(real,real),fun(real,bool),hAPP(fun(real,bool),fun(fun(real,real),fun(real,bool)),combb(real,bool,real),hAPP(real,fun(real,bool),fequal(real),hAPP(complex,real,im,hAPP(complex,complex,sgn_sgn(complex),Z)))),sin))),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),pi)))),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),ord_less_eq(real)),pi)))))) # label(fact_3953_arg__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2675 (all Z_1 hAPP(complex,complex,cnj,hAPP(int,complex,ring_1_of_int(complex),Z_1)) = hAPP(int,complex,ring_1_of_int(complex),Z_1)) # label(fact_3601_complex__cnj__of__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2676 (all X_a (preorder(X_a) -> (all Z_1 all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),Z_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Z_1))))))) # label(fact_553_order__le__less__trans) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2677 (all Ya all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Ya)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(int,nat,nat_1,Xa)),hAPP(int,nat,nat_1,Ya))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Xa),Ya)))))) # label(fact_1028_transfer__nat__int__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2678 (all V_2 all V ((-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V))) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_2)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,int,number_number_of(int),V_2)))) & (hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V))) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_2)) = zero_zero(nat)))) # label(fact_1633_div__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2679 (all B all D_3 all A_1 (hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_3),A_1) = one_one(int) -> (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_3),B) -> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_3),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)) = one_one(int)))) # label(fact_4414_coprime__mult__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2680 (all X_a (number_ring(X_a) -> (all W hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),W)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),hAPP(int,X_a,number_number_of(X_a),W))))) # label(fact_6_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2681 (all X_a (real_normed_field(X_a) & banach(X_a) -> (all G all Xa all Ma (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),Xa),Ma)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),exp(X_a)),G)),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,exp(X_a),hAPP(X_a,X_a,G,Xa))),Ma))))))) # label(fact_3884_DERIV__fun__exp) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2682 (all X_a all X_b all A_1 all B hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_1),B)) = ti(X_a,A_1)) # label(fact_3158_fst__conv) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2683 (all X_a (real_normed_field(X_a) -> (all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),Y) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y))))) # label(fact_1939_divide_Ominus) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2684 (all X_a all P_1 all A_3 (hBOOL(hAPP(X_a,bool,P_1,A_3)) -> ((all X_1 (hBOOL(hAPP(X_a,bool,P_1,X_1)) -> ti(X_a,X_1) = ti(X_a,A_3))) -> hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1) = ti(X_a,A_3)))) # label(fact_4814_some__equality) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2685 (all M all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,suc,M)),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),hAPP(nat,nat,suc,N))) # label(fact_1336_add__Suc__shift) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2686 (all X_a all X_b all A_2 all B_2 all F all Xa (ti(X_a,B_2) = hAPP(X_b,X_a,F,Xa) -> (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Xa),A_2)) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2)))))) # label(fact_3654_image__eqI) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2687 (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),pls)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Xa)),zero_zero(X_a))))))) # label(fact_49_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2688 (all M M = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),M)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) # label(fact_1173_add__self__div__2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2689 (all X_a (comm_ring_1(X_a) -> (all X hAPP(X_a,X_a,uminus_uminus(X_a),X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),X)))) # label(fact_1966_comm__ring__1__class_Onormalizing__ring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2690 (all P_1 all A_3 all B_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2)) -> (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),minus_minus(real)),hAPP(fun(nat,product_prod(real,real)),fun(nat,real),hAPP(fun(product_prod(real,real),real),fun(fun(nat,product_prod(real,real)),fun(nat,real)),combb(product_prod(real,real),real,nat),product_fst(real,real)),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2)))),hAPP(fun(nat,product_prod(real,real)),fun(nat,real),hAPP(fun(product_prod(real,real),real),fun(fun(nat,product_prod(real,real)),fun(nat,real)),combb(product_prod(real,real),real,nat),product_snd(real,real)),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2)))),zero_zero(real)),sequentially)) -> (exists L_3 ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(product_prod(real,real),real,product_fst(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),N_1))),L_3))) & (all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),L_3),hAPP(product_prod(real,real),real,product_snd(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),N_1))))) & hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,product_prod(real,real)),fun(nat,real),hAPP(fun(product_prod(real,real),real),fun(fun(nat,product_prod(real,real)),fun(nat,real)),combb(product_prod(real,real),real,nat),product_snd(real,real)),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2))),L_3),sequentially)) & hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,product_prod(real,real)),fun(nat,real),hAPP(fun(product_prod(real,real),real),fun(fun(nat,product_prod(real,real)),fun(nat,real)),combb(product_prod(real,real),real,nat),product_fst(real,real)),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2))),L_3),sequentially))))))))) # label(fact_4539_Bolzano__nest__unique) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2691 (all X_a (ab_group_add(X_a) -> (all A_1 zero_zero(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),A_1)))) # label(fact_1960_ab__left__minus) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2692 (all X_a (linordered_field(X_a) -> (all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)))))))) # label(fact_1839_divide__neg__neg) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2693 (all X_b all X_a (real_algebra_1(X_a) & real_normed_vector(X_a) -> (all G all A_2 hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,real),fun(X_b,X_a),hAPP(fun(real,X_a),fun(fun(X_b,real),fun(X_b,X_a)),combb(real,X_a,X_b),of_real(X_a)),G)),A_2) = hAPP(real,X_a,of_real(X_a),hAPP(fun(X_b,bool),real,hAPP(fun(X_b,real),fun(fun(X_b,bool),real),big_co1399186613setsum(X_b,real),G),A_2))))) # label(fact_4211_of__real_Osetsum) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2694 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,fact_fact(nat),N)),hAPP(nat,nat,fact_fact(nat),M)) = zero_zero(nat))) # label(fact_2554_fact__mod) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2695 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),min)),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),arccos),Xa),hAPP(real,real,inverse_inverse(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))))))) # label(fact_2642_DERIV__arccos) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2696 (all X_a (semiring_div(X_a) -> (all D_1 all B_2 all C all A_3 (ti(X_a,A_3) != zero_zero(X_a) -> (ti(X_a,C) != zero_zero(X_a) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_3),B_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),C),D_1)) -> (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),D_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),C) <-> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),B_2),A_3) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),D_1),C))))))))) # label(fact_1120_dvd__div__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2697 (all N all D_3 all A_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_3),A_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A_1),N))))) # label(fact_2874_coprime__exp) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2698 (all X_a all D all B_1 all A_2 all C_2 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),C_2)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),D),B_1)) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),C_2),D)))))) # label(fact_2212_Diff__mono) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2699 (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)),hAPP(X_a,X_a,abs_abs(X_a),A_1)))))) # label(fact_986_abs__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2700 (all X hAPP(complex,complex,cnj,hAPP(complex,complex,inverse_inverse(complex),X)) = hAPP(complex,complex,inverse_inverse(complex),hAPP(complex,complex,cnj,X))) # label(fact_2965_complex__cnj__inverse) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2701 (all X_a (real_normed_field(X_a) -> (all E_3 all F all Xa all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),E_3)) -> ti(X_a,D) = ti(X_a,E_3)))))) # label(fact_2648_DERIV__unique) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2702 (all X_a (ordered_ab_group_add(X_a) -> ti(fun(X_a,fun(X_a,fun(X_a,bool))),ord_gr676655987ssThan(X_a)) = ord_gr676655987ssThan(X_a))) # label(tsy_c_SetInterval_Oord__class_OgreaterThanLessThan_2_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2703 (all B all A_1 all X (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),X)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),X)))) # label(fact_3259_gcd__semilattice__nat_Ole__infI1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2704 (all R_3 all X all Y hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),X)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),Y)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,of_real(complex),R_3)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,X),Y))) # label(fact_2816_complex__of__real__mult__Complex) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2705 (all Y all X (hBOOL(hAPP(int,bool,nat_is_nat,X)) -> (hBOOL(hAPP(int,bool,nat_is_nat,Y)) -> hBOOL(hAPP(int,bool,nat_is_nat,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),Y)))))) # label(fact_4866_Nat__Transfer_Otransfer__int__nat__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2706 (all A_1 all B all P_5 all Q_3 hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(product_prod(int,int),int,twoSqu1094008024sum2sq,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_1),B))),hAPP(product_prod(int,int),int,twoSqu1094008024sum2sq,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_5),Q_3))) = hAPP(product_prod(int,int),int,twoSqu1094008024sum2sq,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),P_5)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_3))),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),Q_3)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),P_5))))) # label(fact_1009_mult__sum2sq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2707 (all L all U hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_gr676655987ssThan(nat),L),U)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),U),hAPP(nat,nat,suc,L))) # label(fact_3701_card__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2708 (all X_a (real_normed_vector(X_a) -> (all A_1 all B all C_1 all D_3 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),D_3)))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),C_1))),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),D_3)))))))) # label(fact_2029_norm__diff__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2709 (all A_2 all Ma (hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),A_2),hAPP(int,fun(fun(int,bool),bool),rsetR,Ma))) & hAPP(fun(int,bool),nat,finite_card(int),A_2) = hAPP(int,nat,phi,Ma) <-> hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),Ma)))) # label(fact_3652_is__RRset__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2710 (all X_a all X_b all Xa all Ya hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),the(product_prod(X_a,X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),hAPP(fun(X_b,bool),fun(X_a,fun(X_b,bool)),hAPP(fun(X_a,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(X_a,fun(X_b,bool))),combc(X_a,fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(bool,bool)),fun(X_a,fun(fun(X_b,bool),fun(X_b,bool))),hAPP(fun(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_a,fun(bool,bool)),fun(X_a,fun(fun(X_b,bool),fun(X_b,bool)))),combb(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),X_a),combb(bool,bool,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(X_a,fun(X_a,bool),fequal(X_a),Xa)))),hAPP(X_b,fun(X_b,bool),fequal(X_b),Ya)))) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),Xa),Ya)) # label(fact_4170_The__split__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2711 (all Xa (Xa = hAPP(int,real,real_1(int),hAPP(real,int,archim2085082626_floor(real),Xa)) <-> (exists N_1 hAPP(int,real,real_1(int),N_1) = Xa))) # label(fact_3351_real__of__int__floor__cancel) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2712 (all K_4 all I_1 all J_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),I_1),J_2)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),J_2),K_4)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),I_1),K_4))))) # label(fact_597_real__le__trans) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2713 (all X_a (ordered_ab_group_add(X_a) -> (all A_3 all B_2 all C all D_1 (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),C),D_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_1)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2))))))) # label(fact_738_diff__eq__diff__less) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2714 (all X_a (division_ring(X_a) -> (all A_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),one_one(X_a)),A_1) = hAPP(X_a,X_a,inverse_inverse(X_a),A_1)))) # label(fact_2602_inverse__eq__divide) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2715 (all Y all X all A_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1)) -> (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),X),Y)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,Y),A_1))))))) # label(fact_3373_powr__less__mono2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2716 (all Na all Ma hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),hAPP(nat,nat,suc,Na)),Ma) = hAPP(nat,nat,hAPP(fun(nat,nat),fun(nat,nat),hAPP(nat,fun(fun(nat,nat),fun(nat,nat)),nat_case(nat),hAPP(nat,nat,suc,Na)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),suc),hAPP(nat,fun(nat,nat),ord_max(nat),Na))),Ma)) # label(fact_4988_max__Suc1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2717 (all I_1 all J_2 hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,uminus_uminus(int),I_1)),J_2) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I_1),J_2)) # label(fact_3468_zgcd__zminus) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2718 (all X_a (ring_1(X_a) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),one_one(X_a)),ring_1_Ints(X_a))))) # label(fact_3825_Ints__1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2719 (all X_a (monoid_mult(X_a) -> (all A_1 all M all N hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M)),N)))) # label(fact_608_power__mult) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2720 (all N all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),N)) -> one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,N),P_5)))) # label(fact_3491_zprime__imp__zrelprime) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2721 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)),zero_zero(int))) -> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) = hAPP(int,int,number_number_of(int),min)))) # label(fact_1127_div__pos__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2722 (all P_5 (hBOOL(hAPP(nat,bool,prime,P_5)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),P_5)))) # label(fact_3076_prime__g__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2723 (all X_a (real_vector(X_a) -> (all X hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(int,real,number_number_of(real),min)),X) = hAPP(X_a,X_a,uminus_uminus(X_a),X)))) # label(fact_4690_scaleR__minus1__left) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2724 (all X_a (real_normed_field(X_a) -> (all Y_2 all B_2 all X_2 all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),X_2),A_3)) -> (hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),Y_2),B_2)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),plus_plus(X_a)),X_2)),Y_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_2)))))))) # label(fact_4115_sums__add) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2725 (all X_a (ring_1(X_a) -> ring_1_of_int(X_a) = ti(fun(int,X_a),ring_1_of_int(X_a)))) # label(tsy_c_Int_Oring__1__class_Oof__int_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2726 (all X_a (archim1727834104eiling(X_a) -> (all Z all Xa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)),Xa)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),hAPP(X_a,int,archim2085082626_floor(X_a),Xa))))))) # label(fact_3618_le__floor__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2727 (all A_3 all 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(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_3),B_2)),zero_zero(int))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),zero_zero(int)))))) # label(fact_1078_pos__imp__zdiv__neg__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2728 (all X_a (division_ring(X_a) -> (all A_1 all B (zero_zero(X_a) != ti(X_a,B) -> hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)))))) # label(fact_1972_nonzero__minus__divide__right) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2729 (all P_1 ((exists X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) & hBOOL(hAPP(int,bool,P_1,X_1)))) <-> (exists X_1 hBOOL(hAPP(int,bool,P_1,hAPP(nat,int,semiring_1_of_nat(int),X_1)))))) # label(fact_266_transfer__int__nat__quantifiers_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2730 (all N all M -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),N)),hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),M))))) # label(fact_2075_not__int__zless__negative) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2731 (all F all Xa all L (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),Xa),L)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),L)) -> (exists D_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_2)) & (all H_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H_1)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),H_1),D_2)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,Xa)),hAPP(real,real,F,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa),H_1)))))))))))) # label(fact_2666_DERIV__pos__inc__right) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2732 (all X_c all X_b all X_a (real_normed_vector(X_a) & topological_space(X_b) & topological_space(X_c) -> (all G all C all F all B_2 all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),B_2),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> (hBOOL(hAPP(filter(X_b),bool,hAPP(X_c,fun(filter(X_b),bool),hAPP(fun(X_b,X_c),fun(X_c,fun(filter(X_b),bool)),tendsto(X_b,X_c),G),C),hAPP(X_b,filter(X_b),at(X_b),B_2))) -> ((exists D_2 ((all X_1 (ti(X_a,A_3) != ti(X_a,X_1) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_1),A_3))),D_2)) -> ti(X_b,B_2) != hAPP(X_a,X_b,F,X_1))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_2)))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_c,fun(filter(X_a),bool),hAPP(fun(X_a,X_c),fun(X_c,fun(filter(X_a),bool)),tendsto(X_a,X_c),hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c)),combb(X_b,X_c,X_a),G),F)),C),hAPP(X_a,filter(X_a),at(X_a),A_3))))))))) # label(fact_4633_LIM__compose2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2733 (all X_b all X_a (comm_monoid_add(X_a) -> (all F all G all A_2 hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,X_a)),combs(X_b,X_a,X_a),hAPP(fun(X_b,X_a),fun(X_b,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_b),plus_plus(X_a)),F)),G)),A_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2)),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),G),A_2))))) # label(fact_4248_setsum__addf) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2734 (all X_a (real_normed_vector(X_a) -> (all Ra all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2)) -> hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra)),X_2)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra),hAPP(fun(nat,X_a),X_a,suminf(X_a),X_2)))))) # label(fact_4744_scaleR__right_Osuminf) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2735 (all K all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),Na)) | zero_zero(nat) = K <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Na))))) # label(fact_842_nat__mult__dvd__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2736 (all X_a (number_ring(X_a) & linord1117847801e_zero(X_a) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),one_one(X_a)),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) # label(fact_1813_half) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2737 (all B all A_1 all C_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),C_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),A_1))))) # label(fact_886_Euler_Oaux2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2738 (all X_a (linordered_field(X_a) -> (all C_1 all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C_1)))))))) # label(fact_1837_divide__strict__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2739 (all K_4 all L_4 hAPP(int,int,bit0,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),L_4)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,bit0,K_4)),L_4)) # label(fact_399_mult__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2740 (all X_a (monoid_mult(X_a) -> (all A_1 all N hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N))))) # label(fact_393_power__commutes) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2741 (all X_a all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2)) <-> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2)),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),B_1))))) # label(fact_2196_Compl__subset__Compl__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2742 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,cos,X)))))) # label(fact_2343_cos__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2743 (all X hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,suc,X)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,X)),hAPP(nat,nat,fact_fact(nat),X))) # label(fact_2546_Fact_Ofact__Suc) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2744 (all N (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N)) -> hAPP(int,int,hAPP(int,fun(int,int),times_times(int),N),hAPP(int,int,fact_fact(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),N),one_one(int)))) = hAPP(int,int,fact_fact(int),N))) # label(fact_3227_fact__reduce__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2745 (all X_a (real_normed_field(X_a) -> (all H_2 all B all A_1 (ti(X_a,A_1) != zero_zero(X_a) -> (ti(X_a,B) != zero_zero(X_a) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B))),H_2) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),H_2))),hAPP(X_a,X_a,inverse_inverse(X_a),B)))))))) # label(fact_2638_DERIV__inverse__lemma) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2746 (all X_a (linordered_field(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)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1))))))) # label(fact_2592_positive__imp__inverse__positive) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2747 (all A_1 all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)))))) # label(fact_3402_ge__one__powr__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2748 (all A_3 all B_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),B_2)) -> -hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),d22set,A_3))))) # label(fact_1221_d22set__le__swap) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2749 (all A_2 (hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2)) -> hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2)) = A_2)) # label(fact_3688_transfer__nat__int__set__return__embed) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2750 (all Z_1 hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Z_1),Z_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Z_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) # label(fact_647_nat__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2751 (all X_a (real_normed_field(X_a) -> (all F all Xa all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),uminus_uminus(X_a)),F)),Xa),hAPP(X_a,X_a,uminus_uminus(X_a),D))))))) # label(fact_3891_DERIV__minus) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2752 (all X_a all X_b (comm_semiring_1(X_b) -> (all F all G all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) -> ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),dvd_dvd(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,G,X_1))))) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),dvd_dvd(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),B_1))))))))) # label(fact_3992_setprod__dvd__setprod__subset2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2753 (all Q_5 all R_4 all A_1 all B all Q_3 all R_3 (hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_3),R_3))) -> (hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_5),R_4))) -> (zero_zero(int) != ti(int,B) -> ti(int,Q_5) = ti(int,Q_3))))) # label(fact_1613_unique__quotient) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2754 (all X_a (linorder(X_a) -> (all T_5 exists Z_2 all X_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_2),X_1)) -> -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),T_5)))))) # label(fact_2857_pinf_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2755 (all Z_1 ti(int,Z_1) = hAPP(int,int,uminus_uminus(int),hAPP(int,int,uminus_uminus(int),Z_1))) # label(fact_2057_zminus__zminus) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2756 (all X_a (preorder(X_a) -> (all X all 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_516_order__less__asym) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2757 (all X_a all A_2 all B_1 hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)),B_1) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)) # label(fact_2243_Diff__idemp) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2758 (all X_a (comm_semiring_1(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C_1)))))))) # label(fact_730_dvd__add) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2759 (all Ma all Xa ((exists Y_1 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Y_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),Xa),Ma))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,Ma),Xa)))) # label(fact_795_QuadRes__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2760 (all X_a (banach(X_a) & real_normed_field(X_a) -> (all Z all F all Xa (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),F)),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa)))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),Z)),hAPP(X_a,real,norm_norm(X_a),Xa))) -> hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),F)),hAPP(X_a,fun(nat,X_a),power_power(X_a),Z)))))))))) # label(fact_4024_powser__insidea) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2761 (all A_1 all B hAPP(complex,complex,inverse_inverse(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A_1),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,uminus_uminus(real),B)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) # label(fact_2768_complex__inverse) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2762 (all X_a all A_2 hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2) = hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(bool,bool),fun(fun(X_a,bool),fun(X_a,bool)),combb(bool,bool,X_a),fNot),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),A_2)))) # label(fact_2240_Compl__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2763 (all N zero_zero(real) = hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),pi))) # label(fact_2376_tan__npi) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2764 (all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1)) -> -(all M_2 hAPP(nat,int,semiring_1_of_nat(int),M_2) != ti(int,Z_1)))) # label(fact_343_nonneg__eq__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2765 (all X all Y hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),X),Y) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),X),hAPP(complex,complex,uminus_uminus(complex),Y))) # label(fact_2056_complex__diff__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2766 (all K all L hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K),L) = hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),hAPP(bool,fun(int,fun(int,int)),if(int),hAPP(int,bool,hAPP(int,fun(int,bool),fequal(int),L),zero_zero(int))),K),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,L),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,abs_abs(int),K)),hAPP(int,int,abs_abs(int),L)))))) # label(fact_3490_zgcd__code) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2767 (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_110_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2768 (all Y exists X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X_1)) & Y = hAPP(real,real,tan,X_1) & (all Y_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) & Y = hAPP(real,real,tan,Y_1) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),Y_1)) -> Y_1 = X_1)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) # label(fact_2459_tan__total) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2769 (all X all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),Y)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,sin,Y)),hAPP(real,real,sin,X))))))) # label(fact_2338_sin__monotone__2pi_H) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2770 (all Xa all P_4 (hAPP(int,int,hAPP(int,fun(int,int),standardRes,P_4),Xa) != zero_zero(int) <-> -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Xa),zero_zero(int)),P_4)))) # label(fact_1416_StandardRes__prop3) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2771 (all X_a (idom(X_a) -> (all C all A_3 all B_2 (ti(X_a,C) = zero_zero(X_a) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_3),B_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B_2))))))) # label(fact_747_dvd__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2772 (all X_a (number_ring(X_a) -> (all Z_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),hAPP(int,X_a,number_number_of(X_a),min)) = hAPP(X_a,X_a,uminus_uminus(X_a),Z_1)))) # label(fact_2014_mult__minus1__right) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2773 (all K hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(nat,fun(nat,bool),ord_atMost(nat),K)))) # label(fact_4960_finite__atMost) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2774 (all N all M hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)))) # label(fact_409_le__add2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2775 (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)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),Y)))))) # label(fact_1081_Divides_Otransfer__nat__int__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2776 (all X hAPP(complex,complex,uminus_uminus(complex),hAPP(complex,complex,cnj,X)) = hAPP(complex,complex,cnj,hAPP(complex,complex,uminus_uminus(complex),X))) # label(fact_2958_cnj_Ominus) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2777 (all A_1 all B (zero_zero(int) != ti(int,B) -> hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,A_1),B))))) # label(fact_3100_divmod__int__correct) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2778 (all Ya hAPP(real,real,arccos,Ya) = hAPP(fun(real,bool),real,the(real),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)))),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),ord_less_eq(real)),pi))),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(real,real),fun(real,fun(real,bool))),combb(real,fun(real,bool),real),fequal(real)),cos)),Ya))))) # label(fact_4164_arccos__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2779 (all M hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),zero_zero(int)) = hAPP(int,int,abs_abs(int),M)) # label(fact_3478_zgcd__0) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2780 (all Ma all D_1 ((exists Q_4 hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),D_1),Q_4) = Ma) <-> zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Ma),D_1))) # label(fact_1534_mod__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2781 (all X_a (ordere142940540dd_abs(X_a) -> (all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_3)),B_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)),B_2)))))) # label(fact_1977_abs__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2782 (all X_a (ring_char_0(X_a) -> (all A_3 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),ring_1_Ints(X_a))) -> (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),A_3) = zero_zero(X_a) <-> zero_zero(X_a) = ti(X_a,A_3)))))) # label(fact_3835_Ints__double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2783 (all X_a (number(X_a) & field_inverse_zero(X_a) -> (all B_2 all C all Wa ((ti(X_a,C) = zero_zero(X_a) -> hAPP(int,X_a,number_number_of(X_a),Wa) = zero_zero(X_a)) & (ti(X_a,C) != zero_zero(X_a) -> ti(X_a,B_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),C)) <-> hAPP(int,X_a,number_number_of(X_a),Wa) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C))))) # label(fact_1778_divide__eq__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2784 (all X_a (ordered_ring(X_a) -> (all C_1 all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B)))))))) # label(fact_435_mult__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2785 (all A_3 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),Ma)) -> hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,A_3),Ma)),hAPP(int,fun(fun(int,bool),bool),rsetR,Ma))))) # label(fact_3667_Bnor__in__RsetR) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2786 (all V ((hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V))) -> one_one(nat) = hAPP(nat,nat,suc,hAPP(int,nat,number_number_of(nat),V))) & (-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V))) -> hAPP(int,nat,number_number_of(nat),hAPP(int,int,succ,V)) = hAPP(nat,nat,suc,hAPP(int,nat,number_number_of(nat),V))))) # label(fact_1627_Suc__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2787 (all X_a (comm_semiring_1(X_a) -> (all A_1 all C_1 all D_3 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),D_3)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),D_3))))) # label(fact_300_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2788 (all X_a (real_n2089651433ebra_1(X_a) -> hAPP(X_a,real,norm_norm(X_a),one_one(X_a)) = one_one(real))) # label(fact_1980_norm__one) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2789 (all Xa hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),sin),Xa))) # label(fact_4927_isCont__sin) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2790 (all Xa all Wa (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Xa),hAPP(int,nat,number_number_of(nat),Wa)))) <-> zero_zero(nat) = hAPP(int,nat,number_number_of(nat),Wa) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Xa)))) # label(fact_152_zero__less__power__nat__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2791 (all X_a (real_normed_field(X_a) -> (all F all Xa all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_a),fun(X_a,bool),isCont(X_a,X_a),F),Xa)))))) # label(fact_4926_DERIV__isCont) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2792 (all Y all X (hBOOL(hAPP(int,bool,nat_is_nat,X)) -> (hBOOL(hAPP(int,bool,nat_is_nat,Y)) -> hBOOL(hAPP(int,bool,nat_is_nat,hAPP(int,int,hAPP(int,fun(int,int),nat_tsub,X),Y)))))) # label(fact_4872_Nat__Transfer_Otransfer__int__nat__function__closures_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2793 (all X_2 all A_3 (hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),X_2),A_3)) -> hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),im),X_2)),hAPP(complex,real,im,A_3))))) # label(fact_4142_Im_Osums) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2794 (all X_b all X_a (topological_space(X_a) & power(X_b) & real_normed_algebra(X_b) -> (all Na all F all L all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(nat,fun(X_a,X_b),hAPP(fun(X_a,fun(nat,X_b)),fun(nat,fun(X_a,X_b)),combc(X_a,nat,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(nat,X_b)),hAPP(fun(X_b,fun(nat,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(nat,X_b))),combb(X_b,fun(nat,X_b),X_a),power_power(X_b)),F)),Na)),hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),L),Na)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4585_LIM__power) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2795 (all X_a (ordered_ring(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)))))))) # label(fact_430_mult__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2796 (all X hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,uminus_uminus(real),hAPP(complex,real,im,X))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,re,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,im,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(complex,real,im,hAPP(complex,complex,inverse_inverse(complex),X))) # label(fact_3017_complex__Im__inverse) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2797 (all A_3 all P_4 hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,A_3),P_4)))) # label(fact_2729_wset__fin) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2798 (all X all Y hAPP(complex,complex,inverse_inverse(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(real,complex,of_real(complex),X)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),ii),hAPP(real,complex,of_real(complex),Y)))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(real,complex,of_real(complex),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),ii),hAPP(real,complex,of_real(complex),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),Y),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))))) # label(fact_2788_complex__inverse__complex__split) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2799 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,K)),min)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),min)))) # label(fact_688_rel__simps_I11_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2800 (all K_4 all I_1 all J_2 (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I_1),J_2) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),I_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),J_2))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),I_1),K_4))))) # label(fact_3488_zrelprime__dvd__mult) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2801 (all K_4 hAPP(product_prod(int,int),rat,frct,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),K_4),zero_zero(int))) = zero_zero(rat)) # label(fact_3726_Frct__code__post_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2802 (all N hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),minus_minus(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,N)),one_one(code_code_numeral)) = N) # label(fact_2980_Suc__code__numeral__minus__one) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2803 (all X_a (number_ring(X_a) -> -hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),min))))) # label(fact_2158_nonzero__number__of__Min) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2804 (all A_3 all B_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,B_2),A_3)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_3),B_2)))) # label(fact_2876_coprime__commute) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2805 (all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (hBOOL(hAPP(nat,bool,even_odd_even(nat),Na)) <-> -hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))))))) # label(fact_2514_even__num__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2806 (all P_4 hAPP(int,fun(int,bool),sRStar,P_4) = hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),P_4)))) # label(fact_1545_SRStar__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2807 (all Xa all P_4 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),sRStar,P_4))) -> ti(int,Xa) = hAPP(int,int,hAPP(int,fun(int,int),standardRes,P_4),Xa))) # label(fact_1530_StandardRes__SRStar__prop3) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2808 (all X_a (archim1727834104eiling(X_a) -> archim2085082626_floor(X_a) = ti(fun(X_a,int),archim2085082626_floor(X_a)))) # label(tsy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2809 (all S ((all M_2 exists N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),N_1)) & hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),N_1),S)))) <-> -hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),S)))) # label(fact_3681_infinite__nat__iff__unbounded) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2810 (all X hAPP(real,real,sqrt,hAPP(real,real,inverse_inverse(real),X)) = hAPP(real,real,inverse_inverse(real),hAPP(real,real,sqrt,X))) # label(fact_2562_real__sqrt__inverse) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2811 (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_706_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2812 (all X_a (archim1727834104eiling(X_a) -> hAPP(X_a,int,archim2085082626_floor(X_a),one_one(X_a)) = one_one(int))) # label(fact_2896_floor__one) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2813 (all Ma all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),Ma)),Z)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),hAPP(int,nat,nat_1,Z))))) # label(fact_1026_zless__nat__eq__int__zless) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2814 (all X_a (archim1727834104eiling(X_a) -> (all Z_1 hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)) = ti(int,Z_1)))) # label(fact_3597_floor__of__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2815 (all X_a (ordered_ab_group_add(X_a) -> (all Xa all Ya hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,X_a),fun(fun(X_a,bool),fun(X_a,bool)),image(X_a,X_a),uminus_uminus(X_a)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),Xa),Ya)) = hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Ya)),hAPP(X_a,X_a,uminus_uminus(X_a),Xa))))) # label(fact_4551_image__uminus__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2816 (all X_a all A_3 all F all Va hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),A_3),F),hAPP(int,nat,number_number_of(nat),Va)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),hAPP(bool,fun(X_a,fun(X_a,X_a)),if(X_a),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),A_3),hAPP(X_a,X_a,hAPP(nat,fun(X_a,X_a),F,hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),A_3),F),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va))))))) # label(fact_4184_nat__rec__number__of) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2817 (all X_b all X_a (real_normed_algebra(X_b) & topological_space(X_a) -> (all C all F all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),hAPP(X_b,fun(X_b,X_b),times_times(X_b),C)),F)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4602_LIM__mult__right__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2818 (all X_b all X_a (topological_space(X_a) & real_n1866405975lgebra(X_b) -> (all F all L_2 all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> (ti(X_b,L_2) != zero_zero(X_b) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),inverse_inverse(X_b)),F)),hAPP(X_b,X_b,inverse_inverse(X_b),L_2)),hAPP(X_a,filter(X_a),at(X_a),A_3)))))))) # label(fact_4613_LIM__inverse) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2819 (all F ((all I_2 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,F,hAPP(nat,nat,suc,I_2))),hAPP(nat,nat,F,I_2)))) -> (exists I_2 all K_1 hAPP(nat,nat,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_2),K_1)) = hAPP(nat,nat,F,I_2)))) # label(fact_3577_weak__decr__stable) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2820 (all Z_1 zero_zero(real) = hAPP(complex,real,im,hAPP(int,complex,ring_1_of_int(complex),Z_1))) # label(fact_3608_complex__Im__of__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2821 (all K hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(nat,fun(nat,bool),ord_lessThan(nat),K)))) # label(fact_4340_finite__lessThan) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2822 (all A_1 all B hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A_1)),B))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) # label(fact_349_zadd__power2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2823 (all X_a (real_normed_vector(X_a) -> zero_zero(X_a) = hAPP(X_a,X_a,sgn_sgn(X_a),zero_zero(X_a)))) # label(fact_2262_sgn__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.75 2824 (all X_a (comm_monoid_add(X_a) -> (all F all Na hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(nat,fun(fun(nat,bool),X_a)),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combs(nat,fun(nat,bool),X_a),hAPP(fun(nat,fun(nat,X_a)),fun(nat,fun(fun(nat,bool),X_a)),hAPP(fun(fun(nat,X_a),fun(fun(nat,bool),X_a)),fun(fun(nat,fun(nat,X_a)),fun(nat,fun(fun(nat,bool),X_a))),combb(fun(nat,X_a),fun(fun(nat,bool),X_a),nat),big_co1399186613setsum(nat,X_a)),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,X_a)),hAPP(fun(fun(nat,nat),fun(nat,X_a)),fun(fun(nat,fun(nat,nat)),fun(nat,fun(nat,X_a))),combb(fun(nat,nat),fun(nat,X_a),nat),hAPP(fun(nat,fun(nat,X_a)),fun(fun(nat,nat),fun(nat,X_a)),combs(nat,nat,X_a),F)),minus_minus(nat)))),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na)) = hAPP(fun(product_prod(nat,nat),bool),X_a,hAPP(fun(product_prod(nat,nat),X_a),fun(fun(product_prod(nat,nat),bool),X_a),big_co1399186613setsum(product_prod(nat,nat),X_a),hAPP(fun(nat,fun(nat,X_a)),fun(product_prod(nat,nat),X_a),product_prod_case(nat,nat,X_a),F)),hAPP(fun(product_prod(nat,nat),bool),fun(product_prod(nat,nat),bool),collect(product_prod(nat,nat)),hAPP(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_prod_case(nat,nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,fun(nat,bool))),fun(nat,fun(nat,fun(nat,bool))),combc(nat,nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,fun(nat,bool))),fun(nat,fun(nat,fun(nat,bool))),hAPP(fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool))),fun(fun(nat,fun(nat,fun(nat,bool))),fun(nat,fun(nat,fun(nat,bool)))),combb(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),nat),combc(nat,nat,bool)),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,fun(nat,bool))),hAPP(fun(fun(nat,nat),fun(nat,fun(nat,bool))),fun(fun(nat,fun(nat,nat)),fun(nat,fun(nat,fun(nat,bool)))),combb(fun(nat,nat),fun(nat,fun(nat,bool)),nat),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),ord_less(nat))),plus_plus(nat)))),Na))))))) # label(fact_4647_setsum__triangle__reindex) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2825 (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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),Ma)),Ma)) <-> Na = one_one(nat)))) # label(fact_880_dvd__mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2826 (all X_a (field_char_0(X_a) & field_inverse_zero(X_a) -> (all A_3 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),field_char_0_Rats(X_a))) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_3)),field_char_0_Rats(X_a))))))) # label(fact_3717_Rats__inverse) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2827 (all X_a (comm_semiring_1(X_a) -> (all X ti(X_a,X) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),one_one(nat))))) # label(fact_335_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2828 (all B_2 all A_3 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),d22set,A_3))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),B_2)))) # label(fact_1223_d22set__g__1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2829 (all L (pls = ti(int,L) <-> pls = hAPP(int,int,bit0,L))) # label(fact_71_rel__simps_I38_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2830 (all X_b all X_a (real_normed_field(X_a) & banach(X_b) -> (all G all F all K (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K)) -> (hBOOL(hAPP(fun(nat,real),bool,summable(real),F)) -> ((all H_1 all N_1 (ti(X_a,H_1) != zero_zero(X_a) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),H_1)),K)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_b,real,norm_norm(X_b),hAPP(nat,X_b,hAPP(X_a,fun(nat,X_b),G,H_1),N_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,F,N_1)),hAPP(X_a,real,norm_norm(X_a),H_1))))))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,fun(nat,X_b)),fun(X_a,X_b),hAPP(fun(fun(nat,X_b),X_b),fun(fun(X_a,fun(nat,X_b)),fun(X_a,X_b)),combb(fun(nat,X_b),X_b,X_a),suminf(X_b)),G)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a)))))))))) # label(fact_4650_lemma__termdiff5) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2831 (all X_a (order(X_a) -> (all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya)) & ti(X_a,Xa) != ti(X_a,Ya))))) # label(fact_569_order__less__le) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2832 (all X_a (metric_space(X_a) -> (all X all Y -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y)),zero_zero(real)))))) # label(fact_3539_dist__not__less__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2833 (all X_a (plus(X_a) & dvd(X_a) & linorder(X_a) -> plus_plus(X_a) = ti(fun(X_a,fun(X_a,X_a)),plus_plus(X_a)))) # label(tsy_c_Groups_Oplus__class_Oplus_0_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2834 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),hAPP(real,nat,natfloor,X))),X)))) # label(fact_2399_real__natfloor__le) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2835 (all N all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),N),X))),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,real_1(int),N)),hAPP(int,real,real_1(int),X))))) # label(fact_3385_real__of__int__div4) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2836 (all X_a (semiring_1(X_a) -> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),zero_zero(X_a)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = zero_zero(X_a))) # label(fact_4_zero__power2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2837 (all X_b all X_a (topological_space(X_a) & real_normed_vector(X_b) -> (all Ra all G all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),Ra)),G)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4761_scaleR__right_OLIM__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2838 (all X_a (real_normed_vector(X_a) -> (all A_1 all B all B_5 hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),B_5)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B_5))))) # label(fact_4699_scaleR_Oadd__right) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2839 (all X_a (comm_monoid_add(X_a) & t2_space(X_a) -> (all F (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(fun(nat,bool),X_a),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combb(fun(nat,bool),X_a,nat),hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F)),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)))),hAPP(fun(nat,X_a),X_a,suminf(X_a),F)),sequentially)))))) # label(fact_4521_summable__sumr__LIMSEQ__suminf) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2840 (all X_a (divisi14063676e_zero(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1))))) # label(fact_2570_inverse__inverse__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2841 (all K_4 hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,number_number_of(int),K_4)),one_one(int)) = hAPP(int,rat,number_number_of(rat),K_4)) # label(fact_3796_rat__number__collapse_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2842 (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_248_zero__le__one) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2843 (all R all S_5 hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R),hAPP(real,real,inverse_inverse(real),S_5)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),R),S_5)) # label(fact_2604_real__divide__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2844 (all X_a (linordered_idom(X_a) -> (all N hAPP(X_a,X_a,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_982_abs__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2845 (all K_4 all M all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N))) # label(fact_3242_gcd__mult__distrib__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2846 (all X_a ti(X_a,one_frac(X_a)) = one_frac(X_a)) # label(tsy_c_Nitpick_Oone__frac_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2847 (all M all P_5 (hBOOL(hAPP(nat,bool,prime,P_5)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),M),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),M))))) # label(fact_3101_prime__dvd__square) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2848 (all X_a all F1 all F2 all Code_numeral hAPP(code_code_numeral,X_a,hAPP(fun(code_code_numeral,X_a),fun(code_code_numeral,X_a),hAPP(X_a,fun(fun(code_code_numeral,X_a),fun(code_code_numeral,X_a)),code_c600343653l_case(X_a),F1),F2),hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,Code_numeral)) = hAPP(code_code_numeral,X_a,F2,Code_numeral)) # label(fact_3556_code__numeral_Osimps_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2849 (all N -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N)))))) # label(fact_2096_not__zle__0__negative) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2850 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,uminus_uminus(real),pi)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),zero_zero(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,tan,X)),zero_zero(real)))))) # label(fact_2451_tan__less__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2851 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> M = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),M)),N))) # label(fact_1166_div__mult__self1__is__m) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2852 (all A_1 all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),A_1)),X)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),hAPP(real,int,archim2085082626_floor(real),X))))) # label(fact_3387_le__floor) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2853 (all Ya all Xa all A_3 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A_3)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Ya)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),log,A_3),Xa)),hAPP(real,real,hAPP(real,fun(real,real),log,A_3),Ya))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya))))))) # label(fact_2036_log__le__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2854 (all D2 all D1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D1)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D2)) -> (exists E (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),E),D1)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),E),D2)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E))))))) # label(fact_2754_real__lbound__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2855 (all A_3 all C all D_1 all B_2 (zero_zero(int) != ti(int,B_2) -> (ti(int,D_1) != zero_zero(int) -> (hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_2)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,C),D_1))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_3),D_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_2),D_1))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),B_2)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_2),D_1)))))))) # label(fact_3801_less__rat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2856 (all K (hAPP(int,int,bit0,K) = pls <-> pls = ti(int,K))) # label(fact_70_rel__simps_I44_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2857 (all X_a (group_add(X_a) -> (all Xa all Ya (zero_zero(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Xa),Ya) <-> hAPP(X_a,X_a,uminus_uminus(X_a),Xa) = ti(X_a,Ya))))) # label(fact_1956_add__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2858 (all X_a all G all A_2 hAPP(complex,real,im,hAPP(fun(X_a,bool),complex,hAPP(fun(X_a,complex),fun(fun(X_a,bool),complex),big_co1399186613setsum(X_a,complex),G),A_2)) = hAPP(fun(X_a,bool),real,hAPP(fun(X_a,real),fun(fun(X_a,bool),real),big_co1399186613setsum(X_a,real),hAPP(fun(X_a,complex),fun(X_a,real),hAPP(fun(complex,real),fun(fun(X_a,complex),fun(X_a,real)),combb(complex,real,X_a),im),G)),A_2)) # label(fact_4274_Im_Osetsum) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2859 (all Na hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less_eq(int)),Na)))))) # label(fact_3851_bdd__int__set__l__le__finite) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2860 (all A_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),one_one(nat)),A_1))) # label(fact_1257_gcd__lcm__complete__lattice__nat_Obot__least) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2861 (all X_a (field(X_a) -> (all X all W all Z_1 all Y (ti(X_a,Y) != zero_zero(X_a) -> (ti(X_a,Z_1) != zero_zero(X_a) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Z_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),W),Y))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),W),Z_1))))))) # label(fact_1884_diff__frac__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2862 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),Xa)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),hAPP(real,nat,natfloor,Xa))))) # label(fact_1674_le__natfloor__eq__one) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2863 (all X_a (archim1727834104eiling(X_a) -> (all Xa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),one_one(X_a))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archimedean_ceiling(X_a),Xa)),one_one(int))))))) # label(fact_3327_ceiling__le__one) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2864 (all X_a (sgn_div_norm(X_a) -> sgn_sgn(X_a) = ti(fun(X_a,X_a),sgn_sgn(X_a)))) # label(tsy_c_Groups_Osgn__class_Osgn_0_res) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2865 (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)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) | ti(X_a,Y) = ti(X_a,X))))) # label(fact_558_order__le__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2866 (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),hAPP(int,int,bit1,K1)),hAPP(int,int,bit1,K2))))) # label(fact_217_less__eq__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2867 (all C_1 all A_1 all B all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,B),C_1),M)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),C_1),M))))) # label(fact_871_zcong__trans) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2868 (all P_1 all Na all K (hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Na),K))) <-> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),zero_zero(int))) -> (all I_2 all J_1 (hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),I_2)),J_1) = ti(int,Na) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),J_1),zero_zero(int))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),J_1)) -> hBOOL(hAPP(int,bool,P_1,J_1))))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K)) -> (all I_2 all J_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),J_1),K)) & ti(int,Na) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),I_2)),J_1) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J_1)) -> hBOOL(hAPP(int,bool,P_1,J_1))))) & (zero_zero(int) = ti(int,K) -> hBOOL(hAPP(int,bool,P_1,Na))))) # label(fact_1473_split__zmod) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2869 (all X_a (real_normed_vector(X_a) -> (all Ra all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2)) -> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra)),X_2))))))) # label(fact_4743_scaleR__right_Osummable) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2870 (all X_a (real_normed_vector(X_a) -> (all A_1 all B all B_5 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B_5)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),B_5))))) # label(fact_4702_scaleR_Odiff__right) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2871 (all T_2 all T_1 (order(T_1) -> order(fun(T_2,T_1)))) # label(arity_fun___Orderings_Oorder) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2872 (all X_b all X_a (real_normed_vector(X_a) & real_normed_vector(X_b) -> (all Ra all F all L_2 all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Ra)) -> (exists S_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),S_1)) & (all X_1 (ti(X_a,A_3) != ti(X_a,X_1) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_1),A_3))),S_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_a,X_b,F,X_1)),L_2))),Ra))))))))))) # label(fact_4670_LIM__D) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2873 (all X all Y (zero_zero(nat) != Y -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y))) # label(fact_3301_gcd__non__0__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2874 (all Na all Va hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(nat,nat,suc,Na)),hAPP(int,nat,number_number_of(nat),Va)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),zero_zero(nat)),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),Na),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va))))))) # label(fact_4821_min__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2875 (all Y all X all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5)) -> (hBOOL(hAPP(int,bool,zprime,P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Y),zero_zero(int)),P_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),X)),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),Y)),P_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),Y),P_5)))))))) # label(fact_1185_MultInv__prop5) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2876 (all X_a (real_normed_vector(X_a) -> (all Ra_1 all R_3 all X hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra_1),X)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra_1),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),X))))) # label(fact_4711_scaleR__right_OscaleR) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2877 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),M)),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N))) # label(fact_3298_gcd__diff2__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2878 (all X_a (real_normed_field(X_a) -> (all F all Xa all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D)) -> (zero_zero(X_a) != hAPP(X_a,X_a,F,Xa) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),inverse_inverse(X_a)),F)),Xa),hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,F,Xa))),D)),hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,F,Xa))))))))))) # label(fact_3888_DERIV__inverse_H) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2879 (all A_3 all B_2 all Wa (hAPP(int,real,number_number_of(real),Wa) = A_3 & B_2 = zero_zero(real) <-> hAPP(int,complex,number_number_of(complex),Wa) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_3),B_2))) # label(fact_2760_Complex__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2880 (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)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ma),Na))))) # label(fact_630_nat__0__less__mult__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2881 (all Na all Ma (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),Na)),hAPP(nat,real,real_1(nat),Ma))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),Ma)))) # label(fact_2384_real__of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2882 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)))))))) # label(fact_247_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2883 (all P_1 all A0 all A1 (hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),negDivAlg_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))) -> ((all A_4 all B_4 (hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),negDivAlg_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_4),B_4))) -> ((-(hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_4),B_4))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_4),zero_zero(int)))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B_4)))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A_4),B_4))))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A0),A1))))) # label(fact_1680_negDivAlg_Opinduct) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2884 (all Xa all Ya all Z (Xa = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),Ya),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Z)) <-> hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),Ya),Z) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Xa))) # label(fact_1805_eq__divide__2__times__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2885 (all X_a all X_b (comm_monoid_mult(X_b) -> (all B_2 all A_3 all S (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),S)) -> hAPP(X_a,X_b,B_2,A_3) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),A_3))),B_2)),one_one(X_b))),S)) & (-hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),S)) -> one_one(X_b) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),A_3))),B_2)),one_one(X_b))),S)))))) # label(fact_3982_setprod__delta) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2886 (all X_a all X_b (comm_monoid_add(X_b) -> (all F all S all T_3 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),T_3)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),S),T_3)) -> ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),T_3),S))) -> zero_zero(X_b) = hAPP(X_a,X_b,F,X_1))) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),T_3) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),S))))))) # label(fact_4322_setsum__mono__zero__right) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2887 (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)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,sqrt,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) # label(fact_2170_real__sqrt__pow2__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2888 (all X_a (dense_linorder(X_a) -> (all A_3 all B_2 all C all D_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),A_3),B_2)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),C),D_1))) <-> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),A_3)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),D_1))))))) # label(fact_4554_atLeastAtMost__subseteq__atLeastLessThan__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2889 (all X_b all X_a (metric_space(X_b) & metric_space(X_a) -> (all F all L_2 all A_3 ((all R_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1)) -> (exists S_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),S_1)) & (all X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X_1),A_3)),S_1)) & ti(X_a,X_1) != ti(X_a,A_3) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,hAPP(X_b,fun(X_b,real),dist_dist(X_b),hAPP(X_a,X_b,F,X_1)),L_2)),R_1)))))))) <-> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4676_LIM__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2890 (all Xa hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),cos)),cos)),Xa),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,Xa))),hAPP(real,real,cos,Xa))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,Xa))),hAPP(real,real,cos,Xa)))))) # label(fact_3913_DERIV__cos__cos__mult) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2891 (all Z_1 all X all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X),Y)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z_1))))) # label(fact_3062_termination__basic__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2892 (all X_a (linord20386208strict(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))))))) # label(fact_474_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2893 (all X_a all C_2 all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),B_1),C_2)) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),C_2))))) # label(fact_2201_subset__psubset__trans) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2894 (all X_a (real_normed_algebra(X_a) -> (all A_1 all B all B_5 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),B_5)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B_5))))) # label(fact_1243_mult_Oadd__right) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2895 (all Na hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),Na)))))) # label(fact_3852_bdd__int__set__l__finite) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2896 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1)) -> hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),A_1))) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A_1)))) # label(fact_1517_pos__zmod__mult__2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2897 (all N all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M)) -> hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(nat,real,real_1(nat),M)),hAPP(nat,real,real_1(nat),N)))) # label(fact_2397_real__of__nat__diff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2898 (all P_4 hAPP(rat,int,archim2085082626_floor(rat),P_4) = hAPP(product_prod(int,int),int,hAPP(fun(int,fun(int,int)),fun(product_prod(int,int),int),product_prod_case(int,int,int),div_div(int)),hAPP(rat,product_prod(int,int),quotient_of,P_4))) # label(fact_4054_rat__floor__code) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2899 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,arccos,Y)))))) # label(fact_2354_arccos__lbound) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2900 (all X_a (zero(X_a) -> (all Diff all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(X_a,fun(nat,real),hAPP(fun(nat,fun(X_a,real)),fun(X_a,fun(nat,real)),combc(nat,X_a,real),Diff),zero_zero(X_a)))),hAPP(real,fun(nat,real),power_power(real),zero_zero(real))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,hAPP(nat,fun(X_a,real),Diff,Na),zero_zero(X_a))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),zero_zero(real)),Na))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))) = hAPP(X_a,real,hAPP(nat,fun(X_a,real),Diff,zero_zero(nat)),zero_zero(X_a)))))) # label(fact_4263_Maclaurin__bi__le__lemma) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2901 (all M all N hAPP(int,int,hAPP(int,fun(int,int),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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N))) # label(fact_23_zadd__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2902 (all X_a all A_2 all B_1 (ti(fun(X_a,bool),B_1) = ti(fun(X_a,bool),A_2) <-> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) & hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2)))) # label(fact_2209_set__eq__subset) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2903 (all M all N hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),M),N)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(nat,real,real_1(nat),M)),N)) # label(fact_2389_real__of__nat__power) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2904 (all X_a all X_c all X_b all F all Xa all G (hAPP(product_prod(X_a,X_c),product_prod(X_a,X_b),hAPP(fun(X_c,X_b),fun(product_prod(X_a,X_c),product_prod(X_a,X_b)),product_apsnd(X_c,X_b,X_a),F),Xa) = hAPP(product_prod(X_a,X_c),product_prod(X_a,X_b),hAPP(fun(X_c,X_b),fun(product_prod(X_a,X_c),product_prod(X_a,X_b)),product_apsnd(X_c,X_b,X_a),G),Xa) <-> hAPP(X_c,X_b,G,hAPP(product_prod(X_a,X_c),X_c,product_snd(X_a,X_c),Xa)) = hAPP(X_c,X_b,F,hAPP(product_prod(X_a,X_c),X_c,product_snd(X_a,X_c),Xa)))) # label(fact_4091_apsnd__eq__conv) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2905 (all A_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_1),B)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),A_1)) -> hAPP(int,int,abs_abs(int),A_1) = hAPP(int,int,abs_abs(int),B)))) # label(fact_985_zdvd__antisym__abs) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2906 (all M all N all S_4 all S_3 all T_6 all T_5 all R_4 all R_3 all K_4 (hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,R_4),R_3) = ti(int,K_4) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),R_3)) -> (exists Sn exists Tn hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),hAPP(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))),hAPP(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))))),xzgcda,M),N),R_4),R_3),S_4),S_3),T_6),T_5) = hAPP(product_prod(int,int),product_prod(int,product_prod(int,int)),hAPP(int,fun(product_prod(int,int),product_prod(int,product_prod(int,int))),product_Pair(int,product_prod(int,int)),K_4),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Sn),Tn)))))) # label(fact_3505_xzgcd__correct__aux1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2907 (all X_a (ring_11004092258visors(X_a) -> (all N all A_1 (zero_zero(X_a) != ti(X_a,A_1) -> zero_zero(X_a) != hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N))))) # label(fact_168_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2908 (all K_4 all N all M hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N)),M)),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)) # label(fact_1537_mod__mult__self3) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2909 (all Y all X (hBOOL(hAPP(int,bool,even_odd_even(int),X)) -> hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y))))) # label(fact_1592_even__times__anything) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2910 (all M hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),zero_zero(nat)) = M) # label(fact_158_Nat_Oadd__0__right) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2911 (all Z_1 all W ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W))) -> hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Z_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,W))) = one_one(int)) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W))) -> hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Z_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,W))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z_1),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Z_1),hAPP(int,nat,number_number_of(nat),W)))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Z_1),hAPP(int,nat,number_number_of(nat),W)))))) # label(fact_3931_zpower__number__of__odd) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2912 (all Y all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Y)) -> hAPP(real,nat,natfloor,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),hAPP(nat,real,real_1(nat),Y))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(real,nat,natfloor,X)),Y)))) # label(fact_2438_natfloor__div__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2913 (all X_a all X_b (ordere779506340up_add(X_b) & comm_monoid_add(X_b) -> (all F all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) -> ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),B_1),A_2))) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(X_a,X_b,F,X_1))))) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),B_1))))))))) # label(fact_4319_setsum__mono3) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2914 (all X hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,sin,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,cos,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = one_one(real)) # label(fact_2296_sin__cos__squared__add) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2915 (all X_a all X_b (ordered_ab_group_add(X_b) -> (all I all F all Sb (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),Sb)) -> ((all I_2 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I_2),Sb)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(X_a,X_b,F,I_2))))) -> (zero_zero(X_b) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),Sb) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I),Sb)) -> zero_zero(X_b) = hAPP(X_a,X_b,F,I)))))))) # label(fact_4327_setsum__nonneg__0) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2916 (all Ma all Na (Na = zero_zero(nat) & zero_zero(nat) = Ma <-> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Ma),Na) = zero_zero(nat))) # label(fact_3281_gcd__zero__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2917 (all X_a (real_algebra_1(X_a) & real_normed_vector(X_a) -> (all X hAPP(real,X_a,of_real(X_a),hAPP(real,real,uminus_uminus(real),X)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(real,X_a,of_real(X_a),X))))) # label(fact_2807_of__real_Ominus) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2918 (all X_a (preorder(X_a) -> (all Z_1 all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),Z_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Z_1))))))) # label(fact_503_order__trans) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2919 (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),N)))))) # label(fact_1012_zero__le__power__abs) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2920 (all X_a (monoid_mult(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),one_one(X_a)),A_1)))) # label(fact_387_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2921 (all M_5 all D_4 (hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M_5),D_4) = zero_zero(int) -> (exists Q_4 hAPP(int,int,hAPP(int,fun(int,int),times_times(int),D_4),Q_4) = ti(int,M_5)))) # label(fact_1521_zmod__eq__0D) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2922 (all X_a (linordered_idom(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,sgn_sgn(X_a),A_3)),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a))))))) # label(fact_2279_sgn__less) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2923 (all X_a (number_ring(X_a) & linord1117847801e_zero(X_a) -> (all R_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),R_3)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),R_3),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))))) # label(fact_1808_half__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2924 (all Na hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),hAPP(nat,real,real_1(nat),Na)),field_char_0_Rats(real)))) # label(fact_3700_Rats__real__nat) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2925 (all F all K all A_3 all B_2 (A_3 != B_2 -> ((all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),K))) -> hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B_2),A_3)),K) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,F,B_2)),hAPP(real,real,F,A_3))))) # label(fact_2669_DERIV__const__ratio__const) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2926 (all A_1 all B hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),ii),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,uminus_uminus(real),B)),A_1)) # label(fact_2785_i__mult__Complex) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2927 (all X_a (ordere223160158up_add(X_a) -> (all C_1 all D_3 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D_3)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D_3)))))))) # label(fact_233_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2928 (all K_4 all I_1 all J_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),J_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K_4)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),I_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),J_2)))))) # label(fact_468_zmult__zless__mono2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2929 (all X_a (linorder(X_a) -> (all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) -> -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)))))) # label(fact_561_leD) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2930 (all K all L (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),L)) <-> (exists N_1 hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),N_1) = L))) # label(fact_1209_le__Suc__ex__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2931 (all X_a (real_normed_algebra(X_a) & banach(X_a) -> (all B_2 all A_3 (hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),A_3))) -> (hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),B_2))) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),A_3)),hAPP(fun(nat,X_a),X_a,suminf(X_a),B_2)) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(nat,fun(fun(nat,bool),X_a)),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combs(nat,fun(nat,bool),X_a),hAPP(fun(nat,fun(nat,X_a)),fun(nat,fun(fun(nat,bool),X_a)),hAPP(fun(fun(nat,X_a),fun(fun(nat,bool),X_a)),fun(fun(nat,fun(nat,X_a)),fun(nat,fun(fun(nat,bool),X_a))),combb(fun(nat,X_a),fun(fun(nat,bool),X_a),nat),big_co1399186613setsum(nat,X_a)),hAPP(fun(nat,fun(nat,X_a)),fun(nat,fun(nat,X_a)),hAPP(fun(fun(nat,X_a),fun(nat,X_a)),fun(fun(nat,fun(nat,X_a)),fun(nat,fun(nat,X_a))),combb(fun(nat,X_a),fun(nat,X_a),nat),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),A_3))),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,X_a)),hAPP(fun(fun(nat,nat),fun(nat,X_a)),fun(fun(nat,fun(nat,nat)),fun(nat,fun(nat,X_a))),combb(fun(nat,nat),fun(nat,X_a),nat),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),B_2)),minus_minus(nat))))),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat))))))))) # label(fact_4646_Cauchy__product) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2932 (all X_b all X_a (real_normed_vector(X_b) & topological_space(X_a) -> (all F all L_2 all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),uminus_uminus(X_b)),F)),hAPP(X_b,X_b,uminus_uminus(X_b),L_2)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4593_LIM__minus) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2933 (all X_a (ordere223160158up_add(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C_1))))))) # label(fact_329_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2934 (all X_a (real_normed_algebra(X_a) -> (all C all F (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) -> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),F)),C))))))) # label(fact_4027_summable__mult2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2935 (all X_a (order(X_a) -> (all X_2 ((all N_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,X_2,N_1)),hAPP(nat,X_a,X_2,hAPP(nat,nat,suc,N_1))))) -> hBOOL(hAPP(fun(nat,X_a),bool,monoseq(X_a),X_2)))))) # label(fact_4167_mono__SucI1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2936 (all P_1 all P1 all D_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_1)) -> ((all X_1 all K_1 (hBOOL(hAPP(int,bool,P1,X_1)) <-> hBOOL(hAPP(int,bool,P1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_1),D_1)))))) -> ((exists Z_2 all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),Z_2)) -> (hBOOL(hAPP(int,bool,P1,X_1)) <-> hBOOL(hAPP(int,bool,P_1,X_1))))) -> ((exists X1 hBOOL(hAPP(int,bool,P1,X1))) -> (exists X1 hBOOL(hAPP(int,bool,P_1,X1)))))))) # label(fact_2869_minusinfinity) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2937 (all X_a all F1 all F2 ti(X_a,F1) = hAPP(code_code_numeral,X_a,hAPP(fun(code_code_numeral,X_a),fun(code_code_numeral,X_a),hAPP(X_a,fun(fun(code_code_numeral,X_a),fun(code_code_numeral,X_a)),code_c600343653l_case(X_a),F1),F2),zero_zero(code_code_numeral))) # label(fact_3574_code__numeral_Osimps_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2938 (all X_a (linordered_semidom(X_a) -> (all Ma all Na all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_3)) -> (Ma = Na <-> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_3),Ma) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_3),Na)))))) # label(fact_184_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2939 (all M hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),M))))) # label(fact_614_le__cube) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2940 (all L all U hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),L),one_one(int))),U) = hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_gr676655987ssThan(int),L),U)) # label(fact_4288_atLeastPlusOneLessThan__greaterThanLessThan__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2941 (all Ma all Na (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Ma),hAPP(int,int,number_number_of(int),Na))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),Ma)),hAPP(int,real,number_number_of(real),Na))))) # label(fact_3415_number__of__le__real__of__int__iff2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2942 (all X_a all X_c all X_b all Z all C all A_3 all B_2 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Z),hAPP(X_c,fun(X_a,bool),hAPP(X_b,fun(X_c,fun(X_a,bool)),C,A_3),B_2))) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Z),hAPP(product_prod(X_b,X_c),fun(X_a,bool),hAPP(fun(X_b,fun(X_c,fun(X_a,bool))),fun(product_prod(X_b,X_c),fun(X_a,bool)),product_prod_case(X_b,X_c,fun(X_a,bool)),C),hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),A_3),B_2)))))) # label(fact_4065_mem__splitI) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2943 (all X_a (real_algebra_1(X_a) -> (all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(real,X_a,of_real(X_a),X)),hAPP(real,X_a,of_real(X_a),Y)) = hAPP(real,X_a,of_real(X_a),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y))))) # label(fact_2804_of__real__diff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2944 (all L all U hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),L),U) = hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),L),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),U),one_one(int)))) # label(fact_4597_atLeastLessThanPlusOne__atLeastAtMost__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2945 (all A_1 all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),A_1)),X)) -> hAPP(real,nat,natfloor,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),hAPP(nat,real,real_1(nat),A_1))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(real,nat,natfloor,X)),A_1))) # label(fact_2413_natfloor__subtract) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2946 (all X_b all X_c all X_a (comm_monoid_add(X_a) -> (all F all B_1 all A_2 hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_c,bool),fun(X_b,X_a),hAPP(fun(X_b,fun(fun(X_c,bool),X_a)),fun(fun(X_c,bool),fun(X_b,X_a)),combc(X_b,fun(X_c,bool),X_a),hAPP(fun(X_b,fun(X_c,X_a)),fun(X_b,fun(fun(X_c,bool),X_a)),hAPP(fun(fun(X_c,X_a),fun(fun(X_c,bool),X_a)),fun(fun(X_b,fun(X_c,X_a)),fun(X_b,fun(fun(X_c,bool),X_a))),combb(fun(X_c,X_a),fun(fun(X_c,bool),X_a),X_b),big_co1399186613setsum(X_c,X_a)),F)),B_1)),A_2) = hAPP(fun(X_c,bool),X_a,hAPP(fun(X_c,X_a),fun(fun(X_c,bool),X_a),big_co1399186613setsum(X_c,X_a),hAPP(fun(X_b,bool),fun(X_c,X_a),hAPP(fun(X_c,fun(fun(X_b,bool),X_a)),fun(fun(X_b,bool),fun(X_c,X_a)),combc(X_c,fun(X_b,bool),X_a),hAPP(fun(X_c,fun(X_b,X_a)),fun(X_c,fun(fun(X_b,bool),X_a)),hAPP(fun(fun(X_b,X_a),fun(fun(X_b,bool),X_a)),fun(fun(X_c,fun(X_b,X_a)),fun(X_c,fun(fun(X_b,bool),X_a))),combb(fun(X_b,X_a),fun(fun(X_b,bool),X_a),X_c),big_co1399186613setsum(X_b,X_a)),hAPP(fun(X_b,fun(X_c,X_a)),fun(X_c,fun(X_b,X_a)),combc(X_b,X_c,X_a),F))),A_2)),B_1)))) # label(fact_4243_setsum__commute) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2947 (all X_a (real_vector(X_a) -> (all A_1 hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),zero_zero(X_a)) = zero_zero(X_a)))) # label(fact_4730_scaleR__right_Ozero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2948 (all G all Xa (hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),G),Xa)) -> hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(real,fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(real,fun(nat,fun(real,real))),combc(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),zero_zero(real))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),G),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),div_div(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),minus_minus(nat)),one_one(nat)))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),Xa)))) # label(fact_4131_sums__if_H) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2949 (all X_a (real_normed_vector(X_a) -> (all X_2 all K (hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),X_2),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K)))) -> hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2)))))) # label(fact_3863_Bseq__offset) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2950 (all P_1 all Xa all Ya (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)) -> (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) -> hBOOL(P_1)))) # label(fact_939_dvd_Oless__imp__triv) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2951 (all N all P_5 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P_5)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),P_5),N)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),hAPP(nat,nat,fact,N)))))) # label(fact_3557_divides__fact) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2952 (all X_a (semiring_div(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)),B) = zero_zero(X_a)))) # label(fact_1358_mod__div__trivial) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2953 (all Ya all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Ya))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Ya))))) # label(fact_2698_real__root__gt__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2954 (all X_a (linordered_semidom(X_a) -> (all B all C_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),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)))))))) # label(fact_179_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2955 (all X_a all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)),hAPP(fun(X_a,bool),nat,finite_card(X_a),B_1))) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1)))))) # label(fact_2734_card__psubset) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2956 (all M all N hAPP(int,int,hAPP(int,fun(int,int),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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N))) # label(fact_640_zmult__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2957 (all X_a (linordered_field(X_a) -> (all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),B)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)))))))) # label(fact_2608_le__imp__inverse__le__neg) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2958 (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)) -> hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,ln,X)),hAPP(real,real,ln,Y))))) # label(fact_1730_ln__mult) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2959 (all R_3 hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,zero_zero(real)),R_3) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),ii),hAPP(real,complex,of_real(complex),R_3))) # label(fact_2821_i__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2960 (all X all Q_3 all N all R_3 (hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Q_3),N)),R_3) = X -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),R_3)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),R_3),N)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N),X)))))) # label(fact_875_divides__div__not) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2961 (all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> (hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,nat_1,Xa))) <-> hBOOL(hAPP(int,bool,even_odd_even(int),Xa))))) # label(fact_2517_pos__int__even__equiv__nat__even) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2962 (all X_a all X_b (real_normed_vector(X_b) -> (all Xa all G all A_3 all A_2 (hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),G),A_3),A_2)) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,real),fun(X_a,fun(X_b,X_b)),hAPP(fun(real,fun(X_b,X_b)),fun(fun(X_a,real),fun(X_a,fun(X_b,X_b))),combb(real,fun(X_b,X_b),X_a),scaleR_scaleR(X_b)),G)),Xa)),hAPP(X_b,X_b,hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),A_3),Xa)),A_2)))))) # label(fact_4752_scaleR__left_Otendsto) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2963 (all Xa (zero_zero(real) = hAPP(real,real,sqrt,Xa) <-> zero_zero(real) = Xa)) # label(fact_2126_real__sqrt__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2964 (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_172_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2965 (all Xa (hAPP(real,real,exp(real),hAPP(real,real,ln,Xa)) = Xa <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)))) # label(fact_1791_exp__ln__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2966 (all U_1 all X (hAPP(real,real,exp(real),U_1) = X -> U_1 = hAPP(real,real,ln,X))) # label(fact_1760_exp__ln__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2967 (all X_a all X_b all S all R_2 ((all X_1 all Xa_1 (hBOOL(hAPP(fun(product_prod(X_a,X_b),bool),bool,hAPP(product_prod(X_a,X_b),fun(fun(product_prod(X_a,X_b),bool),bool),member(product_prod(X_a,X_b)),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X_1),Xa_1)),R_2)) <-> hBOOL(hAPP(fun(product_prod(X_a,X_b),bool),bool,hAPP(product_prod(X_a,X_b),fun(fun(product_prod(X_a,X_b),bool),bool),member(product_prod(X_a,X_b)),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X_1),Xa_1)),S)))) <-> ti(fun(product_prod(X_a,X_b),bool),R_2) = ti(fun(product_prod(X_a,X_b),bool),S))) # label(fact_4365_pred__equals__eq2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2968 (all X_a (banach(X_a) & real_normed_field(X_a) -> (all Xa hAPP(X_a,X_a,exp(X_a),Xa) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa)))))) # label(fact_4776_exp__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2969 (all Na hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(fun(nat,real),fun(nat,fun(real,real))),combs(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),div_div(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),zero_zero(real))))),zero_zero(real))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),one_one(nat)),Na)) = zero_zero(real)) # label(fact_4266_lemma__STAR__cos2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2970 (all X_a (field_char_0(X_a) -> (all B_2 all A_3 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),field_char_0_Rats(X_a))) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),field_char_0_Rats(X_a))) -> (zero_zero(X_a) != ti(X_a,B_2) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),B_2)),field_char_0_Rats(X_a))))))))) # label(fact_3705_nonzero__Rats__divide) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2971 (all X (-hBOOL(hAPP(int,bool,even_odd_even(int),X)) -> hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(int)) = ti(int,X))) # label(fact_1583_two__times__odd__div__two__plus__one) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2972 (all N hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,fact,N)))) # label(fact_3569_fact__lt) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2973 (all X_a all X_b all A_2 all B_1 all P_1 (hBOOL(hAPP(fun(product_prod(fun(X_a,bool),fun(X_b,bool)),bool),bool,hAPP(product_prod(fun(X_a,bool),fun(X_b,bool)),fun(fun(product_prod(fun(X_a,bool),fun(X_b,bool)),bool),bool),member(product_prod(fun(X_a,bool),fun(X_b,bool))),hAPP(fun(X_b,bool),product_prod(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,bool),fun(fun(X_b,bool),product_prod(fun(X_a,bool),fun(X_b,bool))),product_Pair(fun(X_a,bool),fun(X_b,bool)),A_2),B_1)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(fun(X_a,bool),fun(X_b,bool)),bool),bijR(X_a,X_b),P_1))) -> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_1)))) # label(fact_4949_fin__bijRr) # label(axiom) # label(non_clause). [assumption]. 7.70/7.76 2974 (all A_3 all B_2 all K all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Ma)) -> (hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K),Ma) = one_one(int) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),Ma)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),A_3)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),B_2)),Ma)))))) # label(fact_3495_zcong__cancel2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2975 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit0,K)),hAPP(int,int,bit1,L))))) # label(fact_258_rel__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2976 (all X_a all X_b (real_normed_field(X_b) -> (all G all B_2 all F all A_3 all A_2 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),A_3),A_2)) -> (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),B_2),A_2)) -> (ti(X_b,B_2) != zero_zero(X_b) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),inverse_divide(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),inverse_divide(X_b),A_3),B_2)),A_2)))))))) # label(fact_4465_tendsto__divide) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2977 (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)) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,abs_abs(X_a),Y)),X) = hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),X)))))) # label(fact_1010_abs__mult__pos) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2978 (all P_1 all A_3 all B_2 (-((exists D_2 (hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B_2),D_2) = A_3 & -hBOOL(hAPP(nat,bool,P_1,D_2)))) | -hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_3),B_2))) <-> hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A_3),B_2))))) # label(fact_859_nat__diff__split__asm) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2979 (all X_a (group_add(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),zero_zero(X_a))))) # label(fact_736_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2980 (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)) -> hAPP(X_a,X_a,abs_abs(X_a),A_1) = ti(X_a,A_1))))) # label(fact_989_abs__of__pos) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2981 (all P_4 hAPP(int,fun(int,bool),sr,P_4) = hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),P_4)))) # label(fact_1261_SR__def) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2982 (all X_a all X_c all B_1_1 all B_2_1 hAPP(X_a,X_c,B_1_1,B_2_1) = hAPP(X_a,X_c,B_1_1,ti(X_a,B_2_1))) # label(tsy_c_hAPP_arg2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2983 (all X_a (real_algebra_1(X_a) & real_normed_vector(X_a) -> (all R_3 all X hAPP(real,X_a,of_real(X_a),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),R_3),X)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),hAPP(real,X_a,of_real(X_a),X))))) # label(fact_4714_of__real_OscaleR) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2984 (all P_1 all A_3 all B_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2)) -> (all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(product_prod(real,real),real,product_fst(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),N_1))),hAPP(product_prod(real,real),real,product_fst(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),hAPP(nat,nat,suc,N_1)))))))) # label(fact_3167_Bolzano__bisect__fst__le__Suc) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2985 (all X_a (real_normed_vector(X_a) -> (all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2)) -> -(all K_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2)) -> -(all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,X_2,N_1))),K_2))))))))) # label(fact_3526_BseqE) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2986 (all X_a (preorder(X_a) -> (all X all 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_528_order__less__imp__not__less) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2987 (all X_a (archim1804426504_field(X_a) -> (all X exists Z_2 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_2)),X))))) # label(fact_3644_ex__of__int__less) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2988 (all I all J hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,nat),fun(fun(nat,bool),fun(nat,bool)),image(nat,nat),suc),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),I),J)) = hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,suc,I)),hAPP(nat,nat,suc,J))) # label(fact_4568_image__Suc__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2989 (all X_a (ring_n68954251visors(X_a) -> (all A_3 all B_2 (zero_zero(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_2) <-> ti(X_a,A_3) = zero_zero(X_a) | zero_zero(X_a) = ti(X_a,B_2))))) # label(fact_375_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2990 (all K all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Na))))) # label(fact_629_mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2991 (all R_3 all A_1 hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),R_3),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),hAPP(real,real,cos,A_1))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),hAPP(real,real,sin,A_1))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) # label(fact_2295_sin__cos__squared__add2__mult) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2992 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))) -> hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)))) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)))) # label(fact_3136_div__pos__neg) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2993 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(real,real,exp(real),Xa))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Xa)))) # label(fact_1785_one__le__exp__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2994 (all A_3 all B_2 all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),B_2)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),powr,Xa),A_3)),hAPP(real,real,hAPP(real,fun(real,real),powr,Xa),B_2)))))) # label(fact_3379_powr__less__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2995 (all X_a (linordered_idom(X_a) -> (all X all Y (ti(X_a,Y) != ti(X_a,X) -> (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X))))))) # label(fact_155_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2996 (all Xa hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),hAPP(int,real,real_1(int),Xa)),ring_1_Ints(real)))) # label(fact_3827_Ints__real__of__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2997 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N)),M))))) # label(fact_1169_div__less__dividend) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2998 (all M all N hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),M),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),hAPP(nat,nat,suc,M)),hAPP(nat,nat,suc,N))) # label(fact_4991_max__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 2999 (all X_a (field_inverse_zero(X_a) -> (all A_1 all B all N hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)),N) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),N))))) # label(fact_1753_power__divide) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3000 (all F all A_3 all B_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),B_2)) -> ((all X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),X_1)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),B_2)) -> (exists Y_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y_1)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),Y_1)))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,A_3)),hAPP(real,real,F,B_2)))))) # label(fact_2865_DERIV__pos__imp__increasing) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3001 (all X_a (comm_ring_1(X_a) -> (all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y))))) # label(fact_1481_realpow__two__diff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3002 (all X_a (linordered_semidom(X_a) -> (all N 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),hAPP(nat,nat,fact_fact(nat),N))))))) # label(fact_2550_of__nat__fact__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3003 (all X_a (real_normed_field(X_a) -> (all G all E_3 all F all Xa all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),Xa),E_3)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),plus_plus(X_a)),F)),G)),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),D),E_3)))))))) # label(fact_3865_DERIV__add) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3004 (all X all Y (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) -> -(-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X))))) # label(fact_942_dvd_Oless__imp__not__less) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3005 (all X all Y hAPP(rat,real,ratreal,hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(rat,real,ratreal,X)),hAPP(rat,real,ratreal,Y))) # label(fact_3793_real__times__code) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3006 (all A_1 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)) -> hAPP(real,real,hAPP(real,fun(real,real),powr,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y)),A_1) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,Y),A_1))))) # label(fact_3401_powr__mult) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3007 (all P all Q (-hBOOL(Q) | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fimplies,P),Q)))) # label(help_fimplies_2_1_U) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3008 (all X_a (ab_group_add(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)))) # label(fact_1968_ab__diff__minus) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3009 (all Y all X (hAPP(real,real,cos,X) != zero_zero(real) -> (zero_zero(real) != hAPP(real,real,cos,Y) -> hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,tan,X)),hAPP(real,real,tan,Y)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,cos,Y)))))) # label(fact_2436_add__tan__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3010 (all X_a all X_b (linordered_semidom(X_b) -> (all G all F all A_2 ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(X_a,X_b,F,X_1))) & hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,G,X_1))))) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),A_2))))))) # label(fact_4177_setprod__mono) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3011 (all X_a (linord20386208strict(X_a) -> (all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B))))))) # label(fact_447_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3012 (all X_a (comm_monoid_add(X_a) -> (all F all Ma all Na hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),suc)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),Na)) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,suc,Ma)),hAPP(nat,nat,suc,Na)))))) # label(fact_4569_setsum__shift__bounds__cl__Suc__ivl) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3013 (all X_a (ring_char_0(X_a) -> (all Z (ti(int,Z) = zero_zero(int) <-> zero_zero(X_a) = hAPP(int,X_a,ring_1_of_int(X_a),Z))))) # label(fact_3590_of__int__0__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3014 (all Na hAPP(real,real,uminus_uminus(real),hAPP(nat,real,sin_coeff,Na)) = hAPP(nat,real,hAPP(fun(nat,real),fun(nat,real),diffs(real),cos_coeff),Na)) # label(fact_4149_cos__fdiffs2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3015 (all Va (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),Va)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(int,nat,number_number_of(nat),Va))))) # label(fact_164_less__0__number__of) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3016 (all X_a (real_normed_vector(X_a) -> (exists K_2 all A_4 all B_4 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_4),B_4))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),A_4)),hAPP(X_a,real,norm_norm(X_a),B_4))),K_2)))))) # label(fact_4784_scaleR_Obounded) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3017 (all X_a (comm_semiring_1(X_a) -> (all A_1 all M all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),M) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),M)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),M))))) # label(fact_385_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3018 (all Ya all Xa all P_4 (hBOOL(hAPP(int,bool,zprime,P_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),sRStar,P_4))) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ya),hAPP(int,fun(int,bool),sRStar,P_4))) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),standardRes,P_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Xa),Ya))),hAPP(int,fun(int,bool),sRStar,P_4)))))))) # label(fact_1566_SRStar__mult__prop1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3019 (all N all K_4 all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),M)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),N)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)))) # label(fact_816_Nat_Odiff__diff__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3020 (all R_3 exists N_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_3),hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N_1),one_one(int))))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),N_1)),R_3)))) # label(fact_3449_real__lb__ub__int) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3021 (all X_c all X_a all X_b all R_2 all A_3 all B_2 all C (hBOOL(hAPP(X_c,bool,hAPP(product_prod(X_a,X_b),fun(X_c,bool),hAPP(fun(X_a,fun(X_b,fun(X_c,bool))),fun(product_prod(X_a,X_b),fun(X_c,bool)),product_prod_case(X_a,X_b,fun(X_c,bool)),R_2),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_3),B_2)),C)) -> hBOOL(hAPP(X_c,bool,hAPP(X_b,fun(X_c,bool),hAPP(X_a,fun(X_b,fun(X_c,bool)),R_2,A_3),B_2),C)))) # label(fact_4070_splitD_H) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3022 (all X hAPP(complex,real,re,hAPP(complex,complex,uminus_uminus(complex),X)) = hAPP(real,real,uminus_uminus(real),hAPP(complex,real,re,X))) # label(fact_3029_Re_Ominus) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3023 (all M all I_1 all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),M),N)),I_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),I_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),I_1))) # label(fact_4839_min__diff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3024 (all X_a all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)) = hAPP(fun(fun(X_a,bool),bool),nat,finite_card(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),A_2)))) # label(fact_2839_card__Pow) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3025 (all X_a (ring_div(X_a) -> (all B all B_5 all A_1 all C_1 all A_6 (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_6),C_1) -> (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B_5),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C_1) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_6),B_5)),C_1)))))) # label(fact_1298_mod__diff__cong) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3026 (all X_c all X_a all X_b all Z all C all P_4 ((all A_4 all B_4 (ti(product_prod(X_a,X_b),P_4) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_4),B_4) -> hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),Z),hAPP(X_b,fun(X_c,bool),hAPP(X_a,fun(X_b,fun(X_c,bool)),C,A_4),B_4))))) -> hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),Z),hAPP(product_prod(X_a,X_b),fun(X_c,bool),hAPP(fun(X_a,fun(X_b,fun(X_c,bool))),fun(product_prod(X_a,X_b),fun(X_c,bool)),product_prod_case(X_a,X_b,fun(X_c,bool)),C),P_4))))) # label(fact_4085_mem__splitI2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3027 (all C_1 all A_1 all B all Q_3 all R_3 (hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_3),R_3))) -> (ti(int,B) != zero_zero(int) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C_1)) -> hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C_1)),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Q_3),C_1)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Q_3),C_1))),R_3)))))))) # label(fact_1581_zmult2__lemma) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3028 (all Y all X (hAPP(real,real,cos,X) != zero_zero(real) -> (hAPP(real,real,cos,Y) != zero_zero(real) -> (hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)) != zero_zero(real) -> hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,tan,X)),hAPP(real,real,tan,Y))),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,tan,X)),hAPP(real,real,tan,Y)))))))) # label(fact_2442_tan__add) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3029 (all X hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,sin,hAPP(real,real,uminus_uminus(real),X))),hAPP(real,real,sin,X))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,cos,hAPP(real,real,uminus_uminus(real),X))),hAPP(real,real,cos,X))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = zero_zero(real)) # label(fact_2310_sin__cos__minus) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3030 (all X_a (number_ring(X_a) -> (all V ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),V))) -> hAPP(int,X_a,number_number_of(X_a),V) = hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,number_number_of(nat),V))) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),V))) -> zero_zero(X_a) = hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,number_number_of(nat),V))))))) # label(fact_316_of__nat__number__of__lemma) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3031 (all X_a (order(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) -> -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)))))) # label(fact_523_xt1_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3032 (all X_a all X_c all X_b (topological_space(X_b) & real_normed_vector(X_c) -> (all F all Xa all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2)) -> hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,X_c),fun(X_b,bool),isCont(X_b,X_c),hAPP(X_a,fun(X_b,X_c),F,X_1)),Xa)))) -> hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,X_c),fun(X_b,bool),isCont(X_b,X_c),hAPP(fun(X_a,bool),fun(X_b,X_c),hAPP(fun(X_b,fun(fun(X_a,bool),X_c)),fun(fun(X_a,bool),fun(X_b,X_c)),combc(X_b,fun(X_a,bool),X_c),hAPP(fun(X_b,fun(X_a,X_c)),fun(X_b,fun(fun(X_a,bool),X_c)),hAPP(fun(fun(X_a,X_c),fun(fun(X_a,bool),X_c)),fun(fun(X_b,fun(X_a,X_c)),fun(X_b,fun(fun(X_a,bool),X_c))),combb(fun(X_a,X_c),fun(fun(X_a,bool),X_c),X_b),big_co1399186613setsum(X_a,X_c)),hAPP(fun(X_a,fun(X_b,X_c)),fun(X_b,fun(X_a,X_c)),combc(X_a,X_b,X_c),F))),A_2)),Xa))))))) # label(fact_4947_isCont__setsum) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3033 (all W (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W))) -> hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,one_one(int)),hAPP(int,int,number_number_of(int),W))) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),one_one(int)),hAPP(int,int,number_number_of(int),W)))) # label(fact_3135_div__pos__pos__1__number__of) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3034 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),min)),X)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X))),X)))) # label(fact_1732_ln__add__one__self__le__self2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3035 (all X_a (semiring_div(X_a) -> (all A_1 all B (ti(X_a,B) != zero_zero(X_a) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B)),one_one(X_a)))))) # label(fact_1117_div__add__self2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3036 (all A_1 all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int)))) -> one_one(int) != hAPP(int,int,hAPP(int,fun(int,int),inv,P_5),A_1))))) # label(fact_1178_inv__not__1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3037 (all Z_1 all W hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z_1),W) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W),Z_1)) # label(fact_367_zmult__commute) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3038 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) -> hAPP(real,real,exp(real),hAPP(real,real,ln,X)) = X)) # label(fact_1790_exp__ln) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3039 (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),Ma),Na)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),Na)))))) # label(fact_862_nat__mult__dvd__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3040 (all Y all X all N (-hBOOL(hAPP(nat,bool,even_odd_even(nat),N)) -> (hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),N) = X -> hAPP(real,real,hAPP(nat,fun(real,real),root,N),X) = Y))) # label(fact_2678_odd__real__root__unique) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3041 (all C_1 all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) -> (C_1 = B -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C_1))))) # label(fact_951_dvd_Oord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3042 (all X_a (archim1727834104eiling(X_a) -> hAPP(X_a,int,archim2085082626_floor(X_a),zero_zero(X_a)) = zero_zero(int))) # label(fact_2889_floor__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3043 (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)),hAPP(int,int,hAPP(int,fun(int,int),nat_tsub,X),Y)))))) # label(fact_1472_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3044 (all N hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N))))) # label(fact_2486_real__of__nat__fact__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3045 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,suc,N))) -> (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) -> hAPP(nat,nat,suc,N) = M))) # label(fact_1344_le__SucE) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3046 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B all C_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C_1)))))) # label(fact_723_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3047 (all X_a (linord581940658strict(X_a) -> (all A_3 all C all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),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(X_a),A_3),B_2)) | 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),B_2),A_3)))))) # label(fact_441_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3048 (all N all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,nat_1,X)),N) = hAPP(int,nat,nat_1,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),N)))) # label(fact_1048_Nat__Transfer_Otransfer__nat__int__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3049 (all X_a (order(X_a) -> (all Na all Ma all F ((all N_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,F,N_1)),hAPP(nat,X_a,F,hAPP(nat,nat,suc,N_1))))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),Ma)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,F,Na)),hAPP(nat,X_a,F,Ma)))))))) # label(fact_1578_lift__Suc__mono__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3050 (all R_3 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_3),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),hAPP(real,int,archim2085082626_floor(real),R_3))),one_one(real))))) # label(fact_3422_real__of__int__floor__add__one__gt) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3051 (all U all F ((all N_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),hAPP(nat,nat,F,N_1)))) -> hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),ord_less_eq(nat)),F)),U)))))) # label(fact_4190_finite__less__ub) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3052 (all J_2 all A_1 all P_5 all K_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),K_4))),P_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),J_2),K_4)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),K_4))),K_4)),P_5)))) # label(fact_1164_aux______1) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3053 (all A_1 hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,number_number_of(int),min)) = hAPP(int,int,uminus_uminus(int),A_1)) # label(fact_2091_zdiv__minus1__right) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3054 (all X_a (ordere236663937imp_le(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),B))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)))))) # label(fact_325_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3055 (all X_a (linorder(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)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)))))) # label(fact_513_linorder__linear) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3056 (all N all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,real_1(nat),N)),hAPP(nat,real,real_1(nat),X))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N),X)))))) # label(fact_2422_real__of__nat__div2) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3057 (all Xa all Ya all F ((all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),zero_zero(real)))) -> hAPP(real,real,F,Xa) = hAPP(real,real,F,Ya))) # label(fact_2723_DERIV__isconst__all) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3058 (all X_a (real_normed_field(X_a) -> (all G all E_3 all F all Xa all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),hAPP(X_a,X_a,F,Xa)),E_3)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),G),F)),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),E_3),D)))))))) # label(fact_3886_DERIV__chain_H) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3059 (all X_a (boolean_algebra(X_a) -> (all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Xa)),hAPP(X_a,X_a,uminus_uminus(X_a),Ya))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ya),Xa)))))) # label(fact_2987_compl__less__compl__iff) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3060 (all X_a all A_2 all B_1 (hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(fun(X_a,bool),bool),fun(fun(fun(X_a,bool),bool),bool),ord_less_eq(fun(fun(X_a,bool),bool)),A_2),B_1)) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),A_2)),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),B_1))))) # label(fact_1797_Union__mono) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3061 (all Ya all Xa (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ya),zOdd)) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xa),Ya)),zOdd))))) # label(fact_1713_even__minus__odd) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3062 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,K)),hAPP(int,int,bit0,L))))) # label(fact_53_rel__simps_I16_J) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3063 (all Z_1 zero_zero(real) = hAPP(complex,real,im,hAPP(real,complex,of_real(complex),Z_1))) # label(fact_2999_Im__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3064 (all Na all Ma (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),Ma)) -> (Ma = Na <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),hAPP(nat,nat,suc,Ma)))))) # label(fact_1331_not__less__less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3065 (all A_2 hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2)) = A_2) # label(fact_3674_transfer__int__nat__set__return__embed) # label(axiom) # label(non_clause). [assumption]. 7.70/7.77 3066 (all X_a all C all X_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),X_2)) -> hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),hAPP(int,fun(X_a,int),combk(int,X_a),C)),X_2) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(X_a,bool),nat,finite_card(X_a),X_2))))) # label(fact_4289_setsum__const) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3067 (all Xa (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd)) -> -(all K_1 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),K_1)),one_one(int)) != ti(int,Xa)))) # label(fact_1233_zOddE) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3068 (all X_1 ((hBOOL(hAPP(nat,bool,even_odd_even(nat),X_1)) -> hAPP(nat,real,sin_coeff,X_1) = zero_zero(real)) & (-hBOOL(hAPP(nat,bool,even_odd_even(nat),X_1)) -> hAPP(nat,real,sin_coeff,X_1) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),X_1),hAPP(nat,nat,suc,zero_zero(nat)))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),X_1)))))) # label(fact_2533_sin__coeff__def) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3069 (all R_3 hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,R_3),zero_zero(real)) = hAPP(real,complex,of_real(complex),R_3)) # label(fact_2802_complex__of__real__def) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3070 (all X_a (metric_space(X_a) -> (all X_2 ((all E (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E)) -> (exists M_3 all M_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),M_2)) -> (all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),hAPP(nat,X_a,X_2,M_2)),hAPP(nat,X_a,X_2,N_1))),E)))))))) -> hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_2)))))) # label(fact_3550_metric__CauchyI) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3071 (all Y all X (zero_zero(real) != hAPP(real,real,cos,X) -> (hAPP(real,real,cos,Y) != zero_zero(real) -> hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,cos,Y))) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,tan,X)),hAPP(real,real,tan,Y)))))) # label(fact_2443_lemma__tan__add1) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3072 (all X all Y hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),X)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Y),Y)))))) # label(fact_2154_le__real__sqrt__sumsq) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3073 (all X_b all X_c all X_a all F all G ((all X_1 all Y_1 hAPP(product_prod(X_a,X_b),X_c,G,hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X_1),Y_1)) = hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),F,X_1),Y_1)) -> ti(fun(product_prod(X_a,X_b),X_c),G) = hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),F))) # label(fact_4356_cond__split__eta) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3074 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) -> (all Y_1 exists N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y_1),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),X)))))) # label(fact_2464_reals__Archimedean3) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3075 (all X_a (linordered_idom(X_a) -> (all A_3 (ti(X_a,A_3) != zero_zero(X_a) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_3),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))))) # label(fact_98_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3076 (all K_4 all L_4 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),L_4)) -> (exists N_1 L_4 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_4),N_1)))) # label(fact_1692_le__Suc__ex) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3077 (all X_b all X_a all X_c all F all G all A_2 hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),G),A_2)) = hAPP(fun(X_c,bool),fun(X_a,bool),hAPP(fun(X_c,X_a),fun(fun(X_c,bool),fun(X_a,bool)),image(X_c,X_a),hAPP(fun(X_c,X_b),fun(X_c,X_a),hAPP(fun(X_b,X_a),fun(fun(X_c,X_b),fun(X_c,X_a)),combb(X_b,X_a,X_c),F),G)),A_2)) # label(fact_3873_image__image) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3078 (all X_a (archim1727834104eiling(X_a) -> (all Va all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),Va)),hAPP(X_a,int,archimedean_ceiling(X_a),Xa))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,number_number_of(X_a),Va)),one_one(X_a))),Xa)))))) # label(fact_3236_number__of__le__ceiling) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3079 (all Xa hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),cos)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),Xa),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,min))),hAPP(real,real,cos,Xa))),hAPP(real,real,sin,Xa))))) # label(fact_3940_DERIV__cos__realpow2a) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3080 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,number_number_of(nat),pls)),N)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M)),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,pls)))))) # label(fact_1487_Suc__diff__eq__diff__pred) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3081 (all K all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),Na))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)))) # label(fact_130_nat__add__left__cancel__less) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3082 (all Q_3 all R_3 hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_3),R_3)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_3),hAPP(int,int,uminus_uminus(int),R_3))) # label(fact_3097_negateSnd__eq) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3083 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),Xa)),one_one(real))) -> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),inverse_divide(real),one_one(real))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat)))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),Xa)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat))))),zero_zero(real)),sequentially)))) # label(fact_4525_zeroseq__arctan__series) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3084 (all X_a (archim1727834104eiling(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)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),hAPP(X_a,int,archim2085082626_floor(X_a),Y))))))) # label(fact_2891_floor__mono) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3085 (all A_1 all B all C_1 all D_3 all N (hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),D_3),N) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)),C_1) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N)) -> (hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B) = one_one(int) -> (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,B),C_1) -> (hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,C_1),A_1) = one_one(int) -> (exists K_1 exists L_3 exists M_2 (hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),K_1),N) = hAPP(int,int,abs_abs(int),A_1) & hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),L_3),N) = hAPP(int,int,abs_abs(int),B) & hAPP(int,int,abs_abs(int),C_1) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),M_2),N))))))))) # label(fact_3508_int__triple__relprime__power__divisors) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3086 (all F all Ya all G all Xa (hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),G),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),F),Ya)) -> hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(fun(nat,real),fun(nat,fun(real,real))),combs(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),div_div(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),G),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),div_div(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),minus_minus(nat)),one_one(nat)))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa),Ya)))))) # label(fact_4132_sums__if) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3087 (all A_2 all Xa (hBOOL(hAPP(int,bool,nat_is_nat,Xa)) -> (hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),A_2)) <-> hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),hAPP(int,nat,nat_1,Xa)),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2))))))) # label(fact_4884_transfer__int__nat__set__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3088 (all A_3 all B_2 (-(exists P_6 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_6),A_3)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_6),B_2)) & hBOOL(hAPP(int,bool,zprime,P_6)))) <-> one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_3),B_2))) # label(fact_3492_zgcd1__iff__no__common__primedivisor) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3089 (all X_a all X_b all Sb all Ta (ti(product_prod(X_a,X_b),Sb) = ti(product_prod(X_a,X_b),Ta) <-> hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),Ta) = hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),Sb) & hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),Ta) = hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),Sb))) # label(fact_3161_prod__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3090 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> M = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)),N))) # label(fact_1167_div__mult__self__is__m) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3091 (all A_3 all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),A_3)),one_one(real))),Xa)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(real,int,archim2085082626_floor(real),Xa))))) # label(fact_3438_less__floor__eq) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3092 (all X_a (minus(X_a) -> ti(fun(X_a,fun(X_a,X_a)),minus_minus(X_a)) = minus_minus(X_a))) # label(tsy_c_Groups_Ominus__class_Ominus_res) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3093 (all N N = hAPP(int,nat,nat_1,hAPP(nat,int,semiring_1_of_nat(int),N))) # label(fact_984_nat__int) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3094 (all I all J all F ((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),hAPP(nat,nat,F,I_2)),hAPP(nat,nat,F,J_1))))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,F,I)),hAPP(nat,nat,F,J)))))) # label(fact_2727_less__mono__imp__le__mono) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3095 (all P_2 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_2))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) & hBOOL(P_1) <-> hBOOL(P_2) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))))) # label(fact_573_conj__le__cong) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3096 (exists K_2 ((all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(complex,real,re,X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_1)),K_2)))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2)))) # label(fact_3045_Re_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3097 (all X_a (linordered_field(X_a) -> (all X all Y ((all E (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),E)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Y),E))))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)))))) # label(fact_1897_field__le__epsilon) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3098 (all X_a (real_normed_vector(X_a) -> (all X exists K_2 all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),X_1),X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),X_1)),K_2)))))) # label(fact_4783_scaleR__left_Obounded) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3099 (all X_a (comm_semiring_1(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1))))))) # label(fact_727_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3100 (all X all Y all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),Y)) = hAPP(real,real,hAPP(nat,fun(real,real),root,N),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),Y)))) # label(fact_2691_real__root__divide) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3101 (all Nat_1 hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,nat_size,Nat_1)),hAPP(nat,nat,suc,zero_zero(nat))) = hAPP(nat,nat,nat_size,hAPP(nat,nat,suc,Nat_1))) # label(fact_2945_nat_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3102 (all X_a all A_3 all Ra all B_2 (hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,bool),accp(X_a),Ra),B_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),Ra,A_3),B_2)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,bool),accp(X_a),Ra),A_3))))) # label(fact_2943_accp__downward) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3103 (all X_a all X_b all P_4 hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),P_4) = hAPP(product_prod(X_a,X_b),X_a,hAPP(fun(X_a,fun(X_b,X_a)),fun(product_prod(X_a,X_b),X_a),product_prod_case(X_a,X_b,X_a),combk(X_a,X_b)),P_4)) # label(fact_4033_fst__def) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3104 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),N)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),M)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)))))) # label(fact_1440_n__less__m__mult__n) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3105 (all N hAPP(complex,real,re,hAPP(nat,complex,semiring_1_of_nat(complex),N)) = hAPP(nat,real,semiring_1_of_nat(real),N)) # label(fact_3032_complex__Re__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3106 (all X_a (linorder(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)))) # label(fact_4833_min__max_Oinf_Oleft__idem) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3107 (all K_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K_4)) -> (exists N_1 hAPP(nat,int,semiring_1_of_nat(int),N_1) = ti(int,K_4)))) # label(fact_345_zero__le__imp__eq__int) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3108 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) <-> (exists K_1 Na = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),K_1))))) # label(fact_1383_less__iff__Suc__add) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3109 (all X_a (linord581940658strict(X_a) -> (all Xa all Ya (ti(X_a,Xa) != zero_zero(X_a) | ti(X_a,Ya) != zero_zero(X_a) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa),Xa)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ya),Ya)))))))) # label(fact_489_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3110 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) -> zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N))) # label(fact_1159_div__less) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3111 (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),hAPP(nat,nat,fact,M)),hAPP(nat,nat,fact,N))))) # label(fact_3564_fact__mono) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3112 (all X all Y hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(rat,real,ratreal,X)),hAPP(rat,real,ratreal,Y)) = hAPP(rat,real,ratreal,hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),X),Y))) # label(fact_3792_real__divide__code) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3113 (all A_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B))) -> hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),B)) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B))) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,number_number_of(int),min)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B))))) # label(fact_1645_negDivAlg__eqn) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3114 (all A_1 all B all C_1 hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A_1),B)),C_1))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A_1),B))) # label(fact_1546_mod__mult2__eq) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3115 (all X_4 all Y_3 hAPP(real,real,cos,hAPP(real,real,arccos,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X_4),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_4),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y_3),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X_4),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_4),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y_3),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) # label(fact_2361_cos__arccos__lemma1) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3116 (all N all M ((M = zero_zero(nat) -> hAPP(nat,nat,fact_fact(nat),N) = hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N))) & (M != zero_zero(nat) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)),hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N))) = hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N))))) # label(fact_2537_fact__add__num__eq__if2__nat) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3117 (all X_a (order(X_a) -> (all X_2 ((all N_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,X_2,N_1)),hAPP(nat,X_a,X_2,hAPP(nat,nat,suc,N_1))))) | (all N_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,X_2,hAPP(nat,nat,suc,N_1))),hAPP(nat,X_a,X_2,N_1)))) <-> hBOOL(hAPP(fun(nat,X_a),bool,monoseq(X_a),X_2)))))) # label(fact_3949_monoseq__Suc) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3118 (all K_4 all M all N hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K_4),N)) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K_4),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N))) # label(fact_3466_zgcd__left__commute) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3119 (all X_a (comm_ring(X_a) & monoid_mult(X_a) -> (all Xa all Na all Ya hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),hAPP(nat,nat,suc,Na))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Ya),hAPP(nat,nat,suc,Na))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Xa),Ya)),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Ya)),hAPP(nat,fun(nat,nat),minus_minus(nat),Na)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,Na))))))) # label(fact_4254_lemma__realpow__diff__sumr2) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3120 (all X_a (comm_monoid_mult(X_a) & comm_monoid_add(X_a) -> (all F all A_2 (hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2)) -> hAPP(fun(int,bool),X_a,hAPP(fun(int,X_a),fun(fun(int,bool),X_a),big_co302546470etprod(int,X_a),F),A_2) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co302546470etprod(nat,X_a),hAPP(fun(nat,int),fun(nat,X_a),hAPP(fun(int,X_a),fun(fun(nat,int),fun(nat,X_a)),combb(int,X_a,nat),F),semiring_1_of_nat(int))),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2)))))) # label(fact_3972_transfer__int__nat__sum__prod_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3121 (all X_a (comm_monoid_add(X_a) & comm_monoid_mult(X_a) -> (all F all A_2 hAPP(fun(int,bool),X_a,hAPP(fun(int,X_a),fun(fun(int,bool),X_a),big_co1399186613setsum(int,X_a),hAPP(fun(int,nat),fun(int,X_a),hAPP(fun(nat,X_a),fun(fun(int,nat),fun(int,X_a)),combb(nat,X_a,int),F),nat_1)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2)) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),A_2)))) # label(fact_4291_transfer__nat__int__sum__prod_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3122 (all A_1 all N hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(real,complex,cis,A_1)),N) = hAPP(real,complex,cis,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),A_1))) # label(fact_3058_DeMoivre) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3123 (all X hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) # label(fact_650_four__x__squared) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3124 (all A_3 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Ma)) -> (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_3),Ma) -> (exists X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),X_1),Ma)) & (all Y_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),Y_1),Ma)) & hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Y_1),hAPP(int,fun(int,bool),norRRset,Ma))) -> ti(int,X_1) = ti(int,Y_1))) & hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),hAPP(int,fun(int,bool),norRRset,Ma)))))))) # label(fact_3639_norR__mem__unique) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3125 (all Xa hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),cos)),cos)),Xa),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,cos,Xa))),hAPP(real,real,sin,Xa)))))) # label(fact_3935_DERIV__cos__cos__mult3) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3126 (all X hAPP(complex,complex,inverse_inverse(complex),X) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(complex,real,re,X)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,re,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,im,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,uminus_uminus(real),hAPP(complex,real,im,X))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,re,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,im,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) # label(fact_3015_complex__inverse__def) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3127 (all N hAPP(nat,int,semiring_1_of_nat(int),N) = hAPP(real,int,archim2085082626_floor(real),hAPP(nat,real,real_1(nat),N))) # label(fact_2885_floor__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3128 (all A_1 all B hAPP(rat,int,archim2085082626_floor(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)) # label(fact_3769_floor__Fract) # label(axiom) # label(non_clause). [assumption]. 7.81/7.77 3129 (all X_a all P_1 ((exists X1 hBOOL(hAPP(X_a,bool,P_1,X1))) -> hBOOL(hAPP(X_a,bool,P_1,hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1))))) # label(fact_4813_someI__ex) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3130 (all X all Y all C_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C_1),zero_zero(real))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),X)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C_1),hAPP(real,real,abs_abs(real),Y)))) -> zero_zero(real) = X))) # label(fact_1200_rabs__ratiotest__lemma) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3131 (all X_a (semiring_div(X_a) -> (all B all A_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)) = ti(X_a,A_1)))) # label(fact_1411_mod__div__equality2) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3132 (all X_a (order(X_a) -> (all Na all N_6 all F ((all N_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,F,N_1)),hAPP(nat,X_a,F,hAPP(nat,nat,suc,N_1))))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),N_6)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,F,Na)),hAPP(nat,X_a,F,N_6)))))))) # label(fact_1579_lift__Suc__mono__less) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3133 (all Ma all Na (ti(int,Na) = one_one(int) & one_one(int) = ti(int,Ma) | hAPP(int,int,number_number_of(int),min) = ti(int,Na) & ti(int,Ma) = hAPP(int,int,number_number_of(int),min) <-> one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Ma),Na))) # label(fact_691_zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3134 (all Real1_1 all Real2_1 zero_zero(nat) = hAPP(complex,nat,complex_size,hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Real1_1),Real2_1))) # label(fact_2769_complex_Osize_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3135 (all F all L (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),F),L),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),abs_abs(real)),F)),hAPP(real,real,abs_abs(real),L)),sequentially)))) # label(fact_4499_LIMSEQ__imp__rabs) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3136 (all X_a all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1)) & hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1)) <-> hBOOL(hAPP(fun(product_prod(fun(X_a,bool),fun(X_a,bool)),bool),bool,hAPP(product_prod(fun(X_a,bool),fun(X_a,bool)),fun(fun(product_prod(fun(X_a,bool),fun(X_a,bool)),bool),bool),member(product_prod(fun(X_a,bool),fun(X_a,bool))),hAPP(fun(X_a,bool),product_prod(fun(X_a,bool),fun(X_a,bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),product_prod(fun(X_a,bool),fun(X_a,bool))),product_Pair(fun(X_a,bool),fun(X_a,bool)),A_2),B_1)),finite_psubset(X_a))))) # label(fact_3204_in__finite__psubset) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3137 (all Ma all Na (-hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),Na))) <-> -(-hBOOL(hAPP(nat,bool,even_odd_even(nat),Ma)) <-> -hBOOL(hAPP(nat,bool,even_odd_even(nat),Na))))) # label(fact_2468_odd__add) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3138 (all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A_1)) -> -(hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))))) # label(fact_936_dvd_Oless__asym_H) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3139 (all B_1 all A_2 (hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2)) -> (hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,B_1)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,bool),fun(fun(int,bool),bool),ord_less(fun(int,bool)),A_2),B_1)) <-> hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less(fun(nat,bool)),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2)),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),B_1))))))) # label(fact_3682_transfer__int__nat__set__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3140 (all N (one_one(int) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),N) | hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),N) = hAPP(int,int,number_number_of(int),min))) # label(fact_918_neg__one__power) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3141 (all X_a all C_2 all A_2 ((all X_3 (hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),X_3),A_2)) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),X_3),C_2)))) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),A_2)),C_2)))) # label(fact_2722_Union__least) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3142 (all C_1 all A_1 all B (A_1 = B -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),C_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),C_1),B)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),C_1),A_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C_1))))) # label(fact_947_dvd_Oord__eq__less__trans) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3143 (all A_3 all A_2 all Ma (hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),Ma)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),A_2)) -> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_3),Ma) = one_one(int)))) # label(fact_3645_RRset__gcd) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3144 (all X_a (division_ring(X_a) -> (all N all A_1 (zero_zero(X_a) != ti(X_a,A_1) -> hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),N))))) # label(fact_2597_nonzero__power__inverse) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3145 (all Na all Xa ((hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) -> hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa) = hAPP(fun(real,bool),real,the(real),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)))),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(real,real),fun(real,fun(real,bool))),combb(real,fun(real,bool),real),fequal(real)),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),power_power(real)),Na))),Xa)))) & (-hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),zero_zero(real))) -> hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa) = hAPP(real,real,uminus_uminus(real),hAPP(fun(real,bool),real,the(real),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)))),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(real,real),fun(real,fun(real,bool))),combb(real,fun(real,bool),real),fequal(real)),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),power_power(real)),Na))),hAPP(real,real,uminus_uminus(real),Xa)))))) & (-hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),zero_zero(real))) -> hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa) = zero_zero(real))))) # label(fact_4163_root__def) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3146 (all Y_2 all B_2 all X_2 all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),X_2),A_3),sequentially)) -> (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),Y_2),B_2),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),hAPP(fun(nat,real),fun(nat,complex),hAPP(fun(nat,fun(real,complex)),fun(fun(nat,real),fun(nat,complex)),combs(nat,real,complex),hAPP(fun(nat,real),fun(nat,fun(real,complex)),hAPP(fun(real,fun(real,complex)),fun(fun(nat,real),fun(nat,fun(real,complex))),combb(real,fun(real,complex),nat),complex_1),X_2)),Y_2)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_3),B_2)),sequentially))))) # label(fact_4538_LIMSEQ__Complex) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3147 (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)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),Y)))))) # label(fact_262_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3148 (all X all A_1 hAPP(real,int,archim2085082626_floor(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),hAPP(int,real,real_1(int),A_1))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(real,int,archim2085082626_floor(real),X)),A_1)) # label(fact_3396_floor__subtract) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3149 (all T_1 order(filter(T_1))) # label(arity_Limits_Ofilter___Orderings_Oorder) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3150 (all N all M hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N))) # label(fact_1549_mult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3151 (all X_b all X_a (topological_space(X_a) & real_normed_algebra(X_b) -> (all C all F all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),F)),C)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4600_LIM__mult__left__zero) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3152 (all X_a (real_algebra_1(X_a) -> (all N hAPP(nat,X_a,semiring_1_of_nat(X_a),N) = hAPP(real,X_a,of_real(X_a),hAPP(nat,real,semiring_1_of_nat(real),N))))) # label(fact_2811_of__real__of__nat__eq) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3153 (all X hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) # label(fact_312_power2__ge__self) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3154 (all X_a (real_normed_field(X_a) -> (all F all Xa all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D)) <-> hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),inverse_divide(X_a)),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),minus_minus(X_a)),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),F),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Xa)))),hAPP(X_a,X_a,F,Xa)))),combi(X_a))),D),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a)))))))) # label(fact_4645_DERIV__iff) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3155 (all X_a all R_2 all S (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),R_2),S)) <-> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),R_2)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),S))))) # label(fact_4366_pred__subset__eq) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3156 (all Real1_1 all Real2_1 zero_zero(nat) = hAPP(complex,nat,size_size(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Real1_1),Real2_1))) # label(fact_2774_complex_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3157 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 all B all C_1 all D_3 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),D_3)))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),C_1))),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),D_3)))))))) # label(fact_1014_abs__diff__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3158 (all X_a (comm_monoid_add(X_a) -> (all F all Ma all K all Na hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),Ma),Na)) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),K)))))) # label(fact_4207_setsum__shift__bounds__nat__ivl) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3159 (all X_a (comm_monoid_add(X_a) -> (all F all Ma all K all Na hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),Na)) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),K)))))) # label(fact_4570_setsum__shift__bounds__cl__nat__ivl) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3160 (all P_5 (hBOOL(hAPP(nat,bool,prime,P_5)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5)))) # label(fact_3103_prime__ge__2) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3161 (all Diff all F all Na all H (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F -> ((all M_2 all T_4 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),Na)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_4),H)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T_4)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_2)),T_4),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),T_4))))) -> (exists T_4 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),T_4),H)) & hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),H))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,Na),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),H),Na))) = hAPP(real,real,F,H) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),T_4))))))))) # label(fact_4311_Maclaurin) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3162 (all X_a all F all Xa all A_2 ((all I_2 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I_2),A_2)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(X_a,nat,F,I_2)),Xa) = one_one(nat))) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co302546470etprod(X_a,nat),F),A_2)),Xa) = one_one(nat))) # label(fact_4155_setprod__coprime__nat) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3163 (all N all D_3 all A_1 (hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_3),A_1) = one_one(nat) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A_1),N)) = one_one(nat))) # label(fact_3302_coprime__exp__nat) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3164 (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_254_of__nat__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3165 (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_857_zdvd__int) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3166 (all X_a (number_ring(X_a) -> (all V all C_1 all W hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),C_1),hAPP(int,X_a,number_number_of(X_a),W))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),hAPP(int,int,uminus_uminus(int),W)))),C_1)))) # label(fact_2093_add__number__of__diff2) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3167 (all X all Y hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(complex,real,re,X)),hAPP(complex,real,re,Y))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(complex,real,im,X)),hAPP(complex,real,im,Y))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),X),Y)) # label(fact_3039_complex__add__def) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3168 (all X_a (linorder(X_a) -> (all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y)) = ti(X_a,X)))) # label(fact_4975_min__max_Oinf__sup__absorb) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3169 (all N all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),hAPP(nat,nat,suc,M))) -> hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M)),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M)),N)),hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N))))) # label(fact_2507_fact__diff__Suc) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3170 (all X_a (ab_semigroup_mult(X_a) -> times_times(X_a) = ti(fun(X_a,fun(X_a,X_a)),times_times(X_a)))) # label(tsy_c_Groups_Otimes__class_Otimes_5_res) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3171 (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),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) -> ti(int,X) = zero_zero(int) | ti(int,X) = one_one(int)))) # label(fact_336_int__pos__lt__two__imp__zero__or__one) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3172 (all A_1 all N all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A_1),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),N))) -> (zero_zero(nat) != N -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))))) # label(fact_855_divides__rev) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3173 (all A_1 all B hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),B)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) # label(fact_786_zspecial__product) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3174 (all L_4 pls != hAPP(int,int,bit1,L_4)) # label(fact_67_rel__simps_I39_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3175 (all Z_1 hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),Z_1),hAPP(real,complex,of_real(complex),hAPP(complex,real,norm_norm(complex),Z_1))) = hAPP(complex,complex,sgn_sgn(complex),Z_1)) # label(fact_2813_sgn__eq) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3176 (all K_4 all L_4 hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,bit1,K_4)),L_4) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit0,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),L_4))),L_4)) # label(fact_496_mult__Bit1) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3177 (all X_a (real_normed_vector(X_a) -> (all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_2)) -> hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2)))))) # label(fact_3518_Cauchy__Bseq) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3178 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),min)),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),arcsin),Xa),hAPP(real,real,inverse_inverse(real),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))))))) # label(fact_2643_DERIV__arcsin) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3179 (all X_a (comm_monoid_add(X_a) & t2_space(X_a) -> (all F (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),hAPP(fun(nat,X_a),X_a,suminf(X_a),F))))))) # label(fact_4119_summable__sums) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3180 (all P_1 hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),P_1))))) # label(fact_3849_Nat__Transfer_Otransfer__int__nat__set__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3181 (all R_3 all Q_3 all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),R_3),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),Q_3)) = ti(int,A_1) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_3)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_3),one_one(int))))))) # label(fact_576_self__quotient__aux2) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3182 (all X_a all X_b (real_normed_vector(X_b) -> (all F all A_3 all A_2 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),A_3),A_2)) -> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),hAPP(X_b,real,norm_norm(X_b),A_3)),A_2)))))) # label(fact_4487_tendsto__norm) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3183 (all Ya all Xa (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) -> (Ya = Xa <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya))))) # label(fact_955_dvd_Oantisym__conv) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3184 (all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,abs_abs(int),Z)),one_one(int))) <-> ti(int,Z) = zero_zero(int))) # label(fact_1018_zabs__less__one__iff) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3185 (all M all N hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N)),M))) # label(fact_3238_gcd__dvd1__nat) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3186 (all F all K_3 all Na ((all P_6 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),P_6),Na)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,F,P_6)),K_3)))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),Na)),K_3))))) # label(fact_4315_real__setsum__nat__ivl__bounded) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3187 (all X_a (semiring_div(X_a) -> (all A_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),one_one(X_a)) = ti(X_a,A_1)))) # label(fact_1075_div__by__1) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3188 (all X_a (ab_semigroup_add(X_a) -> (all A_1 all B all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C_1)))) # label(fact_297_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3189 (all X_a all F all S (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S)) -> ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),S)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(X_a,nat,F,X_1))))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co302546470etprod(X_a,nat),F),S)))))) # label(fact_3966_setprod__pos__nat__iff) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3190 (all X_b all X_a all F all A_2 all B_1 (ti(fun(X_a,bool),B_1) = hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2) -> hAPP(fun(fun(X_b,bool),bool),fun(fun(X_a,bool),bool),hAPP(fun(fun(X_b,bool),fun(X_a,bool)),fun(fun(fun(X_b,bool),bool),fun(fun(X_a,bool),bool)),image(fun(X_b,bool),fun(X_a,bool)),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F)),hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),pow(X_b),A_2)) = hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),B_1))) # label(fact_3658_image__Pow__surj) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3191 (all A_1 all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5)) -> (hBOOL(hAPP(int,bool,zprime,P_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),legendre,A_1),P_5)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),P_5))))) # label(fact_1135_Euler__Criterion) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3192 (all X_a (group_add(X_a) -> (all A_1 all B ti(X_a,B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B))))) # label(fact_1914_add__minus__cancel) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3193 (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),hAPP(int,int,bit0,K1)),hAPP(int,int,bit0,K2))))) # label(fact_220_less__eq__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3194 (all X_a (archim1727834104eiling(X_a) -> (all X all Z_1 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(X_a,int,archimedean_ceiling(X_a),X)),Z_1) = hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)))))) # label(fact_3622_ceiling__add__of__int) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3195 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) -> hAPP(real,real,sgn_sgn(real),X) = one_one(real))) # label(fact_2302_real__sgn__pos) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3196 (all X_a (real_n2089651433ebra_1(X_a) -> (all Z_1 hAPP(real,real,abs_abs(real),hAPP(int,real,ring_1_of_int(real),Z_1)) = hAPP(X_a,real,norm_norm(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1))))) # label(fact_3599_norm__of__int) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3197 (all B all Q_5 all R_4 all Q_3 all R_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_5)),R_4)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_3)),R_3))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_4),B)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_3),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_5),Q_3))))))) # label(fact_583_unique__quotient__lemma) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3198 (all K_4 all M all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N))) # label(fact_823_diff__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3199 (all Na (zero_zero(nat) = Na <-> hAPP(nat,int,semiring_1_of_nat(int),Na) = zero_zero(int))) # label(fact_27_int__eq__0__conv) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3200 (all P_1 all A0 all A1 (hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),upto_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))) -> ((all I_2 all J_1 (hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),upto_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),I_2),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),P_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),I_2),one_one(int))),J_1))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,I_2),J_1))))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A0),A1))))) # label(fact_3580_upto_Opinduct) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3201 (all A_1 all B all C_1 hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)),C_1) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),C_1)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),C_1))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),C_1)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),C_1))),C_1))) # label(fact_1372_zdiv__zadd1__eq) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3202 (all X_a (real_div_algebra(X_a) & divisi14063676e_zero(X_a) -> (all X hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(real,X_a,of_real(X_a),X)) = hAPP(real,X_a,of_real(X_a),hAPP(real,real,inverse_inverse(real),X))))) # label(fact_2809_of__real__inverse) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3203 (all F (hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),abs_abs(real)),F))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(fun(nat,real),real,suminf(real),F))),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),abs_abs(real)),F)))))) # label(fact_4046_summable__rabs) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3204 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) | Na = Ma <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),hAPP(nat,nat,suc,Na))))) # label(fact_1333_less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3205 (all D_3 all A_1 all B (hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)) = one_one(nat) -> one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_3),A_1))) # label(fact_3296_coprime__lmult__nat) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3206 (all P_5 all R_3 all S_3 all Q_3 (zero_zero(int) != ti(int,Q_3) -> (zero_zero(int) != ti(int,S_3) -> (hAPP(product_prod(int,int),product_prod(int,int),normalize,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),R_3),S_3)) = hAPP(product_prod(int,int),product_prod(int,int),normalize,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_5),Q_3)) -> hAPP(int,int,hAPP(int,fun(int,int),times_times(int),P_5),S_3) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),R_3),Q_3))))) # label(fact_3750_normalize__crossproduct) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3207 (all B_2 all C all A_3 all D_1 (hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_3),D_1) = one_one(int) -> (hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),B_2),C) = one_one(int) -> (hAPP(int,int,abs_abs(int),C) = hAPP(int,int,abs_abs(int),D_1) & hAPP(int,int,abs_abs(int),B_2) = hAPP(int,int,abs_abs(int),A_3) <-> hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,abs_abs(int),A_3)),hAPP(int,int,abs_abs(int),C)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,abs_abs(int),B_2)),hAPP(int,int,abs_abs(int),D_1)))))) # label(fact_4421_coprime__crossproduct__int) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3208 (all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),zero_zero(int))) -> (exists N_1 hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N_1))) = ti(int,X)))) # label(fact_2105_negD) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3209 (all K_4 all L_4 hAPP(int,int,bit1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K_4),L_4)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,bit1,K_4)),hAPP(int,int,bit0,L_4))) # label(fact_766_diff__bin__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3210 (all N hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),hAPP(nat,nat,suc,N)))) # label(fact_1270_lessI) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3211 (all X_a (ring_char_0(X_a) -> (all Z (ti(int,Z) = zero_zero(int) <-> hAPP(int,X_a,ring_1_of_int(X_a),Z) = zero_zero(X_a))))) # label(fact_3591_of__int__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3212 (all X_a (comm_ring_1(X_a) -> (all Z_1 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_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Y),Z_1)))))))) # label(fact_711_dvd__diff) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3213 (all X_a (archim1727834104eiling(X_a) -> (all Xa all Va (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archimedean_ceiling(X_a),Xa)),hAPP(int,int,number_number_of(int),Va))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),hAPP(int,X_a,number_number_of(X_a),Va))))))) # label(fact_3329_ceiling__le__number__of) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3214 (all X_a (ring_div(X_a) -> (all A_1 all B all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C_1))),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),C_1)))) # label(fact_1299_mod__diff__eq) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3215 (all J_2 all K_4 all A_1 all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),J_2),K_4)),A_1),P_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),J_2)),J_2)),K_4)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),J_2)),A_1)),P_5)))) # label(fact_1163_aux______3) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3216 (all Xa (hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa)) <-> (exists Y_1 Xa = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))),Y_1)))) # label(fact_2496_even__nat__equiv__def2) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3217 (all X_a all A_2 hAPP(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool),collect(fun(X_a,bool)),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),bool),hAPP(fun(fun(fun(X_a,bool),bool),bool),fun(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),bool)),combb(fun(fun(X_a,bool),bool),bool,fun(X_a,bool)),fEx(fun(X_a,bool))),hAPP(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),hAPP(fun(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool))),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(fun(X_a,bool),bool))),combc(fun(X_a,bool),fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool)),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),fun(bool,bool))),fun(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool))),hAPP(fun(fun(fun(X_a,bool),fun(bool,bool)),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool))),fun(fun(fun(X_a,bool),fun(fun(X_a,bool),fun(bool,bool))),fun(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool)))),combb(fun(fun(X_a,bool),fun(bool,bool)),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool)),fun(X_a,bool)),combs(fun(X_a,bool),bool,bool)),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(fun(X_a,bool),fun(bool,bool))),hAPP(fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(bool,bool))),fun(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(fun(X_a,bool),fun(bool,bool)))),combb(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(bool,bool)),fun(X_a,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(bool,bool))),combb(bool,fun(bool,bool),fun(X_a,bool)),fconj)),hAPP(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),hAPP(fun(fun(X_a,bool),fun(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_a,bool),bool))),fun(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_a,bool),fun(fun(X_a,bool),bool))),combc(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_a,bool),bool)),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_a,bool),bool))),hAPP(fun(fun(fun(X_a,bool),bool),fun(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_a,bool),bool))),fun(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_a,bool),bool)))),combb(fun(fun(X_a,bool),bool),fun(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_a,bool),bool)),fun(X_a,bool)),combb(fun(X_a,bool),bool,fun(X_a,bool))),fequal(fun(X_a,bool)))),uminus_uminus(fun(X_a,bool)))))),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),bool),hAPP(fun(fun(fun(X_a,bool),bool),bool),fun(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),bool)),combb(fun(fun(X_a,bool),bool),bool,fun(X_a,bool)),hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),A_2)),pow(X_a))))) = hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2))) # label(fact_4453_Pow__Compl) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3218 (all X_a (linord1117847801e_zero(X_a) -> (all Xa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),zero_zero(X_a))) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),Xa)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),Xa)),one_one(X_a))))))) # label(fact_2629_inverse__less__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3219 (all Na (Na = zero_zero(nat) <-> zero_zero(real) = hAPP(nat,real,real_1(nat),Na))) # label(fact_2381_real__of__nat__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3220 (all X all 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_eq(nat),X),Y)))) # label(fact_3066_termination__basic__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3221 (all A_2 (hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2)) -> (hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2))) <-> hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),A_2))))) # label(fact_3689_transfer__int__nat__set__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3222 (all Na all Va ((-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))) -> hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va))) = Na) & -hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))) <-> hAPP(nat,nat,suc,Na) = hAPP(int,nat,number_number_of(nat),Va))) # label(fact_3915_Suc__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3223 (all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) -> hAPP(int,real,real_1(int),X) = hAPP(nat,real,real_1(nat),hAPP(int,nat,nat_1,X)))) # label(fact_3419_real__nat__eq__real) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3224 (all X all N_4 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),N),N_4)) -> (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),X),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N_4),X)))))))) # label(fact_2717_real__root__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3225 (exists X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) & zero_zero(real) = hAPP(real,real,cos,X_1) & (all Y_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y_1),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) & hAPP(real,real,cos,Y_1) = zero_zero(real) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y_1)) -> Y_1 = X_1)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X_1)))) # label(fact_2349_cos__is__zero) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3226 (all K_4 ti(int,K_4) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K_4),pls)) # label(fact_742_diff__bin__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3227 (all X all N (-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),N))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(real,nat,natceiling,X)),hAPP(int,nat,number_number_of(nat),N)) = hAPP(real,nat,natceiling,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(int,real,number_number_of(real),N)))))) # label(fact_1656_natceiling__add__number__of) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3228 (all N all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),N)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),N)),one_one(real)))) -> hAPP(real,int,archimedean_ceiling(real),X) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N),one_one(int))))) # label(fact_3444_ceiling__eq) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3229 (all I all P_1 all K (hBOOL(hAPP(int,bool,P_1,K)) -> ((all I_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),I_2)) -> (hBOOL(hAPP(int,bool,P_1,I_2)) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),I_2),one_one(int))))))) -> ((all I_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_2),K)) -> (hBOOL(hAPP(int,bool,P_1,I_2)) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),I_2),one_one(int))))))) -> hBOOL(hAPP(int,bool,P_1,I)))))) # label(fact_2868_int__induct) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3230 (all B zero_zero(int) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),zero_zero(int)),B)) # label(fact_1307_zmod__zero) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3231 (all Z_1 all X all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Z_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Y),Z_1)))))) # label(fact_3265_gcd__semilattice__nat_Oinf__greatest) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3232 (exists K_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2)) & (all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,cnj,X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_1)),K_2)))))) # label(fact_2992_cnj_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3233 (all M all N all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),N))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),N)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),M))))) # label(fact_912_zprime__zdvd__zmult__better) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3234 (all X_a all F all F_1 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),F_1)) -> (hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),F_1) = zero_zero(nat) <-> (all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),F_1)) -> zero_zero(nat) = hAPP(X_a,nat,F,X_1)))))) # label(fact_4281_setsum__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3235 (all X_a (metric_space(X_a) -> (all X all Y all A_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),A_1),X)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),A_1),Y))))))) # label(fact_3532_dist__triangle3) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3236 (all X_a (division_ring(X_a) -> (all A_1 (zero_zero(X_a) != ti(X_a,A_1) -> hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)) = hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)))))) # label(fact_2598_nonzero__inverse__minus__eq) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3237 (all X_a (linord219039673up_add(X_a) -> (all A_3 (zero_zero(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),A_3) <-> ti(X_a,A_3) = zero_zero(X_a))))) # label(fact_321_double__zero__sym) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3238 (all P_5 (hBOOL(hAPP(nat,bool,prime,P_5)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),hAPP(nat,nat,suc,zero_zero(nat)))))) # label(fact_3090_prime__nd__one) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3239 (all X_a (semiring_1(X_a) -> (all Na hAPP(nat,X_a,semiring_1_of_nat(X_a),Na) = hAPP(X_a,X_a,hAPP(nat,fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(nat,fun(X_a,X_a)),semiri532925092at_aux(X_a),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),plus_plus(X_a)),one_one(X_a))),Na),zero_zero(X_a))))) # label(fact_4185_semiring__1__class_Oof__nat__code) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3240 (all X_a all X_b all X_c all F ti(fun(product_prod(X_a,X_b),X_c),F) = hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),hAPP(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,X_c)),hAPP(fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,X_c)),fun(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,X_c))),combb(fun(X_b,product_prod(X_a,X_b)),fun(X_b,X_c),X_a),hAPP(fun(product_prod(X_a,X_b),X_c),fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,X_c)),combb(product_prod(X_a,X_b),X_c,X_b),F)),product_Pair(X_a,X_b)))) # label(fact_4023_split__eta) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3241 (all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> hAPP(nat,nat,fact_fact(nat),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))))) # label(fact_2556_fact__reduce__nat) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3242 (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)) -> (hAPP(int,nat,nat_1,Xa) = hAPP(int,nat,nat_1,Ya) <-> ti(int,Xa) = ti(int,Ya))))) # label(fact_1001_transfer__nat__int__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3243 (all X all Y hAPP(int,int,hAPP(int,fun(int,int),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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),X),Y))) # label(fact_641_Nat__Transfer_Otransfer__int__nat__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3244 (all A_1 all B hAPP(int,rat,hAPP(int,fun(int,rat),fract,B),A_1) = hAPP(rat,rat,inverse_inverse(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B))) # label(fact_3765_inverse__rat) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3245 (all X hAPP(real,real,sin,hAPP(real,real,uminus_uminus(real),X)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,X))) # label(fact_2254_sin__minus) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3246 (all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Z)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(int,nat,nat_1,Z))))) # label(fact_1488_one__less__nat__eq) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3247 (all X all Y hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),X),Y)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),X),Y))) # label(fact_888_diff__square) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3248 (all X_a all X_b (real_normed_algebra(X_b) -> (all Ya all G all A_3 all A_2 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),A_3),A_2)) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),G)),Ya)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_3),Ya)),A_2)))))) # label(fact_4480_mult__left_Otendsto) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3249 (all N all M hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N)))),hAPP(nat,int,semiring_1_of_nat(int),M)))) # label(fact_2055_negative__zless) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3250 (all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),plus_plus(real)),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),sin)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),cos)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X_1),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,cos,X_1))),hAPP(real,real,sin,X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,cos,X_1))),hAPP(real,real,sin,X_1)))))) # label(fact_3945_DERIV__sin__circle__all) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3251 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) -> (hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),one_one(real)) = hAPP(real,real,ln,X) -> one_one(real) = X))) # label(fact_1716_ln__eq__minus__one) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3252 (all X_a (field(X_a) -> (all N all M all A_1 (ti(X_a,A_1) != zero_zero(X_a) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M)) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N))))))) # label(fact_1798_power__diff) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3253 (all A_3 all 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)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_3),B_2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_3))))) # label(fact_1112_pos__imp__zdiv__nonneg__iff) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3254 (all X_a (number_ring(X_a) -> (all W hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a))),hAPP(int,X_a,number_number_of(X_a),W)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,W))))) # label(fact_539_double__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3255 (all K_4 all M all N (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),N))) -> (zero_zero(int) != ti(int,K_4) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M),N))))) # label(fact_759_zdvd__mult__cancel) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3256 (all R_3 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),hAPP(real,int,archim2085082626_floor(real),R_3))),R_3))) # label(fact_3369_real__of__int__floor__le) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3257 (all X_b all X_a (topological_space(X_b) & real_normed_vector(X_a) -> (all F all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_a),fun(X_a,X_b),hAPP(fun(X_a,X_b),fun(fun(X_a,X_a),fun(X_a,X_b)),combb(X_a,X_b,X_a),F),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3))),hAPP(X_a,X_b,F,A_3)),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a)))) <-> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),hAPP(X_a,X_b,F,A_3)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4603_LIM__isCont__iff) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3258 (all X_a (order(X_a) -> (all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya)) <-> ti(X_a,Ya) = ti(X_a,Xa) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya)))))) # label(fact_567_order__le__less) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3259 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all Ya (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),Ya)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Ya))))))) # label(fact_50_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3260 (all P_1 all A_3 all B_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2)) -> (all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(product_prod(real,real),real,product_fst(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),N_1))),hAPP(product_prod(real,real),real,product_snd(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),N_1))))))) # label(fact_3165_Bolzano__bisect__le) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3261 (all Ya all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Ya))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),Ya))))) # label(fact_2701_real__root__ge__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3262 (all X_a (linordered_field(X_a) -> (all A_3 all B_2 all C (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),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)))))))) # label(fact_1873_neg__less__divide__eq) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3263 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y)) -> (exists X_1 (Y = hAPP(real,real,tan,X_1) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X_1)))))) # label(fact_2457_tan__total__pos) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3264 (all K_4 all M all N (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),M),N))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),N)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),M))))) # label(fact_714_zdvd__zdiffD) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3265 (all R_3 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,real_1(int),hAPP(real,int,archimedean_ceiling(real),R_3))),one_one(real))),R_3))) # label(fact_3428_real__of__int__ceiling__diff__one__le) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3266 (all K_4 min != hAPP(int,int,bit0,K_4)) # label(fact_671_rel__simps_I45_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3267 (all X_a (number_ring(X_a) -> one_one(X_a) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls)))) # label(fact_12_semiring__norm_I110_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3268 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),zero_zero(int))) -> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A_1)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),one_one(int))),A_1))),one_one(int)))) # label(fact_1263_neg__zmod__mult__2) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3269 (all X all Y hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_lcm,hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),X))),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),Y)))) = hAPP(int,int,hAPP(int,fun(int,int),int_lcm,X),Y)) # label(fact_3698_Nitpick_Oint__lcm__def) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3270 (all X (zero_zero(real) != hAPP(real,real,cos,X) -> hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,tan,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,inverse_inverse(real),hAPP(real,real,cos,X))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) # label(fact_2558_tan__sec) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3271 (all Q_2 all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),Q_2)),Ma)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),hAPP(nat,nat,suc,Q_2)))) <-> Q_2 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),Ma),Na)))) # label(fact_1495_split__div__lemma) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3272 (all X_b all X_a (topological_space(X_a) & real_normed_vector(X_b) -> (all G all Ma all F all L all Xa (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),Xa))) -> (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),Ma),hAPP(X_a,filter(X_a),at(X_a),Xa))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),plus_plus(X_b)),F)),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),uminus_uminus(X_b)),G))),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),L),hAPP(X_b,X_b,uminus_uminus(X_b),Ma))),hAPP(X_a,filter(X_a),at(X_a),Xa)))))))) # label(fact_4612_LIM__add__minus) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3273 (all Na all Xa (Xa != one_one(real) -> hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(real,fun(nat,real),power_power(real),Xa)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),Na)),one_one(real))),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Xa),one_one(real))))) # label(fact_4233_sumr__geometric) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3274 (all C_1 all D_3 all A_1 all B all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,C_1),D_3),M)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),C_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),D_3)),M))))) # label(fact_893_zcong__zmult) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3275 (all A_1 hAPP(real,complex,hAPP(real,fun(real,complex),rcis,one_one(real)),A_1) = hAPP(real,complex,cis,A_1)) # label(fact_3192_cis__rcis__eq) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3276 (all X_a (linorder(X_a) -> (all T_5 exists Z_2 all X_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_2),X_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),T_5),X_1)))))) # label(fact_2852_pinf_I8_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3277 (all X all Y all Z_1 hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y)),Z_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Y),Z_1))) # label(fact_3243_gcd__semilattice__nat_Oinf__assoc) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3278 (all B all Q_5 all R_4 all Q_3 all R_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_5)),R_4)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_3)),R_3))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_3),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_4)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_3),Q_5))))))) # label(fact_585_unique__quotient__lemma__neg) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3279 (all K all Ma all Na (hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),Na) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),Ma) <-> Ma = Na)) # label(fact_128_nat__add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 7.81/7.78 3280 (all X_a (comm_semiring_1(X_a) -> (all Na hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)),one_one(X_a))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a))),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),semiring_1_of_nat(X_a)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),one_one(nat)),Na)))))) # label(fact_4642_gauss__sum) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3281 (all X_a (real_normed_field(X_a) -> (all Y exists K_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2)) & (all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X_1),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_1)),K_2)))))))) # label(fact_2053_divide_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3282 (all X_a (topological_space(X_a) -> (all F all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,real),fun(X_a,bool),isCont(X_a,real),F),A_3)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,real),fun(X_a,bool),isCont(X_a,real),hAPP(fun(X_a,real),fun(X_a,real),hAPP(fun(real,real),fun(fun(X_a,real),fun(X_a,real)),combb(real,real,X_a),abs_abs(real)),F)),A_3)))))) # label(fact_4923_isCont__rabs) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3283 (all X_a (comm_semiring_1(X_a) -> (all B all A_1 all C_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1))))))) # label(fact_726_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3284 (all M all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N)) # label(fact_3273_gcd__add1__nat) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3285 (all X_a (plus(X_a) & dvd(X_a) & linorder(X_a) -> (all D_1 all Sb exists Z_2 all X_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Z_2)) -> (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),Sb))) <-> -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),Sb)))))))) # label(fact_1203_minf_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3286 (all X all Y all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),M)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),standardRes,M),X)),hAPP(int,int,hAPP(int,fun(int,int),standardRes,M),Y))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y)),M)))) # label(fact_1265_StandardRes__prop4) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3287 (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),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)))))))) # label(fact_538_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3288 (all X_a (number_ring(X_a) -> hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a)))) # label(fact_8_one__add__one__is__two) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3289 (all X_a (archim1727834104eiling(X_a) -> (all X hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(X_a,int,archim2085082626_floor(X_a),X))),X))))) # label(fact_3610_of__int__floor__le) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3290 (all S (hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),S)) -> (exists K_1 hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less_eq(fun(nat,bool)),S),hAPP(nat,fun(nat,bool),ord_lessThan(nat),K_1)))))) # label(fact_4352_finite__nat__bounded) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3291 (all X all Y (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),hAPP(real,real,exp(real),X)),hAPP(real,real,exp(real),Y))))) # label(fact_1747_exp__less__mono) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3292 (all X_a (topological_space(X_a) -> (all F all K all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),A_3),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),A_3),sequentially)))))) # label(fact_4504_LIMSEQ__offset) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3293 (all Sb all Ta all A_3 all B_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_3),B_2)) -> hBOOL(hAPP(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),bool,hAPP(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),hAPP(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),hAPP(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A_3),Sb)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B_2),Ta))),pair_leq)))) # label(fact_3514_pair__leqI1) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3294 (all A_1 all B hAPP(rat,product_prod(int,int),quotient_of,hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)) = hAPP(product_prod(int,int),product_prod(int,int),normalize,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_1),B))) # label(fact_3799_quotient__of__Fract) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3295 (all Na all Ma (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),Na)),hAPP(int,real,real_1(int),Ma))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),Na)),Ma)))) # label(fact_3413_number__of__less__real__of__int__iff) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3296 (all Xa all Na exists T_4 hAPP(real,real,sin,Xa) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(real,fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(real,fun(nat,fun(real,real))),combc(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),zero_zero(real))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),div_div(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),minus_minus(nat)),hAPP(nat,nat,suc,zero_zero(nat))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat)))))),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),T_4),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,real_1(nat),Na))),pi)))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),Na)))) # label(fact_4294_Maclaurin__sin__expansion) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3297 (all A_3 all X_2 all Xa (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),X_2),Xa),sequentially)) -> ((exists N_2 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,X_2,N_1)),A_3)))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),A_3))))) # label(fact_4518_LIMSEQ__le__const2) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3298 (all X_a (linordered_field(X_a) -> (all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1))))))) # label(fact_2585_inverse__less__imp__less__neg) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3299 (all A_1 all B all C_1 all D_3 hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),D_3)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C_1)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,C_1),D_3))) # label(fact_3767_divide__rat) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3300 (all N hAPP(int,int,ring_1_of_int(int),hAPP(nat,int,semiring_1_of_nat(int),N)) = hAPP(nat,int,semiring_1_of_nat(int),N)) # label(fact_3596_of__int__int__eq) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3301 (all X_a (real_normed_field(X_a) & banach(X_a) -> one_one(X_a) = hAPP(X_a,X_a,exp(X_a),zero_zero(X_a)))) # label(fact_1773_exp__zero) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3302 (all Xa all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma)) -> (hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,Xa),Ma) = one_one(int) -> hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),noXRRset,Ma),Xa)) = hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),norRRset,Ma))))) # label(fact_3640_card__nor__eq__noX) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3303 (all X_b all X_c all X_a all F1 all A_3 all B_2 hAPP(product_prod(X_b,X_c),X_a,hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),product_prod_case(X_b,X_c,X_a),F1),hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),A_3),B_2)) = hAPP(X_c,X_a,hAPP(X_b,fun(X_c,X_a),F1,A_3),B_2)) # label(fact_4022_Product__Type_Oprod_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3304 (all X_a (monoid_add(X_a) -> (all A_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),zero_zero(X_a)) = ti(X_a,A_1)))) # label(fact_320_add__0__right) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3305 (all A_3 hAPP(int,int,zfact,A_3) = hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),hAPP(int,fun(int,bool),d22set,A_3))) # label(fact_3957_d22set__prod__zfact) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3306 (all K all F all Na ((all I_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),Na)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_2),one_one(nat)))),hAPP(nat,int,F,I_2)))),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_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),Na)) & ti(int,K) = hAPP(nat,int,F,I_2))))))) # label(fact_1065_nat0__intermed__int__val) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3307 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)))) # label(fact_358_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3308 (all Na all Diff all F all H (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H)) & (all M_2 all T_4 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_4),H)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T_4)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),Na)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_2)),T_4),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),T_4))))) & hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F -> (exists T_4 (hAPP(real,real,F,H) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),H))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,Na),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),H),Na))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_4),H)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),T_4)))))) # label(fact_4307_Maclaurin2__objl) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3309 (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),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),A_1)))))) # label(fact_1976_abs__ge__minus__self) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3310 (all X_a (real_normed_field(X_a) -> (all Y_2 all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2)) -> (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),Y_2)) -> hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),plus_plus(X_a)),X_2)),Y_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),X_2)),hAPP(fun(nat,X_a),X_a,suminf(X_a),Y_2))))))) # label(fact_4009_suminf__add) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3311 (all X_a (real_normed_algebra(X_a) -> (all X exists K_2 all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_1)),K_2)))))) # label(fact_2111_mult__right_Obounded) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3312 (all X_a all A_2 all C_2 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),C_2))) -> -(all X_3 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),X_3)) -> -hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),X_3),C_2)))))) # label(fact_2113_UnionE) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3313 (all X_a (real_vector(X_a) -> (all A_1 all B all X hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),B),X)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_1),B)),X)))) # label(fact_4693_real__vector_Oscale__left__distrib) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3314 (all X all A_1 hAPP(real,real,exp(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_1),hAPP(real,real,ln,X))) = hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)) # label(fact_3407_powr__def) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3315 (all Na all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),Ma)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,real_1(nat),Na)),one_one(real))),hAPP(nat,real,real_1(nat),Ma))))) # label(fact_2406_nat__less__real__le) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3316 (all M all N hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,N),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M),N))) # label(fact_3461_zgcd__eq) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3317 (all A_3 all B_2 all C all Xa (hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),C),hAPP(int,nat,nat_1,Xa)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_3),B_2) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),hAPP(int,nat,nat_1,Xa))) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd)) -> (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_3),B_2) -> (exists K_1 hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),K_1),hAPP(int,nat,nat_1,Xa)) = ti(int,A_3))))))) # label(fact_3499_int__relprime__odd__power__divisors) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3318 (all Ma all Na (one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ma),Na) <-> Ma = one_one(nat) & one_one(nat) = Na)) # label(fact_619_nat__mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3319 (all X_1 hAPP(nat,fun(nat,bool),ord_less(nat),X_1) = hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,X_1))) # label(fact_3856_less__eq__Suc__le__raw) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3320 (all X_a (divisi14063676e_zero(X_a) & real_n1866405975lgebra(X_a) -> (all A_1 hAPP(real,real,inverse_inverse(real),hAPP(X_a,real,norm_norm(X_a),A_1)) = hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1))))) # label(fact_2581_norm__inverse) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3321 (all X all Y zero_zero(real) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,X)),hAPP(real,real,cos,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,sin,Y))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y))),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,cos,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,X)),hAPP(real,real,sin,Y))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) # label(fact_2309_sin__cos__add) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3322 (all A_1 all B all C_1 all N (hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),C_1),N) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N)) -> (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B) -> (exists K_1 hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),K_1),N) = hAPP(int,int,abs_abs(int),A_1)))))) # label(fact_3501_int__relprime__power__divisors) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3323 (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),N))))))) # label(fact_732_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3324 (all Y all X all A_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_1),zero_zero(real))) -> (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),X),Y)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),powr,Y),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1))))))) # label(fact_3374_powr__less__mono2__neg) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3325 (all B_5 all A_6 all A_1 all B (zero_zero(int) != hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B) -> (ti(int,A_1) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_6),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B)) -> (hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_5),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B)) = ti(int,B) -> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_6),B_5) = one_one(int))))) # label(fact_4418_gcd__coprime__int) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3326 (all A_1 hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,cos,A_1)),hAPP(real,real,sin,A_1)) = hAPP(real,complex,cis,A_1)) # label(fact_3057_cis__def) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3327 (all Y all X all Z_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X),Z_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z_1))))) # label(fact_3067_termination__basic__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3328 (all F all K all A_3 all B_2 (A_3 != B_2 -> ((all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),K))) -> K = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,F,B_2)),hAPP(real,real,F,A_3))),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B_2),A_3))))) # label(fact_2668_DERIV__const__ratio__const2) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3329 (all N (zero_zero(nat) != N -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)),one_one(nat))))) # label(fact_1514_Suc__mult__two__diff__one) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3330 (all X_b all X_a (semiring_0(X_a) -> (all F all A_2 all Ra hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(X_a,fun(X_b,X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(X_a,fun(X_b,X_a)),combc(X_b,X_a,X_a),hAPP(fun(X_b,X_a),fun(X_b,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_b),times_times(X_a)),F)),Ra)),A_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2)),Ra)))) # label(fact_4241_setsum__left__distrib) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3331 (all Na (hBOOL(hAPP(nat,bool,even_odd_even(nat),Na)) <-> (exists M_2 Na = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),M_2)))) # label(fact_2522_even__mult__two__ex) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3332 (all X_a (linorder(X_a) -> (all X all Y (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(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),Y),X))))))) # label(fact_515_linorder__cases) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3333 (all P_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),P_4)) & (all M_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M_2)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M_2),P_4)) -> one_one(int) = ti(int,M_2) | ti(int,M_2) = ti(int,P_4))) <-> hBOOL(hAPP(int,bool,zprime,P_4)))) # label(fact_798_zprime__def) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3334 (all L all U hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),hAPP(nat,nat,suc,L)),U) = hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_gr676655987ssThan(nat),L),U)) # label(fact_4249_atLeastSucLessThan__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3335 (all X_a all B_1 all A_2 (hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),B_1),A_2)) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),A_2))))) # label(fact_1796_Union__upper) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3336 (all X all A_1 hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1) != zero_zero(real)) # label(fact_3349_powr__not__zero) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3337 (all X_a all A_2 (hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,finite_finite_1(fun(X_a,bool)),A_2)) -> ((all M_3 (hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),M_3),A_2)) -> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),M_3)))) -> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),A_2)))))) # label(fact_2750_finite__Union) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3338 (all N all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M),N)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N),M)) -> ti(int,M) = ti(int,N)))))) # label(fact_761_zdvd__antisym__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3339 (all A_3 all B_2 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma)) -> (hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B_2),Ma) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_3),Ma) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),Ma))))) # label(fact_1426_zcong__zmod__eq) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3340 (all X_a (archim1727834104eiling(X_a) -> (all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archimedean_ceiling(X_a),Xa)),zero_zero(int))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),hAPP(int,X_a,number_number_of(X_a),min))))))) # label(fact_3334_ceiling__less__zero) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3341 (all N zero_zero(real) = hAPP(real,real,hAPP(nat,fun(real,real),root,N),zero_zero(real))) # label(fact_2674_real__root__zero) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3342 (all A_3 all A_2 all Ma ((hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),Ma)) & hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),A_2)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Ma)) -> hAPP(fun(int,bool),int,hilbert_Eps(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3)),Ma))),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(fun(int,bool),bool)),fun(fun(int,bool),fun(int,bool)),combc(int,fun(int,bool),bool),member(int)),hAPP(int,fun(int,bool),norRRset,Ma)))) = hAPP(int,int,hAPP(int,fun(int,int),hAPP(fun(int,bool),fun(int,fun(int,int)),rRset2norRR,A_2),Ma),A_3)) & (-(hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Ma)) & hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),Ma)) & hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),A_2))) -> hAPP(int,int,hAPP(int,fun(int,int),hAPP(fun(int,bool),fun(int,fun(int,int)),rRset2norRR,A_2),Ma),A_3) = zero_zero(int)))) # label(fact_4180_RRset2norRR__def) # label(axiom) # label(non_clause). [assumption]. 7.81/7.79 3343 (all K_4 all I_1 all J_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),J_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_4)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),K_4)),I_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),K_4)),J_2)))))) # label(fact_541_zmult__zless__mono2__lemma) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3344 (all X_a all P_1 all Q_1 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fdisj),P_1)),Q_1)))) <-> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),P_1))) & hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),Q_1))))) # label(fact_3875_finite__Collect__disjI) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3345 (all X_b all X_a (ab_group_add(X_a) -> (all F all A_2 hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),uminus_uminus(X_a)),F)),A_2)))) # label(fact_4238_setsum__negf) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3346 (all X_a (ordered_ring(X_a) -> (all C_1 all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1)))))))) # label(fact_434_mult__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3347 (all Y all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(int,nat,nat_1,X)),hAPP(int,nat,nat_1,Y)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),Y)))) # label(fact_1172_nat__div__distrib) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3348 (all X_a (number_ring(X_a) -> (all N hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,number_number_of(X_a),min)),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N))) = hAPP(int,X_a,number_number_of(X_a),min)))) # label(fact_1513_power__m1__odd) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3349 (all Q_3 all P_5 (hBOOL(hAPP(nat,bool,prime,P_5)) -> (hBOOL(hAPP(nat,bool,prime,Q_3)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),Q_3)) -> Q_3 = P_5)))) # label(fact_3071_primes__eq) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3350 (all X_a (real_normed_algebra(X_a) -> (all C all F (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) -> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),times_times(X_a),C)),F))))))) # label(fact_4026_summable__mult) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3351 (all X_a (real_normed_algebra(X_a) -> (all A_1 all R_3 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),B)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))))) # label(fact_4718_mult_OscaleR__right) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3352 (all X_a (number_ring(X_a) -> (all V all W all Z_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),W)),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),V),W))),Z_1)))) # label(fact_369_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3353 (all X_a (order(X_a) -> ti(fun(fun(nat,fun(nat,bool)),fun(fun(nat,X_a),bool)),mono(nat,X_a)) = mono(nat,X_a))) # label(tsy_c_Orderings_Oorder_Omono_res) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3354 (all N (hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))) -> -hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) # label(fact_2529_lemma__odd__mod__4__div__2) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3355 (all A_3 all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_3),hAPP(real,int,archimedean_ceiling(real),Xa))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,real_1(int),A_3)),one_one(real))),Xa)))) # label(fact_3442_le__ceiling__eq) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3356 (all A_3 all C all D_1 all B_2 (ti(int,B_2) != zero_zero(int) -> (ti(int,D_1) != zero_zero(int) -> (hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_2)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,C),D_1))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_3),D_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_2),D_1))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),B_2)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_2),D_1)))))))) # label(fact_3806_le__rat) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3357 (all A_1 all B (B != A_1 -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))))) # label(fact_957_dvd_Oneq__le__trans) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3358 (all K_4 ti(int,K_4) = hAPP(int,int,number_number_of(int),K_4)) # label(fact_123_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3359 (all N all D_3 all A_1 (hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_3),A_1) = one_one(int) -> one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_3),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),N)))) # label(fact_4398_coprime__exp__int) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3360 (all X_a (real_algebra_1(X_a) & real_normed_vector(X_a) -> (all X_2 all A_3 (hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),X_2),A_3)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,real),fun(nat,X_a),hAPP(fun(real,X_a),fun(fun(nat,real),fun(nat,X_a)),combb(real,X_a,nat),of_real(X_a)),X_2)),hAPP(real,X_a,of_real(X_a),A_3))))))) # label(fact_4111_of__real_Osums) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3361 (all X_a all S all T_3 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),S),T_3)) -> (-hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S)) -> -hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),T_3))))) # label(fact_3063_infinite__super) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3362 (all A_1 (one_one(int) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) -> one_one(int) = hAPP(int,int,abs_abs(int),A_1))) # label(fact_1057_power2__eq1__iff) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3363 (all X hAPP(real,real,uminus_uminus(real),hAPP(real,real,arctan,X)) = hAPP(real,real,arctan,hAPP(real,real,uminus_uminus(real),X))) # label(fact_2136_arctan__minus) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3364 (all X_a (real_normed_algebra(X_a) & banach(X_a) -> (all B_2 all A_3 (hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),A_3))) -> (hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),B_2))) -> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(nat,fun(fun(nat,bool),X_a)),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combs(nat,fun(nat,bool),X_a),hAPP(fun(nat,fun(nat,X_a)),fun(nat,fun(fun(nat,bool),X_a)),hAPP(fun(fun(nat,X_a),fun(fun(nat,bool),X_a)),fun(fun(nat,fun(nat,X_a)),fun(nat,fun(fun(nat,bool),X_a))),combb(fun(nat,X_a),fun(fun(nat,bool),X_a),nat),big_co1399186613setsum(nat,X_a)),hAPP(fun(nat,fun(nat,X_a)),fun(nat,fun(nat,X_a)),hAPP(fun(fun(nat,X_a),fun(nat,X_a)),fun(fun(nat,fun(nat,X_a)),fun(nat,fun(nat,X_a))),combb(fun(nat,X_a),fun(nat,X_a),nat),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),A_3))),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,X_a)),hAPP(fun(fun(nat,nat),fun(nat,X_a)),fun(fun(nat,fun(nat,nat)),fun(nat,fun(nat,X_a))),combb(fun(nat,nat),fun(nat,X_a),nat),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),B_2)),minus_minus(nat))))),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),A_3)),hAPP(fun(nat,X_a),X_a,suminf(X_a),B_2))))))))) # label(fact_4648_Cauchy__product__sums) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3365 (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N))))))) # label(fact_785_dvd__power) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3366 (all X_b all X_c all X_a all F all P_4 hAPP(X_c,X_a,hAPP(X_b,fun(X_c,X_a),F,hAPP(product_prod(X_b,X_c),X_b,product_fst(X_b,X_c),P_4)),hAPP(product_prod(X_b,X_c),X_c,product_snd(X_b,X_c),P_4)) = hAPP(product_prod(X_b,X_c),X_a,hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),product_prod_case(X_b,X_c,X_a),F),P_4)) # label(fact_4018_prod__case__beta) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3367 (all Code_numeral_4 hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,Code_numeral_4) != zero_zero(code_code_numeral)) # label(fact_2971_code__numeral_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3368 (all B all A_1 (zero_zero(nat) != A_1 -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),A_1)))) # label(fact_3286_gcd__le1__nat) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3369 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,ln,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),one_one(real)),X))))) # label(fact_1719_ln__gt__zero__imp__gt__one) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3370 (all N hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),N))) # label(fact_2899_coprime__plus1) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3371 (all W all V (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,number_number_of(int),V))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),W)),hAPP(int,int,number_number_of(int),V)))) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,number_number_of(int),min)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),W)),hAPP(int,int,number_number_of(int),V))) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,hAPP(int,int,number_number_of(int),W)),hAPP(int,int,number_number_of(int),V))) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),W)),hAPP(int,int,number_number_of(int),V)))) -> hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,hAPP(int,int,number_number_of(int),V)),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,hAPP(int,int,number_number_of(int),W)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,number_number_of(int),V)))) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,hAPP(int,int,number_number_of(int),W)),hAPP(int,int,number_number_of(int),V))))) # label(fact_1644_negDivAlg__eqn__number__of) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3372 (all X_a all F all A_2 hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co302546470etprod(X_a,nat),F),A_2)) = hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co302546470etprod(X_a,int),hAPP(fun(X_a,nat),fun(X_a,int),hAPP(fun(nat,int),fun(fun(X_a,nat),fun(X_a,int)),combb(nat,int,X_a),semiring_1_of_nat(int)),F)),A_2)) # label(fact_3967_int__setprod) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3373 (all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) -> hAPP(nat,nat,fact_fact(nat),hAPP(int,nat,nat_1,X)) = hAPP(int,nat,nat_1,hAPP(int,int,fact_fact(int),X)))) # label(fact_3224_transfer__nat__int__factorial) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3374 (all I_1 all K_4 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),J_2)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),I_1)),K_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),K_4)),I_1))) # label(fact_854_diff__add__assoc2) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3375 (exists K_2 ((all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(complex,real,im,X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_1)),K_2)))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2)))) # label(fact_3012_Im_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3376 (all P_2 all P_1 ((all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) -> (hBOOL(hAPP(int,bool,P_1,X_1)) <-> hBOOL(hAPP(int,bool,P_2,X_1))))) -> hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),P_2)) = hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),P_1)))) # label(fact_4361_transfer__nat__int__set__cong) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3377 (all Na all Xa exists T_4 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),T_4)),hAPP(real,real,abs_abs(real),Xa))) & hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,exp(real),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),Na))) = hAPP(real,real,exp(real),Xa))) # label(fact_4297_Maclaurin__exp__le) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3378 (all X all N hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,abs_abs(int),X)),N)))) # label(fact_1025_zero__le__zpower__abs) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3379 (all X_a (monoid_mult(X_a) & comm_semiring_0(X_a) -> (all Xa all Ya all Na hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ya),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Ya)),hAPP(nat,fun(nat,nat),minus_minus(nat),Na)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,Na)))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Ya)),hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,Na))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,Na)))))) # label(fact_4202_lemma__realpow__diff__sumr) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3380 (all B_2 all A_3 all P_4 (hBOOL(hAPP(int,bool,zprime,P_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int)))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),B_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),A_3)) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,A_3),P_4)))))))) # label(fact_1210_wset__mem) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3381 (all N all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),N))))))) # label(fact_2557_Ln_Oaux1) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3382 (all P_4 hAPP(product_prod(int,int),product_prod(int,int),hAPP(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_prod_case(int,int,product_prod(int,int)),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int)))),combs(int,fun(int,product_prod(int,int)),fun(int,product_prod(int,int))),hAPP(fun(int,fun(product_prod(int,int),product_prod(int,int))),fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),hAPP(fun(fun(product_prod(int,int),product_prod(int,int)),fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),fun(fun(int,fun(product_prod(int,int),product_prod(int,int))),fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int))))),combb(fun(product_prod(int,int),product_prod(int,int)),fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int))),int),combb(product_prod(int,int),product_prod(int,int),int)),hAPP(product_prod(int,int),fun(int,fun(product_prod(int,int),product_prod(int,int))),hAPP(fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),fun(product_prod(int,int),fun(int,fun(product_prod(int,int),product_prod(int,int)))),combc(int,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),hAPP(fun(int,bool),fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),hAPP(fun(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),fun(fun(int,bool),fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))),combb(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),int),if(product_prod(int,int))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),fequal(int)),zero_zero(int)))),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))))),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,fun(int,product_prod(int,int)))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combs(int,int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,fun(int,product_prod(int,int)))),fun(int,fun(int,fun(int,product_prod(int,int)))),hAPP(fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int)))),fun(fun(int,fun(int,fun(int,product_prod(int,int)))),fun(int,fun(int,fun(int,product_prod(int,int))))),combb(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int))),int),combc(int,int,product_prod(int,int))),hAPP(fun(int,fun(int,int)),fun(int,fun(int,fun(int,product_prod(int,int)))),hAPP(fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),fun(fun(int,fun(int,int)),fun(int,fun(int,fun(int,product_prod(int,int))))),combb(fun(int,int),fun(int,fun(int,product_prod(int,int))),int),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int))),hAPP(fun(int,int),fun(int,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(int,int),fun(int,fun(int,int))),combb(int,fun(int,int),int),times_times(int)),sgn_sgn(int))))),abs_abs(int)))),hAPP(rat,product_prod(int,int),quotient_of,P_4)) = hAPP(rat,product_prod(int,int),quotient_of,hAPP(rat,rat,inverse_inverse(rat),P_4))) # label(fact_4060_rat__inverse__code) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3383 (all X_a (semiring_div(X_a) -> div_div(X_a) = ti(fun(X_a,fun(X_a,X_a)),div_div(X_a)))) # label(tsy_c_Divides_Odiv__class_Odiv_res) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3384 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all X ((hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),X)),zero_zero(X_a))) -> hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(int,X_a,number_number_of(X_a),X)) = hAPP(X_a,X_a,abs_abs(X_a),hAPP(int,X_a,number_number_of(X_a),X))) & (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),X)),zero_zero(X_a))) -> hAPP(X_a,X_a,abs_abs(X_a),hAPP(int,X_a,number_number_of(X_a),X)) = hAPP(int,X_a,number_number_of(X_a),X)))))) # label(fact_2027_abs__number__of) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3385 (all N hAPP(nat,real,real_1(nat),N) = hAPP(int,real,real_1(int),hAPP(nat,int,semiring_1_of_nat(int),N))) # label(fact_3363_real__of__int__of__nat__eq) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3386 (all X_a (dense_linorder(X_a) -> (all Z_1 all Y ((all X_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Z_1)))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),Z_1)))))) # label(fact_2858_dense__le) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3387 (all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),Z)),Z)),zero_zero(int))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z),zero_zero(int))))) # label(fact_64_odd__less__0) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3388 (all R1 all A_1 all R2 all B hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R1),A_1)),hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R2),B)) = hAPP(real,complex,hAPP(real,fun(real,complex),rcis,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R1),R2)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_1),B))) # label(fact_3196_rcis__mult) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3389 (all Na all N_3 ((all X_1 (hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_1),N_3)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_1),Na)))) -> hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),N_3)))) # label(fact_3693_bounded__nat__set__is__finite) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3390 (all Xa (zero_zero(real) = Xa <-> -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Xa),Xa))))) # label(fact_631_not__real__square__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3391 (all X_a (divisi14063676e_zero(X_a) -> (all A_1 hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1))))) # label(fact_2561_inverse__minus__eq) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3392 (all X all Y hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y)),Y))) # label(fact_3254_gcd__semilattice__nat_Oinf__le2) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3393 (all X_b all X_a (metric_space(X_a) & real_normed_vector(X_b) -> (all F all L all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),minus_minus(X_b)),F)),L)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) <-> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4607_LIM__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3394 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) -> hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X))),one_one(real))) = hAPP(real,real,tan,X)))) # label(fact_2364_tan__half) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3395 (all R_3 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),R_3)) -> (exists N_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_3),hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N_1),one_one(int))))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),N_1)),R_3)))))) # label(fact_3447_reals__Archimedean__6b__int) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3396 (all Y all U_1 all X1_1 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)),X1_1)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X1_1),Y)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),U_1))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Y),hAPP(real,real,inverse_inverse(real),X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),U_1),hAPP(real,real,inverse_inverse(real),X1_1)))))))) # label(fact_2623_real__mult__inverse__cancel2) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3397 (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),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),zero_zero(X_a)))))) # label(fact_101_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3398 (all A_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,hAPP(nat,nat,suc,zero_zero(nat))),A_1))) # label(fact_2888_coprime__Suc0_H) # label(axiom) # label(non_clause). [assumption]. 7.82/7.79 3399 (all X_a (real_algebra_1(X_a) -> (all X all N hAPP(real,X_a,of_real(X_a),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(real,X_a,of_real(X_a),X)),N)))) # label(fact_2800_of__real__power) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3400 (all X all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(real,nat,natfloor,X)),hAPP(real,nat,natfloor,Y))))) # label(fact_1664_natfloor__mono) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3401 (all X_a (order(X_a) -> (all X all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) -> ti(X_a,X) != ti(X_a,Y))))) # label(fact_527_order__less__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3402 (all X_b all X_a (real_normed_vector(X_b) & topological_space(X_a) -> (all G all F all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3)) -> (hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),G),A_3)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),minus_minus(X_b)),F)),G)),A_3))))))) # label(fact_4924_isCont__diff) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3403 (all X_a (real_normed_field(X_a) -> ti(fun(fun(X_a,X_a),fun(X_a,fun(X_a,bool))),deriv(X_a)) = deriv(X_a))) # label(tsy_c_Deriv_Oderiv_res) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3404 (all X_a (number_ring(X_a) -> (all W hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,uminus_uminus(int),W)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(int,X_a,number_number_of(X_a),W))))) # label(fact_1927_arith__simps_I30_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3405 (all C_2 hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),C_2)))) # label(fact_3686_Nat__Transfer_Otransfer__int__nat__set__function__closures_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3406 (all A_2 all P_2 all P_1 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D)) -> ((exists Z_2 all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_2),X_1)) -> (hBOOL(hAPP(int,bool,P_1,X_1)) <-> hBOOL(hAPP(int,bool,P_2,X_1))))) -> ((all X_1 ((all Xa_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D))) -> (all Xb (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2)) -> ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa_1))))) -> (hBOOL(hAPP(int,bool,P_1,X_1)) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),D)))))) -> ((all X_1 all K_1 (hBOOL(hAPP(int,bool,P_2,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_1),D)))) <-> hBOOL(hAPP(int,bool,P_2,X_1)))) -> ((exists X1 hBOOL(hAPP(int,bool,P_1,X1))) <-> (exists X_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D))) & hBOOL(hAPP(int,bool,P_2,X_1)))) | (exists X_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D))) & (exists Xa_1 (hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xa_1),X_1))) & hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),A_2)))))))))))) # label(fact_4958_cppi) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3407 (all Ma all D_1 ((exists Q_4 hAPP(int,int,hAPP(int,fun(int,int),times_times(int),D_1),Q_4) = ti(int,Ma)) <-> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Ma),D_1) = zero_zero(int))) # label(fact_1361_zmod__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3408 (all X_a (linorder(X_a) -> (all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ya),Xa)) <-> -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya)))))) # label(fact_572_linorder__not__less) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3409 (all X_a (ring_1(X_a) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),zero_zero(X_a)),ring_1_Ints(X_a))))) # label(fact_3831_Ints__0) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3410 (all X_a (division_ring(X_a) -> one_one(X_a) = hAPP(X_a,X_a,inverse_inverse(X_a),one_one(X_a)))) # label(fact_2565_inverse__1) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3411 (all X_a all X_b all A_2 all B_1 (hBOOL(hAPP(fun(sum_sum(X_a,X_b),bool),bool,finite_finite_1(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_2),B_1))) -> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_1)))) # label(fact_3210_finite__PlusD_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3412 (all X_a (division_ring(X_a) -> (all A_1 (zero_zero(X_a) != ti(X_a,A_1) -> zero_zero(X_a) != hAPP(X_a,X_a,inverse_inverse(X_a),A_1))))) # label(fact_2575_nonzero__imp__inverse__nonzero) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3413 (all F all Xa all L (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),Xa),L)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),L)) -> (exists D_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_2)) & (all H_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H_1)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),H_1),D_2)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Xa),H_1))),hAPP(real,real,F,Xa))))))))))) # label(fact_2664_DERIV__pos__inc__left) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3414 (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),hAPP(int,int,bit0,K1)),hAPP(int,int,bit0,K2))))) # label(fact_33_less__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3415 (all X_a (real_normed_field(X_a) -> (all E_3 all F all Xa all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D)) -> (ti(X_a,D) = ti(X_a,E_3) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),E_3))))))) # label(fact_2646_Deriv_Olemma__DERIV__subst) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3416 (all B all A_1 all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> (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),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),P_5)) & -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,B),zero_zero(int)),P_5)) -> -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)),zero_zero(int)),P_5)))))) # label(fact_920_zcong__zprime__prod__zero__contra) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3417 (all X_b (division_ring(X_b) & number(X_b) -> (all W hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),inverse_divide(X_b),one_one(X_b)),hAPP(int,X_b,number_number_of(X_b),W)) = hAPP(X_b,X_b,inverse_inverse(X_b),hAPP(int,X_b,number_number_of(X_b),W))))) # label(fact_2622_inverse__eq__divide__number__of) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3418 (all X_a ti(fun(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a))),nat_rec(X_a)) = nat_rec(X_a)) # label(tsy_c_Nat_Onat_Onat__rec_res) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3419 (all X_a (comple44059174attice(X_a) -> (all Xa all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa),A_2)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),hAPP(fun(X_a,bool),X_a,complete_Sup_Sup(X_a),A_2))))))) # label(fact_1815_complete__lattice__class_OSup__upper) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3420 (all K all L (hAPP(int,int,bit1,L) = hAPP(int,int,bit1,K) <-> ti(int,L) = ti(int,K))) # label(fact_42_rel__simps_I51_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3421 (all X_a (order(X_a) -> (all Z_1 all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z_1),Y)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z_1),X))))))) # label(fact_501_xt1_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3422 (all Nat_1 hAPP(nat,nat,size_size(nat),hAPP(nat,nat,suc,Nat_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,size_size(nat),Nat_1)),hAPP(nat,nat,suc,zero_zero(nat)))) # label(fact_2946_nat_Osize_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3423 (all X_a (linordered_idom(X_a) -> (all Xa all Na (-hBOOL(hAPP(nat,bool,even_odd_even(nat),Na)) -> (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)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),Na)))))))) # label(fact_2490_zero__le__odd__power) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3424 (all X_a (real_normed_field(X_a) -> (all G all E_3 all F all Xa all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),Xa),E_3)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),minus_minus(X_a)),F)),G)),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),D),E_3)))))))) # label(fact_3866_DERIV__diff) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3425 (all N all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,real_1(nat),N)),hAPP(nat,real,real_1(nat),X))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N),X)))),one_one(real)))) # label(fact_2424_real__of__nat__div3) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3426 (all Z_1 hAPP(int,int,hAPP(int,fun(int,int),times_times(int),one_one(int)),Z_1) = ti(int,Z_1)) # label(fact_405_zmult__1) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3427 (all X_a (order(X_a) -> (all Z_1 all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),Y)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),X))))))) # label(fact_552_xt1_I8_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3428 (all X_a (linordered_field(X_a) -> (all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),zero_zero(X_a)))))))) # label(fact_1841_divide__pos__neg) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3429 (all N all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),N)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),N)),one_one(real)))) -> ti(int,N) = hAPP(real,int,archim2085082626_floor(real),X)))) # label(fact_3437_floor__eq2) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3430 (all N all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),hAPP(nat,nat,fact_fact(nat),N)))))) # label(fact_2552_dvd__fact__nat) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3431 (all X_a (real_n1866405975lgebra(X_a) -> (all X_2 all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially)) -> (zero_zero(X_a) != ti(X_a,A_3) -> hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),inverse_inverse(X_a)),X_2)))))))) # label(fact_4458_Bseq__inverse) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3432 (all N hAPP(real,int,archim2085082626_floor(real),hAPP(real,real,uminus_uminus(real),hAPP(int,real,real_1(int),N))) = hAPP(int,int,uminus_uminus(int),N)) # label(fact_3398_floor__minus__real__of__int) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3433 (all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),N)) & -hBOOL(hAPP(nat,bool,prime,N)) -> (exists M_2 exists K_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),M_2)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),K_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),N)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_1),N)) & N = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_2),K_1))))) # label(fact_3116_not__prime__ex__mk) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3434 (all X_b all X_a (semiring_1(X_a) -> (all Ya all A_2 hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(X_a,fun(X_b,X_a),combk(X_a,X_b),Ya)),A_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_2))),Ya)))) # label(fact_4255_setsum__constant) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3435 (all X_a (number_ring(X_a) -> (all V hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),hAPP(int,int,bit1,pls))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),one_one(X_a))))) # label(fact_7_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3436 (all X_a (linordered_semidom(X_a) -> (all A_1 all N all N_4 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),N_4)) -> (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_4)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N))))))))) # label(fact_237_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3437 (all X_a all X_b all X_c combs(X_a,X_b,X_c) = ti(fun(fun(X_a,fun(X_b,X_c)),fun(fun(X_a,X_b),fun(X_a,X_c))),combs(X_a,X_b,X_c))) # label(tsy_c_COMBS_res) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3438 (all X hAPP(complex,complex,uminus_uminus(complex),X) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),ii),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),ii),X))) # label(fact_2780_complex__i__mult__minus) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3439 (all X_a (real_vector(X_a) -> (all A_3 all Xa (ti(X_a,Xa) = zero_zero(X_a) | A_3 = zero_zero(real) <-> hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_3),Xa) = zero_zero(X_a))))) # label(fact_4736_scaleR__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3440 (all A_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,one_one(nat)),A_1))) # label(fact_2882_coprime__1_H) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3441 (all X_a (ring_1(X_a) -> (all W all Z_1 hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,ring_1_of_int(X_a),W)),hAPP(int,X_a,ring_1_of_int(X_a),Z_1))))) # label(fact_3587_of__int__mult) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3442 (all X all Y (hAPP(complex,real,re,X) = hAPP(complex,real,re,Y) -> (hAPP(complex,real,im,Y) = hAPP(complex,real,im,X) -> X = Y))) # label(fact_3021_complex__eqI) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3443 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),Xa)) -> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(real,fun(nat,real),power_power(real),Xa))),zero_zero(real)),sequentially)))) # label(fact_4457_LIMSEQ__inverse__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3444 (all X_a (linord1117847801e_zero(X_a) -> (all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),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_2)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_3)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),B_2)),zero_zero(X_a))))))) # label(fact_1844_divide__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3445 (all N hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,real_1(nat),N)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)))) # label(fact_2430_two__realpow__gt) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3446 (all X_a (ordere223160158up_add(X_a) -> (all C_1 all D_3 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D_3)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D_3)))))))) # label(fact_327_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3447 (all X_a (linorder(X_a) -> (all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)) = ti(X_a,X)))) # label(fact_4974_min__max_Osup__inf__absorb) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3448 (all A_3 all B_2 (hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),norm_frac_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_3),B_2))) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_2),zero_zero(int))) -> (ti(int,B_2) = zero_zero(int) | ti(int,A_3) = zero_zero(int) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,A_3),B_2) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))) & (-(zero_zero(int) = ti(int,A_3) | ti(int,B_2) = zero_zero(int)) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,A_3),B_2) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),int_gcd,A_3),B_2))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B_2),hAPP(int,int,hAPP(int,fun(int,int),int_gcd,A_3),B_2))))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_2),zero_zero(int))) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,hAPP(int,int,uminus_uminus(int),A_3)),hAPP(int,int,uminus_uminus(int),B_2)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,A_3),B_2)))) # label(fact_3946_norm__frac_Opsimps) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3449 (all X hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),X))) = hAPP(int,int,abs_abs(int),X)) # label(fact_1218_int__nat__abs) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3450 (all K_4 hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,number_number_of(int),K_4)),one_one(int)) = hAPP(int,rat,number_number_of(rat),K_4)) # label(fact_3797_rat__number__expand_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3451 (all X_a (power(X_a) -> (all A_1 all N hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,N))))) # label(fact_1371_power__Suc) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3452 (all Na (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),Na)),zero_zero(real))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Na),zero_zero(int))))) # label(fact_3408_real__of__int__lt__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3453 (all K_4 ((K_4 != zero_zero(code_code_numeral) -> (hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),K_4),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) != zero_zero(code_code_numeral) -> hAPP(code_code_numeral,int,code_int_of,K_4) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(code_code_numeral,int,code_int_of,hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),K_4),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),one_one(int))) & (hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),K_4),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = zero_zero(code_code_numeral) -> hAPP(code_code_numeral,int,code_int_of,K_4) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(code_code_numeral,int,code_int_of,hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),K_4),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) & (zero_zero(code_code_numeral) = K_4 -> zero_zero(int) = hAPP(code_code_numeral,int,code_int_of,K_4)))) # label(fact_1684_Code__Numeral_Oint__of__code) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3454 (all X all Y ((zero_zero(nat) = Y -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = X) & (Y != zero_zero(nat) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y)))) # label(fact_3300_gcd__nat_Osimps) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3455 (all Sb (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Sb)) -> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),powr),real_1(nat))),hAPP(real,real,uminus_uminus(real),Sb))),zero_zero(real)),sequentially)))) # label(fact_4513_LIMSEQ__neg__powr) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3456 (all X all N_4 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),N),N_4)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N_4),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X))))))) # label(fact_2703_real__root__decreasing) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3457 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit0,K)),min)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),min)))) # label(fact_684_rel__simps_I28_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3458 (all X_a (order(X_a) -> (all A_1 all B (ti(X_a,B) != ti(X_a,A_1) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1))))))) # label(fact_562_xt1_I12_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3459 (all X_a (real_normed_algebra(X_a) -> (all A_1 all B all B_5 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),B_5)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B_5))))) # label(fact_1250_mult_Odiff__right) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3460 (all X all A_1 all B (hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B) = one_one(nat) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),A_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),B)) -> X = one_one(nat))))) # label(fact_3289_coprime__common__divisor__nat) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3461 (all A_1 hAPP(complex,real,im,hAPP(real,complex,cis,A_1)) = hAPP(real,real,sin,A_1)) # label(fact_3052_Im__cis) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3462 (all N all A_1 hAPP(complex,real,re,hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(real,complex,cis,A_1)),N)) = hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),A_1))) # label(fact_3050_cos__n__Re__cis__pow__n) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3463 (all A_3 all B_2 (hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_3),B_2) = one_one(int) <-> (all D_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),D_2)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_2),B_2)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_2),A_3)) <-> one_one(int) = ti(int,D_2))))) # label(fact_4432_coprime__int) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3464 (all X_a (group_add(X_a) -> (all A_1 hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)) = ti(X_a,A_1)))) # label(fact_1921_minus__minus) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3465 (all Z_1 ti(int,Z_1) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),zero_zero(int)),Z_1)) # label(fact_83_zadd__0) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3466 (all L_4 hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),min),hAPP(int,int,bit1,L_4)) = hAPP(int,int,bit0,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),min),L_4))) # label(fact_781_diff__bin__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3467 (all X_a (topological_space(X_a) & comm_monoid_add(X_a) -> (all F all Sb (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(fun(nat,bool),X_a),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combb(fun(nat,bool),X_a,nat),hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F)),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)))),Sb),sequentially)) <-> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),Sb)))))) # label(fact_4522_sums__def) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3468 (all X all Y hAPP(complex,real,re,hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,re,X)),hAPP(complex,real,re,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,im,X)),hAPP(complex,real,im,Y)))) # label(fact_3036_complex__Re__mult) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3469 (all X_b all X_a (real_normed_vector(X_a) & topological_space(X_b) -> (all L all F all G all A_3 all R_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_2)) -> ((all X_1 (ti(X_a,X_1) != ti(X_a,A_3) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_1),A_3))),R_2)) -> hAPP(X_a,X_b,F,X_1) = hAPP(X_a,X_b,G,X_1)))) -> (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3))))))))) # label(fact_4671_LIM__equal2) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3470 (all M_5 all D_4 (hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_5),D_4) = zero_zero(nat) -> (exists Q_4 M_5 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),D_4),Q_4)))) # label(fact_1576_mod__eq__0D) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3471 (all D all Xa (zero_zero(real) != Xa -> ((hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) -> hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,inverse_inverse(real),hAPP(real,real,sqrt,Xa))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = D) -> ((hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),zero_zero(real))) -> hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,inverse_inverse(real),hAPP(real,real,sqrt,Xa)))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = D) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),sqrt),Xa),D)))))) # label(fact_2657_DERIV__real__sqrt__generic) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3472 (all X_a all X_b product_snd(X_a,X_b) = ti(fun(product_prod(X_a,X_b),X_b),product_snd(X_a,X_b))) # label(tsy_c_Product__Type_Osnd_res) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3473 (all X_a (number(X_a) & linord1117847801e_zero(X_a) -> (all B_2 all C all Wa ((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),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),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(X_a),C),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Wa)))) & (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_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),C)),B_2)))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)),hAPP(int,X_a,number_number_of(X_a),Wa))))))) # label(fact_1802_divide__le__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3474 (all X_b all X_a (topological_space(X_a) & real_normed_vector(X_b) -> (all F all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4634_LIM__norm__zero__cancel) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3475 (all X_a (ord(X_a) -> (all I all K (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),I),K)) <-> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I),hAPP(X_a,fun(X_a,bool),ord_lessThan(X_a),K))))))) # label(fact_4341_lessThan__iff) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3476 (all X_a all F all F_1 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),F_1)) -> ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),F_1)) -> hAPP(X_a,nat,F,X_1) = one_one(nat))) <-> one_one(nat) = hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co302546470etprod(X_a,nat),F),F_1)))) # label(fact_3965_setprod__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3477 (all N hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N))) # label(fact_397_less__eq__nat_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3478 (all X all Y hAPP(int,int,hAPP(int,fun(int,int),nat_tsub,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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),X),Y))) # label(fact_1482_Nat__Transfer_Otransfer__int__nat__functions_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3479 (all X_a (field_inverse_zero(X_a) -> (all B_2 all C all A_3 (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C) = ti(X_a,A_3) <-> (zero_zero(X_a) != ti(X_a,C) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C) = ti(X_a,B_2)) & (ti(X_a,C) = zero_zero(X_a) -> zero_zero(X_a) = ti(X_a,A_3)))))) # label(fact_1847_divide__eq__eq) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3480 (all X_a (comm_semiring_1(X_a) -> (all X all P_5 all Q_3 hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),P_5),Q_3)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),P_5)),Q_3)))) # label(fact_607_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3481 (all X_a all X_b (comm_monoid_add(X_b) -> (all G all F all S all T_3 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),T_3)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),S),T_3)) -> ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),T_3),S))) -> hAPP(X_a,X_b,F,X_1) = zero_zero(X_b))) -> ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),S)) -> hAPP(X_a,X_b,F,X_1) = hAPP(X_a,X_b,G,X_1))) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),T_3) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),S)))))))) # label(fact_4372_setsum__mono__zero__cong__right) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3482 (all A_3 all B_2 all C all D_1 all Xa (hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),D_1),hAPP(int,nat,nat_1,Xa)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_3),B_2)),C) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),hAPP(int,nat,nat_1,Xa))) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd)) -> (hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_3),B_2) = one_one(int) -> (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,B_2),C) -> (hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,C),A_3) = one_one(int) -> (exists K_1 exists L_3 exists M_2 (ti(int,A_3) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),K_1),hAPP(int,nat,nat_1,Xa)) & ti(int,C) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),M_2),hAPP(int,nat,nat_1,Xa)) & ti(int,B_2) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),L_3),hAPP(int,nat,nat_1,Xa))))))))))) # label(fact_3506_int__triple__relprime__odd__power__divisors) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3483 (all X_a all X_b (real_normed_vector(X_b) -> (all G all B_2 all F all A_3 all A_2 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),A_3),A_2)) -> (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),B_2),A_2)) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),minus_minus(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),A_3),B_2)),A_2))))))) # label(fact_4501_tendsto__diff) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3484 (all Z_1 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),Z_1),Y)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),Z_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Z_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z_1),X))))) # label(fact_935_dvd_Oless__trans) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3485 (all X_a (linord219039673up_add(X_a) -> (all A_3 (ti(X_a,A_3) = zero_zero(X_a) <-> hAPP(X_a,X_a,uminus_uminus(X_a),A_3) = ti(X_a,A_3))))) # label(fact_1935_equal__neg__zero) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3486 (all X_a (field(X_a) -> (all B all A_1 (zero_zero(X_a) != ti(X_a,A_1) -> (zero_zero(X_a) != ti(X_a,B) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1))),hAPP(X_a,X_a,inverse_inverse(X_a),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B))))))) # label(fact_2612_inverse__add) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3487 (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),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)),X)))))))) # label(fact_490_mult__right__le__one__le) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3488 (all X_a all Na all F all C all H ((all N_1 hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),C),H),N_1) = hAPP(nat,X_a,F,N_1)) -> hAPP(X_a,X_a,hAPP(nat,fun(X_a,X_a),H,Na),hAPP(nat,X_a,F,Na)) = hAPP(nat,X_a,F,hAPP(nat,nat,suc,Na)))) # label(fact_4357_def__nat__rec__Suc) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3489 (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),hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1))),zero_zero(X_a)))))) # label(fact_2001_abs__minus__le__zero) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3490 (all B_2 all C all A_3 all D_1 (hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_3),D_1) = one_one(nat) -> (hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),B_2),C) = one_one(nat) -> (B_2 = A_3 & D_1 = C <-> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_3),C) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B_2),D_1))))) # label(fact_3295_coprime__crossproduct__nat) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3491 (all X_a (linorder(X_a) -> (all B 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),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),B))))))) # label(fact_4986_min__max_Oless__supI1) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3492 (all Ta all B_1 all D_1 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),D)) -> (all X_1 ((all Xa_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D))) -> (all Xb (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1)) -> ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa_1))))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),Ta))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),D)),Ta)))))))) # label(fact_4674_bset_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3493 (all X_a (archim1727834104eiling(X_a) -> (all Xa all Z (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),hAPP(int,X_a,ring_1_of_int(X_a),Z))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archimedean_ceiling(X_a),Xa)),Z)))))) # label(fact_3621_ceiling__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3494 (all N all M (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),hAPP(nat,nat,suc,M))) -> M = N))) # label(fact_1330_less__antisym) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3495 (all X hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),pi))) = hAPP(real,real,tan,X)) # label(fact_2440_tan__periodic) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3496 (all Z all Ya all Xa all X_2 (hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),Xa),X_2)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Ya),Xa)) -> ((all X_1 (hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_1),X_2)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),Z)))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Ya),hAPP(fun(real,bool),real,complete_Sup_Sup(real),X_2))))))) # label(fact_3732_SupInf_OSup__upper2) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3497 (all Xa all Ya (Xa = Ya <-> hAPP(real,real,exp(real),Ya) = hAPP(real,real,exp(real),Xa))) # label(fact_1749_exp__inj__iff) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3498 (all X_a all Xa all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa),A_2)) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa),B_1))))) # label(fact_2204_set__mp) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3499 (all P_1 (-hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) -> ((exists X1 hBOOL(hAPP(nat,bool,P_1,X1))) -> (exists N_1 (-hBOOL(hAPP(nat,bool,P_1,N_1)) & hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,suc,N_1)))))))) # label(fact_2756_exists__least__lemma) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3500 (all X_a (sgn_if(X_a) -> (all X ((zero_zero(X_a) != ti(X_a,X) -> (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X)) -> hAPP(X_a,X_a,sgn_sgn(X_a),X) = hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))) & (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X)) -> one_one(X_a) = hAPP(X_a,X_a,sgn_sgn(X_a),X))) & (zero_zero(X_a) = ti(X_a,X) -> zero_zero(X_a) = hAPP(X_a,X_a,sgn_sgn(X_a),X)))))) # label(fact_2315_sgn__if) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3501 (all X_a (semiring_div(X_a) -> (all A_1 all B all C_1 (zero_zero(X_a) != ti(X_a,C_1) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B))))) # label(fact_1085_div__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3502 (all X_b all X_a all F all Xa all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa),A_2)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(X_a,X_b,F,Xa)),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),F),A_2))))) # label(fact_3660_imageI) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3503 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) -> hAPP(real,real,uminus_uminus(real),hAPP(real,real,ln,X)) = hAPP(real,real,ln,hAPP(real,real,inverse_inverse(real),X)))) # label(fact_2637_ln__inverse) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3504 (all X_a all Xa all P_1 all Q_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),P_1),Q_1)) -> (hBOOL(hAPP(X_a,bool,P_1,Xa)) -> hBOOL(hAPP(X_a,bool,Q_1,Xa))))) # label(fact_4363_predicate1D) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3505 (all X_a (linordered_semidom(X_a) -> (all A_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),one_one(X_a))))))) # label(fact_183_less__add__one) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3506 (all G all Xa all Ma (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),G),Xa),Ma)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),sin),G)),Xa),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,hAPP(real,real,G,Xa))),Ma))))) # label(fact_3883_DERIV__fun__sin) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3507 (all K_4 all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M),K_4)),M)))) # label(fact_1362_zmod__le__nonneg__dividend) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3508 (all Wa all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,nat_1,Wa)),hAPP(int,nat,nat_1,Z))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z))))) # label(fact_1020_nat__mono__iff) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3509 (all M all P_5 (hBOOL(hAPP(nat,bool,prime,P_5)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),M),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),M))))) # label(fact_3069_prime__dvd__power__two) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3510 (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),M),hAPP(nat,nat,suc,N))))) # label(fact_1343_le__SucI) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3511 (all M all K_4 (hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),K_4) != M -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),K_4))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),K_4))))) # label(fact_1443_one__less__k) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3512 (all X_a (semiring_div(X_a) -> (all A_1 all C_1 all B (ti(X_a,B) != zero_zero(X_a) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B))),B))))) # label(fact_1116_div__mult__self1) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3513 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,M)),hAPP(nat,nat,suc,N))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)))) # label(fact_1324_Suc__less__SucD) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3514 (all X_a (ordered_semiring(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C_1)))))))) # label(fact_431_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3515 (all X one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),one_one(nat)),X)) # label(fact_3278_gcd__lcm__complete__lattice__nat_Oinf__bot__left) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3516 (all B_2 all A_3 all F all G all Xa all D (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),hAPP(real,real,G,Xa)),D)) -> (D != zero_zero(real) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),B_2)) -> ((all Y_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y_1),B_2)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),Y_1)) -> hAPP(real,real,F,hAPP(real,real,G,Y_1)) = Y_1)) -> (hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),G),Xa)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),G),Xa),hAPP(real,real,inverse_inverse(real),D)))))))))) # label(fact_4946_DERIV__inverse__function) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3517 (all X_a (linordered_semidom(X_a) -> (all A_1 all N all N_4 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N_4)) -> (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_4)))))))) # label(fact_187_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3518 (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)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),Y)))))) # label(fact_1363_Divides_Otransfer__nat__int__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3519 (all X_a (ring_1(X_a) -> (all Na hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na)),ring_1_Ints(X_a)))))) # label(fact_3834_Ints__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3520 (all X_a (linorder(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)) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y) = ti(X_a,X))))) # label(fact_4849_min__max_Oinf__absorb1) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3521 (all X_a all X_b all B_1 all F all A_2 ((exists AA (hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),AA) = ti(fun(X_a,bool),B_1) & hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),AA),A_2)))) <-> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2))))) # label(fact_3663_subset__image__iff) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3522 (all M all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),hAPP(nat,nat,suc,N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N)) # label(fact_1402_diff__Suc__eq__diff__pred) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3523 (all X_a (real_normed_algebra(X_a) -> (all C all F all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),A_3)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),times_times(X_a),C)),F)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_3))))))) # label(fact_4107_sums__mult) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3524 (all B_2 all A_3 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),d22set,A_3))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),A_3)))) # label(fact_1222_d22set__le) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3525 (all K all Ma all Na (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),Ma),Na))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),Ma)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),Na)))) # label(fact_4404_gcd__greatest__iff__int) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3526 (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),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,tan,X)))))) # label(fact_2446_tan__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3527 (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_eq(nat),hAPP(nat,nat,suc,Na)),Ma)))) # label(fact_1340_not__less__eq__eq) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3528 (all X -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,exp(real),X)),zero_zero(real)))) # label(fact_1768_not__exp__less__zero) # label(axiom) # label(non_clause). [assumption]. 7.82/7.80 3529 (all A_1 all B all N hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),N)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),N)) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B)),N)) # label(fact_3462_zgcd__power__distrib) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3530 (all X_a (inverse(X_a) & real_normed_vector(X_a) -> inverse_divide(X_a) = ti(fun(X_a,fun(X_a,X_a)),inverse_divide(X_a)))) # label(tsy_c_Fields_Oinverse__class_Odivide_0_res) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3531 (all Y all X (hBOOL(hAPP(int,bool,even_odd_even(int),X)) -> (-hBOOL(hAPP(int,bool,even_odd_even(int),Y)) -> -hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),Y)))))) # label(fact_1596_Parity_Oeven__plus__odd) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3532 (all G all F all Xa all D_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_1)) -> ((all Z_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),Xa))),D_1)) -> hAPP(real,real,G,hAPP(real,real,F,Z_2)) = Z_2)) -> ((all Z_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),Xa))),D_1)) -> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),F),Z_2)))) -> (exists Z_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,Xa)),hAPP(real,real,F,Z_2))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),Xa))),D_1)))))))) # label(fact_4955_lemma__isCont__inj) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3533 (all N -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N))) # label(fact_120_less__not__refl) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3534 (all X_b all X_a (real_normed_vector(X_a) -> (all F all A_2 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2))),hAPP(fun(X_b,bool),real,hAPP(fun(X_b,real),fun(fun(X_b,bool),real),big_co1399186613setsum(X_b,real),hAPP(fun(X_b,X_a),fun(X_b,real),hAPP(fun(X_a,real),fun(fun(X_b,X_a),fun(X_b,real)),combb(X_a,real,X_b),norm_norm(X_a)),F)),A_2)))))) # label(fact_4206_norm__setsum) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3535 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),X)),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),one_one(real))))) # label(fact_1129_less__one__imp__sqr__less__one) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3536 (all Ta all B_1 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ta),B_1)) -> (all X_1 ((all Xa_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D))) -> (all Xb (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1)) -> ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa_1))))) -> (ti(int,X_1) != ti(int,Ta) -> ti(int,Ta) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),D))))))) # label(fact_4667_bset_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3537 (all X all M hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,M))),pi)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),M)),pi)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) # label(fact_2367_cos__expansion__lemma) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3538 (all X_a (real_normed_algebra(X_a) -> (all Xa all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2)) -> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa)),X_2))))))) # label(fact_4028_mult__right_Osummable) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3539 (all X_a (archim1727834104eiling(X_a) -> (all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(X_a,int,archim2085082626_floor(X_a),X))),X)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),one_one(int))))))))) # label(fact_3635_floor__correct) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3540 (all Z_1 all X all D_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_3)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),Z_1))),one_one(int))),D_3)))))) # label(fact_1059_incr__lemma) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3541 (all K_4 all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),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_4),M)))))) # label(fact_802_dvd__diffD) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3542 (all X_a all X_b (minus(X_b) -> (all A_2 all B_1 all X_1 hAPP(X_a,X_b,hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),minus_minus(fun(X_a,X_b)),A_2),B_1),X_1) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_a,X_b,A_2,X_1)),hAPP(X_a,X_b,B_1,X_1))))) # label(fact_2989_fun__diff__def) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3543 (all X_a (real_normed_vector(X_a) -> (all Xa all X_2 all A_3 (hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),X_2),A_3)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),X_2)),Xa)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_3),Xa))))))) # label(fact_4747_scaleR__left_Osums) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3544 (all X_a (linorder(X_a) -> hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,X_a)),max(X_a),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),ord_less_eq(X_a))) = ord_min(X_a))) # label(fact_4859_dual__max) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3545 (all X_a (real_normed_field(X_a) -> (all F all Xa all D (hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),inverse_divide(X_a)),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),minus_minus(X_a)),F)),hAPP(X_a,X_a,F,Xa)))),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),minus_minus(X_a)),Xa))),D),hAPP(X_a,filter(X_a),at(X_a),Xa))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D)))))) # label(fact_4626_DERIV__iff2) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3546 (all Z_1 hAPP(complex,complex,expi,Z_1) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,of_real(complex),hAPP(real,real,exp(real),hAPP(complex,real,re,Z_1)))),hAPP(real,complex,cis,hAPP(complex,real,im,Z_1)))) # label(fact_3048_expi__def) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3547 (all X_a (field_inverse_zero(X_a) & number_ring(X_a) -> (all X zero_zero(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),hAPP(int,X_a,number_number_of(X_a),pls))))) # label(fact_1781_divide__Numeral0) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3548 (all X_a (field_inverse_zero(X_a) -> zero_zero(X_a) = hAPP(X_a,X_a,inverse_inverse(X_a),zero_zero(X_a)))) # label(fact_2572_field__inverse__zero) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3549 (all X_a (real_normed_vector(X_a) -> (all R_3 all X all Xa_2 hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),R_3),X)),Xa_2) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),X),Xa_2))))) # label(fact_4709_scaleR__left_OscaleR) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3550 (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]. 7.82/7.81 3551 (all X_a all X_b all P_1 hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),hilbert_Eps(product_prod(X_a,X_b)),hAPP(fun(product_prod(X_a,X_b),X_b),fun(product_prod(X_a,X_b),bool),hAPP(fun(product_prod(X_a,X_b),fun(X_b,bool)),fun(fun(product_prod(X_a,X_b),X_b),fun(product_prod(X_a,X_b),bool)),combs(product_prod(X_a,X_b),X_b,bool),hAPP(fun(product_prod(X_a,X_b),X_a),fun(product_prod(X_a,X_b),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,bool)),fun(fun(product_prod(X_a,X_b),X_a),fun(product_prod(X_a,X_b),fun(X_b,bool))),combb(X_a,fun(X_b,bool),product_prod(X_a,X_b)),P_1),product_fst(X_a,X_b))),product_snd(X_a,X_b))) = hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),hilbert_Eps(product_prod(X_a,X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),P_1))) # label(fact_4807_Eps__split) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3552 (all A_3 all B_2 all C (C != zero_zero(real) -> (hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_3),C) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B_2),C) <-> B_2 = A_3))) # label(fact_610_real__mult__right__cancel) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3553 (all X_a (preorder(X_a) -> (all X hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),X))))) # label(fact_347_order__refl) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3554 (all M all N all K_4 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N)))) # label(fact_3240_gcd__dvd__prod__nat) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3555 (all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(real,real,sqrt,Ya))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),Ya)))) # label(fact_2149_real__sqrt__ge__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3556 (all X_b all X_a (real_normed_vector(X_b) & topological_space(X_a) -> (all F all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),uminus_uminus(X_b)),F)),A_3)))))) # label(fact_4908_isCont__minus) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3557 (all Z_1 ((all N_1 hAPP(nat,int,semiring_1_of_nat(int),N_1) != ti(int,Z_1)) -> -(all N_1 hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N_1))) != ti(int,Z_1)))) # label(fact_3553_int__cases) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3558 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 all B hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B))),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B))))))) # label(fact_995_abs__triangle__ineq2) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3559 (all K_4 hAPP(int,rat,hAPP(int,fun(int,rat),fract,K_4),zero_zero(int)) = zero_zero(rat)) # label(fact_3763_rat__number__collapse_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3560 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) -> (exists N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_1)))),X))))) # label(fact_2641_reals__Archimedean) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3561 (all A_1 all B hAPP(rat,rat,abs_abs(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,abs_abs(int),A_1)),hAPP(int,int,abs_abs(int),B))) # label(fact_3779_abs__rat) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3562 (all Xa (hAPP(real,real,cos,Xa) != zero_zero(real) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),tan),Xa),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,cos,Xa)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) # label(fact_2655_DERIV__tan) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3563 (all K all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Ma),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),Ma)))) # label(fact_3256_gcd__greatest__iff__nat) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3564 (all X_a (field(X_a) -> (all A_1 hAPP(X_a,X_a,inverse_inverse(X_a),A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),one_one(X_a)),A_1)))) # label(fact_2603_field__class_Onormalizing__field__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3565 (all X_b all X_a (real_normed_vector(X_b) & topological_space(X_a) -> (all F all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4635_LIM__norm__zero) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3566 (all X all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Y)) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)))),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)))),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),X),Y))))) = hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,X),Y))) # label(fact_3139_bezw__non__0) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3567 (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),zero_zero(X_a))) -> hAPP(X_a,X_a,uminus_uminus(X_a),A_1) = hAPP(X_a,X_a,abs_abs(X_a),A_1))))) # label(fact_2000_abs__of__nonpos) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3568 (all A_1 all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_5),A_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,zfact,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int)))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),P_5))))))) # label(fact_1132_zfact__prop) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3569 (all Ma all Na (hAPP(nat,nat,suc,zero_zero(nat)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),Na) <-> Na = zero_zero(nat) & hAPP(nat,nat,suc,zero_zero(nat)) = Ma | hAPP(nat,nat,suc,zero_zero(nat)) = Na & zero_zero(nat) = Ma)) # label(fact_1379_one__is__add) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3570 (all U_1 all V all X all Y (Y = X -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),U_1)),V)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),U_1)),Y))),V))))) # label(fact_1197_sin__bound__lemma) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3571 (all Y all X (hBOOL(hAPP(int,bool,nat_is_nat,X)) -> (hBOOL(hAPP(int,bool,nat_is_nat,Y)) -> hBOOL(hAPP(int,bool,nat_is_nat,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y)))))) # label(fact_4873_Nat__Transfer_Otransfer__int__nat__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3572 (all X_b all X_c all X_a combb(X_b,X_c,X_a) = ti(fun(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c))),combb(X_b,X_c,X_a))) # label(tsy_c_COMBB_res) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3573 (all P all Q (-hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q)) | hBOOL(Q))) # label(help_fconj_3_1_U) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3574 (all X_a (ordere453448008miring(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a))) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),zero_zero(X_a))))))) # label(fact_439_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3575 (all Xa hAPP(fun(int,bool),int,the(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(rat,fun(int,bool),hAPP(fun(int,fun(rat,bool)),fun(rat,fun(int,bool)),combc(int,rat,bool),hAPP(fun(int,rat),fun(int,fun(rat,bool)),hAPP(fun(rat,fun(rat,bool)),fun(fun(int,rat),fun(int,fun(rat,bool))),combb(rat,fun(rat,bool),int),ord_less_eq(rat)),ring_1_of_int(rat))),Xa))),hAPP(fun(int,rat),fun(int,bool),hAPP(fun(rat,bool),fun(fun(int,rat),fun(int,bool)),combb(rat,bool,int),hAPP(rat,fun(rat,bool),ord_less(rat),Xa)),hAPP(fun(int,int),fun(int,rat),hAPP(fun(int,rat),fun(fun(int,int),fun(int,rat)),combb(int,rat,int),ring_1_of_int(rat)),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),plus_plus(int)),one_one(int)))))) = hAPP(rat,int,archim2085082626_floor(rat),Xa)) # label(fact_4174_floor__rat__def) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3576 (all X_a (semiring_div(X_a) -> (all A_1 all C_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B))),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)))) # label(fact_1355_mod__mult__self1) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3577 (all X_a all C all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2))) <-> -hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C),A_2)))) # label(fact_2238_Compl__iff) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3578 (all X_a (order(X_a) -> (all C_1 all A_1 all B (ti(X_a,A_1) = ti(X_a,B) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),A_1))))))) # label(fact_507_xt1_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3579 (all N all K_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K_4)) -> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),N)),K_4) = ti(int,K_4))) # label(fact_3479_zgcd__zmult__eq__self2) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3580 (all N (N != zero_zero(nat) -> (exists M_2 hAPP(nat,nat,suc,M_2) = N))) # label(fact_2790_not0__implies__Suc) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3581 (all A_1 all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> hAPP(real,nat,natceiling,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(nat,real,real_1(nat),A_1))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(real,nat,natceiling,X)),A_1))) # label(fact_2429_natceiling__add) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3582 (all Na all S (hBOOL(hAPP(fun(fun(int,bool),bool),bool,finite_finite_1(fun(int,bool)),S)) -> ((all X_1 (hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),X_1),S)) -> hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),X_1)))) -> ((all X_1 (hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),X_1),S)) -> Na = hAPP(fun(int,bool),nat,finite_card(int),X_1))) -> hAPP(fun(fun(int,bool),bool),nat,hAPP(fun(fun(int,bool),nat),fun(fun(fun(int,bool),bool),nat),big_co1399186613setsum(fun(int,bool),nat),hAPP(nat,fun(fun(int,bool),nat),combk(nat,fun(int,bool)),Na)),S) = hAPP(fun(fun(int,bool),bool),nat,hAPP(fun(fun(int,bool),nat),fun(fun(fun(int,bool),bool),nat),big_co1399186613setsum(fun(int,bool),nat),finite_card(int)),S))))) # label(fact_4443_card__setsum__aux) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3583 (all X all Y (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),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),Y)))) # label(fact_1810_real__gt__half__sum) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3584 (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),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,cos,X)))))) # label(fact_2339_cos__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3585 (all X all Y hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sqrt,X)),hAPP(real,real,sqrt,Y)) = hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),Y))) # label(fact_2120_real__sqrt__divide) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3586 (all Xa all Na (-hBOOL(hAPP(nat,bool,even_odd_even(nat),Na)) -> (zero_zero(real) != Xa -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),root,Na)),Xa),hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),Na)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,zero_zero(nat))))))))))) # label(fact_2720_DERIV__odd__real__root) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3587 (all Z_1 all W hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z_1),hAPP(int,int,uminus_uminus(int),W)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z_1),W)) # label(fact_2077_diff__int__def) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3588 (all M all N M = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)),N)) # label(fact_810_diff__add__inverse2) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3589 (all F all A_2 (hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2)) -> ((all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,F,X_1))))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),F),A_2)))))) # label(fact_3964_transfer__nat__int__sum__prod__closure_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3590 (all K all F (zero_zero(nat) = hAPP(nat,nat,F,zero_zero(nat)) -> hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)),K)) = hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,suc,zero_zero(nat))),K)))) # label(fact_4614_setsum__shift__lb__Suc0__0) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3591 (all X_a (ring_1(X_a) -> (all B_2 all A_3 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),ring_1_Ints(X_a))) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),ring_1_Ints(X_a))) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_2)),ring_1_Ints(X_a)))))))) # label(fact_3830_Ints__mult) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3592 (all X_a all X_b all X_c all X_1 all Xa_1 hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),X_1,hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),Xa_1)),hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),Xa_1)) = hAPP(product_prod(X_a,X_b),X_c,hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),X_1),Xa_1)) # label(fact_4041_split__def) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3593 (all Va all Na hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(int,nat,number_number_of(nat),Va)),hAPP(nat,nat,suc,Na)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),zero_zero(nat)),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),Na)))) # label(fact_4822_min__Suc__number__of) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3594 (all X all A_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(nat,real,real_1(nat),A_1))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(real,nat,natceiling,X)),A_1)))) # label(fact_2402_natceiling__le) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3595 (all Xa hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),cos_coeff)),hAPP(real,fun(nat,real),power_power(real),Xa))))) # label(fact_4049_summable__cos) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3596 (all X_a all X_b product_fst(X_a,X_b) = ti(fun(product_prod(X_a,X_b),X_a),product_fst(X_a,X_b))) # label(tsy_c_Product__Type_Ofst_res) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3597 (all X_a (linord1117847801e_zero(X_a) & number(X_a) -> (all Wa all B_2 all C (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),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(X_a),C),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),zero_zero(X_a)))) & (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),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),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(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),C)),B_2))))))) # label(fact_1793_less__divide__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3598 (all A_3 hBOOL(hAPP(filter(complex),bool,hAPP(complex,fun(filter(complex),bool),hAPP(fun(complex,complex),fun(complex,fun(filter(complex),bool)),tendsto(complex,complex),cnj),hAPP(complex,complex,cnj,A_3)),hAPP(complex,filter(complex),at(complex),A_3)))) # label(fact_4573_cnj_Ocont) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3599 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 all B hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)) = hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),A_1))))) # label(fact_977_abs__minus__commute) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3600 (all X_a (linorder(X_a) -> (all X all Y all Z_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Z_1))))) # label(fact_4836_min__max_Oinf__left__commute) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3601 (all X_b all X_a all P_5 hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),P_5)),hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),P_5)) = ti(product_prod(X_a,X_b),P_5)) # label(fact_3149_pair__collapse) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3602 (all X_a (real_normed_vector(X_a) & real_algebra_1(X_a) -> (exists K_2 all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(real,X_a,of_real(X_a),X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),X_1)),K_2)))))) # label(fact_2830_of__real_Obounded) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3603 (all A_1 all B all Q_3 all R_3 (hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_3),R_3))) -> (zero_zero(int) != ti(int,B) -> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = ti(int,R_3)))) # label(fact_1619_divmod__int__rel__mod) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3604 (all N (hBOOL(hAPP(nat,bool,even_odd_even(nat),N)) -> (exists X_1 hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),X_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) # label(fact_2534_even__square) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3605 (all X_a (archim1727834104eiling(X_a) -> ti(fun(X_a,int),archimedean_ceiling(X_a)) = archimedean_ceiling(X_a))) # label(tsy_c_Archimedean__Field_Oceiling_res) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3606 (all A_1 all B (ti(int,B) != zero_zero(int) -> hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)))))) # label(fact_1623_divmod__int__rel__div__mod) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3607 (all R_3 all X hAPP(complex,real,re,hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),R_3),X)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),hAPP(complex,real,re,X))) # label(fact_4726_complex__Re__scaleR) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3608 (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)) -> hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),Y)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(int,nat,nat_1,X)),hAPP(int,nat,nat_1,Y))))) # label(fact_1171_Divides_Otransfer__nat__int__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3609 (all X_a all X_b all X_c all X_d all F all G hAPP(fun(product_prod(X_a,X_b),X_b),fun(product_prod(X_a,X_b),X_c),hAPP(fun(product_prod(X_a,X_b),fun(X_b,X_c)),fun(fun(product_prod(X_a,X_b),X_b),fun(product_prod(X_a,X_b),X_c)),combs(product_prod(X_a,X_b),X_b,X_c),hAPP(fun(product_prod(X_a,X_b),X_d),fun(product_prod(X_a,X_b),fun(X_b,X_c)),hAPP(fun(X_d,fun(X_b,X_c)),fun(fun(product_prod(X_a,X_b),X_d),fun(product_prod(X_a,X_b),fun(X_b,X_c))),combb(X_d,fun(X_b,X_c),product_prod(X_a,X_b)),F),hAPP(fun(product_prod(X_a,X_b),X_a),fun(product_prod(X_a,X_b),X_d),hAPP(fun(X_a,X_d),fun(fun(product_prod(X_a,X_b),X_a),fun(product_prod(X_a,X_b),X_d)),combb(X_a,X_d,product_prod(X_a,X_b)),G),product_fst(X_a,X_b)))),product_snd(X_a,X_b)) = hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),hAPP(fun(X_a,X_d),fun(X_a,fun(X_b,X_c)),hAPP(fun(X_d,fun(X_b,X_c)),fun(fun(X_a,X_d),fun(X_a,fun(X_b,X_c))),combb(X_d,fun(X_b,X_c),X_a),F),G))) # label(fact_4042_split__comp__eq) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3610 (all M all K_4 all N (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K_4),N) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M)),N)),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N))))) # label(fact_3486_zgcd__zmult__zdvd__zgcd) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3611 (all W all Z1 all Z2 hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z1),Z2)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W),Z1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W),Z2))) # label(fact_745_zdiff__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3612 (all X_a all X_b all C all P_4 ((all A_4 all B_4 (hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_4),B_4) = ti(product_prod(X_a,X_b),P_4) -> hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),C,A_4),B_4)))) -> hBOOL(hAPP(product_prod(X_a,X_b),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),C),P_4)))) # label(fact_4089_splitI2) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3613 (all X all B all A_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1)) -> (A_1 != one_one(real) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)) -> (B != one_one(real) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) -> hAPP(real,real,hAPP(real,fun(real,real),log,A_1),X) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,ln,B)),hAPP(real,real,ln,A_1))),hAPP(real,real,hAPP(real,fun(real,real),log,B),X)))))))) # label(fact_2046_log__eq__div__ln__mult__log) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3614 (all X_b (ord(X_b) -> ti(fun(X_b,fun(X_b,bool)),ord_less_eq(X_b)) = ord_less_eq(X_b))) # label(tsy_c_Orderings_Oord__class_Oless__eq_res) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3615 (all X_a (semiring_0(X_a) & power(X_a) -> (all W ((hAPP(int,nat,number_number_of(nat),W) != zero_zero(nat) -> zero_zero(X_a) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),zero_zero(X_a)),hAPP(int,nat,number_number_of(nat),W))) & (zero_zero(nat) = hAPP(int,nat,number_number_of(nat),W) -> one_one(X_a) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),zero_zero(X_a)),hAPP(int,nat,number_number_of(nat),W))))))) # label(fact_11_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3616 (all I_1 all J_2 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I_1),J_2)))) # label(fact_3471_zgcd__pos) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3617 (all X_a (linord1117847801e_zero(X_a) -> (all B_2 all C all A_3 ((-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),C),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)),B_2))) & (-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),zero_zero(X_a)),A_3)))) & (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C)),A_3)))))) # label(fact_1868_divide__less__eq) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3618 (all M all K_4 all N (hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),K_4),N) = one_one(nat) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M)),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N))) # label(fact_3293_gcd__mult__cancel__nat) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3619 (all X_a (real_normed_algebra(X_a) -> (all A_1 zero_zero(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),zero_zero(X_a))))) # label(fact_1241_mult_Ozero__right) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3620 (all X_b all X_a all X all Y all A_1 (ti(X_a,A_1) = hAPP(product_prod(X_b,X_a),X_a,product_snd(X_b,X_a),hAPP(X_a,product_prod(X_b,X_a),hAPP(X_b,fun(X_a,product_prod(X_b,X_a)),product_Pair(X_b,X_a),X),Y)) -> ti(X_a,Y) = ti(X_a,A_1))) # label(fact_3157_snd__eqD) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3621 (all K_4 hAPP(int,real,number_number_of(real),K_4) = hAPP(rat,real,ratreal,hAPP(int,rat,number_number_of(rat),K_4))) # label(fact_3784_number__of__real__code) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3622 (all Xa (one_one(int) = hAPP(int,int,abs_abs(int),Xa) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Xa),one_one(int))))) # label(fact_1008_zdvd1__eq) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3623 (all F all A_3 all B_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2)) -> ((all X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),B_2)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),X_1)) -> (exists Y_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y_1)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),Y_1)))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,F,A_3)),hAPP(real,real,F,B_2)))))) # label(fact_3551_DERIV__nonneg__imp__nonincreasing) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3624 (all X_a (topological_space(X_a) -> (all G all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),zero_zero(complex)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,complex),fun(X_a,real),hAPP(fun(complex,real),fun(fun(X_a,complex),fun(X_a,real)),combb(complex,real,X_a),im),G)),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4641_Im_OLIM__zero) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3625 (all X_a all X_b (comm_monoid_add(X_b) -> (all B_2 all A_3 all S (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),S)) -> hAPP(X_a,X_b,B_2,A_3) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),fequal(X_a),A_3))),B_2)),zero_zero(X_b))),S)) & (-hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),S)) -> zero_zero(X_b) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),fequal(X_a),A_3))),B_2)),zero_zero(X_b))),S)))))) # label(fact_4237_setsum__delta_H) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3626 (all X all Y 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),X),Y)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),Y)))))) # label(fact_2685_real__root__less__mono) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3627 (all X_a (t2_space(X_a) & comm_monoid_add(X_a) -> (all F all Sb (hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),Sb)) -> hAPP(fun(nat,X_a),X_a,suminf(X_a),F) = ti(X_a,Sb))))) # label(fact_4116_sums__unique) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3628 (all P_1 all Na ((all X_1 (hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_1),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)),Na))) -> hBOOL(hAPP(nat,bool,P_1,X_1)))) <-> (all M_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_2),Na)) -> hBOOL(hAPP(nat,bool,P_1,M_2)))))) # label(fact_4561_all__nat__less) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3629 (all X all Y hAPP(complex,real,hAPP(complex,fun(complex,real),dist_dist(complex),X),Y) = hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),X),Y))) # label(fact_3545_dist__complex__def) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3630 (all X_a (real_normed_algebra(X_a) -> (all Ya all X_2 all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),X_2)),Ya)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),Ya)),sequentially)))))) # label(fact_4483_mult__left_OLIMSEQ) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3631 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,uminus_uminus(real),X)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),X))))))) # label(fact_2048_ln__one__minus__pos__lower__bound) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3632 (all A_3 all P_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int)))),one_one(int)),P_4)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))),P_4)))) # label(fact_717_inv__not__p__minus__1__aux) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3633 (all B all A_1 ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,A_1),B)) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B))) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,A_1),B))) & (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_eq(int),zero_zero(int)),B)) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,A_1),B)) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) -> (ti(int,A_1) != zero_zero(int) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,A_1),B) = hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)))) & (ti(int,A_1) = zero_zero(int) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),zero_zero(int))))))) # label(fact_3070_divmod__int__def) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3634 (all X_a (ordered_semiring(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B)))))))) # label(fact_432_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3635 (all X_a (linord20386208strict(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)))))))) # label(fact_444_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3636 (all Xa (hBOOL(hAPP(int,bool,even_odd_even(int),Xa)) <-> hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,uminus_uminus(int),Xa))))) # label(fact_2072_even__neg) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3637 (all X_a (real_normed_vector(X_a) -> (all X hAPP(X_a,real,norm_norm(X_a),X) = hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),zero_zero(X_a))))) # label(fact_3549_norm__conv__dist) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3638 (all X_2 all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),X_2),A_3),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),re),X_2)),hAPP(complex,real,re,A_3)),sequentially)))) # label(fact_4536_Re_OLIMSEQ) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3639 (all Y_2 all X_2 (hBOOL(hAPP(fun(nat,rat),bool,vanishes,X_2)) -> (hBOOL(hAPP(fun(nat,rat),bool,vanishes,Y_2)) -> hBOOL(hAPP(fun(nat,rat),bool,vanishes,hAPP(fun(nat,rat),fun(nat,rat),hAPP(fun(nat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,rat)),combs(nat,rat,rat),hAPP(fun(nat,rat),fun(nat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,fun(rat,rat))),combb(rat,fun(rat,rat),nat),plus_plus(rat)),X_2)),Y_2)))))) # label(fact_4879_vanishes__add) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3640 (all X_a (linordered_semidom(X_a) -> (all Xa all Ya all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),B_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Xa),Ya)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B_2),Xa)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B_2),Ya)))))))) # label(fact_239_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3641 (all A_1 all B hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,uminus_uminus(real),A_1)),hAPP(real,real,uminus_uminus(real),B)) = hAPP(complex,complex,uminus_uminus(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B))) # label(fact_2764_complex__minus) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3642 (all X_a (real_algebra_1(X_a) & real_normed_vector(X_a) -> (all X_2 (hBOOL(hAPP(fun(nat,real),bool,summable(real),X_2)) -> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,real),fun(nat,X_a),hAPP(fun(real,X_a),fun(fun(nat,real),fun(nat,X_a)),combb(real,X_a,nat),of_real(X_a)),X_2))))))) # label(fact_4031_of__real_Osummable) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3643 (all X_a (real_normed_vector(X_a) -> (all A_1 all A_6 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_6),B)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_1),A_6)),B)))) # label(fact_4694_scaleR_Oadd__left) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3644 (all N -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat)))) # label(fact_149_not__less0) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3645 (all Na all Ma (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)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),Ma))))) # label(fact_831_zero__less__diff) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3646 (all X_a all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> (((exists X_1 hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))) -> zero_zero(nat) = one_one(nat)) -> hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2) = zero_zero(nat)))) # label(fact_2751_card_Oneutral) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3647 (all W ((hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),W))) -> hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,W)) = zero_zero(nat)) & (-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),W))) -> hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),W)),hAPP(int,nat,number_number_of(nat),W))) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,W))))) # label(fact_3919_nat__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3648 (all Va (hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))) <-> zero_zero(nat) = hAPP(int,nat,number_number_of(nat),Va))) # label(fact_2936_neg__number__of__pred__iff__0) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3649 (all V all W hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,int,number_number_of(int),W)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,V))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,W)))) # label(fact_1082_zdiv__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3650 (all E_1 all B all D_3 all A_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_3),A_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),E_1),B)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_1),B)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_3),E_1)))))) # label(fact_2877_coprime__divisors) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3651 (all B_2 all A_3 all P_4 (hBOOL(hAPP(int,bool,zprime,P_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int)))) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,A_3),P_4))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),B_2)))))) # label(fact_1206_wset__g__1) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3652 (all X_a (group_add(X_a) -> (all A_3 all B_2 (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_2) = zero_zero(X_a) <-> hAPP(X_a,X_a,uminus_uminus(X_a),B_2) = ti(X_a,A_3))))) # label(fact_1958_eq__neg__iff__add__eq__0) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3653 (all N all M hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),M)))) # label(fact_410_le__add1) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3654 (all X_a (power(X_a) -> (all A_1 one_one(X_a) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),zero_zero(nat))))) # label(fact_190_power__0) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3655 (all A_3 all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),hAPP(nat,real,real_1(nat),A_3))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(real,nat,natceiling,Xa)),A_3))))) # label(fact_2411_natceiling__le__eq) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3656 (all Ya all Xa (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ya),zEven)) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xa),Ya)),zEven))))) # label(fact_1698_EvenOdd_Oeven__plus__even) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3657 (all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A_3),zero_zero(real)),sequentially)) -> ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,A_3,N_1)))) -> ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,A_3,hAPP(nat,nat,suc,N_1))),hAPP(nat,real,A_3,N_1)))) -> hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))))))) # label(fact_4556_summable) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3658 (all X_a (linorder(X_a) -> (all A_1 all X all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),B))))))) # label(fact_4985_min__max_Oless__supI2) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3659 (all K_4 all M all N (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M),N))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),N)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),M))))) # label(fact_1322_zdvd__zmod__imp__zdvd) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3660 (all X_a (power(X_a) & no_zero_divisors(X_a) & zero_neq_one(X_a) & mult_zero(X_a) -> (all A_3 all Wa (hAPP(int,nat,number_number_of(nat),Wa) != zero_zero(nat) & ti(X_a,A_3) = zero_zero(X_a) <-> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_3),hAPP(int,nat,number_number_of(nat),Wa)) = zero_zero(X_a))))) # label(fact_89_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3661 (all X_b all X_a all P_1 all Ya all Xa (hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,Ya),Xa)) -> hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,hAPP(product_prod(X_b,X_a),X_a,product_snd(X_b,X_a),hAPP(X_a,product_prod(X_b,X_a),hAPP(X_b,fun(X_a,product_prod(X_b,X_a)),product_Pair(X_b,X_a),Xa),Ya))),hAPP(product_prod(X_b,X_a),X_b,product_fst(X_b,X_a),hAPP(X_a,product_prod(X_b,X_a),hAPP(X_b,fun(X_a,product_prod(X_b,X_a)),product_Pair(X_b,X_a),Xa),Ya)))))) # label(fact_3163_exI__realizer) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3662 (all K all Ma all Na (Na = Ma <-> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K)),Ma) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K)),Na))) # label(fact_1346_Suc__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3663 (all Ma all K all F ((all M_2 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),N_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,F,M_2)),hAPP(nat,nat,F,N_1))))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,F,Ma)),K)),hAPP(nat,nat,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),K)))))) # label(fact_658_mono__nat__linear__lb) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3664 (all Xa all Na hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,sin,Xa)),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(real,fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(real,fun(nat,fun(real,real))),combc(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),zero_zero(real))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),div_div(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),minus_minus(nat)),hAPP(nat,nat,suc,zero_zero(nat))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat)))))),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,abs_abs(real),Xa)),Na))))) # label(fact_4189_Maclaurin__sin__bound) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3665 (all X all Y hAPP(int,int,hAPP(int,fun(int,int),div_mod(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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y))) # label(fact_1540_Divides_Otransfer__int__nat__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3666 (all X_a (ring_1(X_a) -> hAPP(int,X_a,ring_1_of_int(X_a),zero_zero(int)) = zero_zero(X_a))) # label(fact_3589_of__int__0) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3667 (all X_a (semiring_div(X_a) -> (all N all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Y),X)) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),X),Y)),N))))) # label(fact_1097_div__power) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3668 (all N (one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) -> hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) # label(fact_2528_lemma__even__mod__4__div__2) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3669 (all A_1 all B all P_5 all Q_3 (hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_5),Q_3) = hAPP(product_prod(int,int),product_prod(int,int),normalize,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_1),B)) -> hAPP(int,rat,hAPP(int,fun(int,rat),fract,P_5),Q_3) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B))) # label(fact_3780_normalize__eq) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3670 (all X_a (semiring_div(X_a) -> (all A_1 all C_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C_1)),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C_1)))) # label(fact_1288_zmod__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3671 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,ln,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_eq(real),one_one(real)),X))))) # label(fact_1729_ln__ge__zero__imp__ge__one) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3672 (all V hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),V)) = hAPP(int,nat,number_number_of(nat),V)) # label(fact_1004_nat__number__of__def) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3673 (all X_a (comm_monoid_add(X_a) & topological_space(X_a) -> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(X_a,fun(nat,X_a),combk(X_a,nat),zero_zero(X_a)))))) # label(fact_4007_summable__zero) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3674 (all N one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),N),one_one(int))),N)) # label(fact_4395_coprime__minus__one__int) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3675 (all Ya all Xa (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ya),zOdd)) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Xa),Ya)),zOdd))))) # label(fact_1219_EvenOdd_Oodd__times__odd) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3676 (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_180_not__one__less__zero) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3677 (all X_a (linordered_semidom(X_a) -> (all Xa all Ya all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),B_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Xa),Ya)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B_2),Xa)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B_2),Ya)))))))) # label(fact_185_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3678 (all X_a (banach(X_a) -> (all F (hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),F))) -> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)))))) # label(fact_4032_summable__norm__cancel) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3679 (all X 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_eq(real),zero_zero(real)),X)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)))))) # label(fact_2697_real__root__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3680 (all X_a (finite_finite(X_a) -> (all A_2 hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))))) # label(fact_2739_finite__code) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3681 (all A_1 all B hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),A_1)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_1),B)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),A_1)) # label(fact_1811_real__average__minus__first) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3682 (all X_a all M_1 (hBOOL(hAPP(fun(list(X_a),bool),bool,finite_finite_1(list(X_a)),M_1)) -> (exists N_1 all X_1 (hBOOL(hAPP(fun(list(X_a),bool),bool,hAPP(list(X_a),fun(fun(list(X_a),bool),bool),member(list(X_a)),X_1),M_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(list(X_a),nat,size_size(list(X_a)),X_1)),N_1)))))) # label(fact_3582_finite__maxlen) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3683 (all W all V ((-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V))) -> hAPP(int,nat,nat_1,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,nat,number_number_of(nat),W))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),W))) & (hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V))) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),W)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),zero_zero(nat)),hAPP(int,nat,number_number_of(nat),W))))) # label(fact_1632_power__nat__number__of__number__of) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3684 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,exp(real),X)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))))) # label(fact_1819_exp__bound) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3685 (all B hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,zero_zero(int)),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),zero_zero(int))) # label(fact_1652_posDivAlg__0) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3686 (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_402_less__imp__le__nat) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3687 (all N (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),ord_less(int),zero_zero(int)),hAPP(int,int,fact_fact(int),N))))) # label(fact_3220_fact__gt__zero__int) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3688 (all X_a (real_algebra_1(X_a) -> (all F all X_1 hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),hAPP(fun(nat,real),fun(nat,X_a),hAPP(fun(real,X_a),fun(fun(nat,real),fun(nat,X_a)),combb(real,X_a,nat),of_real(X_a)),F)),X_1) = hAPP(real,X_a,of_real(X_a),hAPP(nat,real,hAPP(fun(nat,real),fun(nat,real),diffs(real),F),X_1))))) # label(fact_4137_diffs__of__real) # label(axiom) # label(non_clause). [assumption]. 7.82/7.81 3689 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)),zero_zero(int))) -> hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)))) # label(fact_1455_mod__pos__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 7.82/7.82 3690 (all X_a (real_n2089651433ebra_1(X_a) -> (all R_3 hAPP(real,X_a,of_real(X_a),hAPP(real,real,sgn_sgn(real),R_3)) = hAPP(X_a,X_a,sgn_sgn(X_a),hAPP(real,X_a,of_real(X_a),R_3))))) # label(fact_2812_sgn__of__real) # label(axiom) # label(non_clause). [assumption]. 7.82/7.82 3691 (all X_a (ab_group_add(X_a) -> (all F all Ma all Na ((-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) -> hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),F)),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),one_one(nat))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),Na)) = zero_zero(X_a)) & (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,F,Ma)),hAPP(nat,X_a,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),F)),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),one_one(nat))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),Ma),Na))))))) # label(fact_4643_setsum__natinterval__difff) # label(axiom) # label(non_clause). [assumption]. 7.82/7.82 3692 (all X_a (linord581940658strict(X_a) -> (all A_3 all B_2 all C (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B_2))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),A_3))))))) # label(fact_449_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 7.82/7.82 3693 (all T_2 all T_1 (finite_finite(T_2) & finite_finite(T_1) -> finite_finite(sum_sum(T_2,T_1)))) # label(arity_sum___Finite__Set_Ofinite) # label(axiom) # label(non_clause). [assumption]. 7.82/7.82 3694 (all X hAPP(int,int,abs_abs(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,abs_abs(int),X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))) # label(fact_1034_abs__power3__distrib) # label(axiom) # label(non_clause). [assumption]. 7.82/7.82 3695 (all N all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),N)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),N)),one_one(real)))) -> hAPP(real,int,archim2085082626_floor(real),X) = ti(int,N)))) # label(fact_3423_floor__eq) # label(axiom) # label(non_clause). [assumption]. 7.82/7.82 3696 (all A_1 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)) -> hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,Y),A_1)) = hAPP(real,real,hAPP(real,fun(real,real),powr,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),Y)),A_1)))) # label(fact_3405_powr__divide) # label(axiom) # label(non_clause). [assumption]. 7.82/7.82 3697 (all X_a (linordered_idom(X_a) -> (all X all Y all N (hBOOL(hAPP(nat,bool,even_odd_even(nat),N)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),X)),hAPP(X_a,X_a,abs_abs(X_a),Y))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),N)))))))) # label(fact_2495_power__mono__even) # label(axiom) # label(non_clause). [assumption]. 7.82/7.82 3698 (all M all N all K_4 (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,N),K_4) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),N))) -> (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),dvd_dvd(int),K_4),M)))))) # label(fact_3497_zrelprime__zdvd__zmult__aux) # label(axiom) # label(non_clause). [assumption]. 7.82/7.82 3699 one_one(int) = t -> (exists X_1 exists Y_1 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Y_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) # label(fact_594__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]. 7.82/7.82 3700 (all X_a (real_normed_vector(X_a) -> (all X all Y hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,norm_norm(X_a),X)),hAPP(X_a,real,norm_norm(X_a),Y))))))) # label(fact_2007_norm__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 7.82/7.82 3701 (all A_1 all B all C_1 hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),C_1))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),B),C_1))),C_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)),C_1)) # label(fact_1547_div__mult1__eq) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3702 (all P_1 all K all I (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),I)) -> (hBOOL(hAPP(int,bool,P_1,K)) -> ((all I_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),I_2)) -> (hBOOL(hAPP(int,bool,P_1,I_2)) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),I_2),one_one(int))))))) -> hBOOL(hAPP(int,bool,P_1,I)))))) # label(fact_346_int__ge__induct) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3703 (all Ma hAPP(int,fun(int,bool),norRRset,Ma) = hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Ma),one_one(int))),Ma)) # label(fact_3585_norRRset__def) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3704 (all K_4 all L_4 hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),hAPP(int,rat,number_number_of(rat),K_4)),hAPP(int,rat,number_number_of(rat),L_4)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,number_number_of(int),K_4)),hAPP(int,int,number_number_of(int),L_4))) # label(fact_3798_Fract__number__of__quotient) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3705 (all X_b all X_a all F all P_1 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),P_1))) -> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,fun(X_a,bool)),fun(X_b,bool),hAPP(fun(fun(X_a,bool),bool),fun(fun(X_b,fun(X_a,bool)),fun(X_b,bool)),combb(fun(X_a,bool),bool,X_b),fEx(X_a)),hAPP(fun(X_a,bool),fun(X_b,fun(X_a,bool)),hAPP(fun(X_b,fun(fun(X_a,bool),fun(X_a,bool))),fun(fun(X_a,bool),fun(X_b,fun(X_a,bool))),combc(X_b,fun(X_a,bool),fun(X_a,bool)),hAPP(fun(X_b,fun(X_a,fun(bool,bool))),fun(X_b,fun(fun(X_a,bool),fun(X_a,bool))),hAPP(fun(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool))),fun(fun(X_b,fun(X_a,fun(bool,bool))),fun(X_b,fun(fun(X_a,bool),fun(X_a,bool)))),combb(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),X_b),combs(X_a,bool,bool)),hAPP(fun(X_b,fun(X_a,bool)),fun(X_b,fun(X_a,fun(bool,bool))),hAPP(fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),fun(fun(X_b,fun(X_a,bool)),fun(X_b,fun(X_a,fun(bool,bool)))),combb(fun(X_a,bool),fun(X_a,fun(bool,bool)),X_b),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj)),hAPP(fun(X_a,X_b),fun(X_b,fun(X_a,bool)),hAPP(fun(X_b,fun(fun(X_a,X_b),fun(X_a,bool))),fun(fun(X_a,X_b),fun(X_b,fun(X_a,bool))),combc(X_b,fun(X_a,X_b),fun(X_a,bool)),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_a,X_b),fun(X_a,bool))),hAPP(fun(fun(X_b,bool),fun(fun(X_a,X_b),fun(X_a,bool))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_a,X_b),fun(X_a,bool)))),combb(fun(X_b,bool),fun(fun(X_a,X_b),fun(X_a,bool)),X_b),combb(X_b,bool,X_a)),fequal(X_b))),F)))),P_1))))))) # label(fact_4447_finite__image__set) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3706 (all X_a undefined(X_a) = ti(X_a,undefined(X_a))) # label(tsy_c_HOL_Oundefined_res) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3707 (all J_2 all A_1 all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),zero_zero(int)),P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_5),A_1)) -> -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),J_2))),P_5)))))))) # label(fact_1142_MultInvPair__distinct) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3708 (all X_a (comm_semiring_1(X_a) -> (all X all N hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N))))) # label(fact_473_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3709 (all K_4 hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit1,K_4)) = hAPP(int,int,bit1,hAPP(int,int,pred,hAPP(int,int,uminus_uminus(int),K_4)))) # label(fact_2931_minus__Bit1) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3710 (all I all K (zero_zero(int) = ti(int,K) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),I)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I),K)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I)) <-> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),I),K) = zero_zero(int))) # label(fact_1113_zdiv__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3711 (all Xa all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),Ya)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Ya),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Ya),Xa)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,tan,Ya)),hAPP(real,real,tan,Xa))))))))) # label(fact_2447_tan__monotone_H) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3712 (all P_5 hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) = hAPP(product_prod(int,int),product_prod(int,int),normalize,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_5),zero_zero(int)))) # label(fact_3748_normalize__denom__zero) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3713 (all X_a (ordere453448008miring(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),zero_zero(X_a)))))))) # label(fact_429_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3714 (all X_a (ordered_semiring(X_a) -> (all C_1 all D_3 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),D_3)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D_3)))))))))) # label(fact_436_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3715 (all A_1 all B hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(nat,int,semiring_1_of_nat(int),A_1)),hAPP(nat,int,semiring_1_of_nat(int),B)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A_1),B))) # label(fact_1161_zdiv__int) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3716 (all A_1 all B hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B)))) # label(fact_4380_Fract__coprime) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3717 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) -> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),ln),Xa)))) # label(fact_4905_isCont__ln) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3718 (all X_a (order(X_a) -> (all Xa all Ya all F (hBOOL(hAPP(fun(nat,X_a),bool,hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,X_a),bool),strict_mono(nat,X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(fun(nat,bool),fun(nat,bool))),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool))),combs(nat,fun(nat,bool),fun(nat,bool)),hAPP(fun(nat,fun(nat,fun(bool,bool))),fun(nat,fun(fun(nat,bool),fun(nat,bool))),hAPP(fun(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool))),fun(fun(nat,fun(nat,fun(bool,bool))),fun(nat,fun(fun(nat,bool),fun(nat,bool)))),combb(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),nat),combs(nat,bool,bool)),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,fun(bool,bool))),hAPP(fun(fun(nat,bool),fun(nat,fun(bool,bool))),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,fun(bool,bool)))),combb(fun(nat,bool),fun(nat,fun(bool,bool)),nat),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj)),dvd_dvd(nat)))),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),hAPP(fun(fun(nat,bool),fun(nat,bool)),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool))),combb(fun(nat,bool),fun(nat,bool),nat),hAPP(fun(bool,bool),fun(fun(nat,bool),fun(nat,bool)),combb(bool,bool,nat),fNot)),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),dvd_dvd(nat))))),F)) -> (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(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,F,Xa)),hAPP(nat,X_a,F,Ya)))))))) # label(fact_4815_dvd_Ostrict__monoD) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3719 (all X_b all X_a (topological_space(X_a) & real_normed_vector(X_b) -> (all G all Ma all F all L all Xa (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),Xa))) -> (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),Ma),hAPP(X_a,filter(X_a),at(X_a),Xa))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),minus_minus(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),L),Ma)),hAPP(X_a,filter(X_a),at(X_a),Xa)))))))) # label(fact_4576_LIM__diff) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3720 (all N all A_1 all B all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),B)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),P_5),N)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),P_5),N)),A_1)))))) # label(fact_784_zprime__power__zdvd__cancel__right) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3721 (all X_a (real_algebra_1(X_a) -> one_one(X_a) = hAPP(real,X_a,of_real(X_a),one_one(real)))) # label(fact_2799_of__real__1) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3722 (all Xa (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)) -> ti(int,Xa) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Xa),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) # label(fact_1737_even__div__2__prop2) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3723 (all X_a all X_b (real_normed_vector(X_b) -> (all F all A_2 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),A_2)) -> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),zero_zero(real)),A_2)))))) # label(fact_4474_tendsto__norm__zero) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3724 (all A_3 all B_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_3),B_2)) <-> (all D_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_2),B_2)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_2),A_3)) <-> one_one(nat) = D_2)))) # label(fact_2897_coprime) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3725 (all X_a all X_b (topological_space(X_b) -> (all F all L all A_2 all A_7 (hBOOL(hAPP(filter(X_a),bool,hAPP(filter(X_a),fun(filter(X_a),bool),ord_less_eq(filter(X_a)),A_2),A_7)) -> (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),A_7)) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),A_2))))))) # label(fact_4516_tendsto__mono) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3726 (all X_1 ((-hBOOL(hAPP(nat,bool,even_odd_even(nat),X_1)) -> zero_zero(real) = hAPP(nat,real,cos_coeff,X_1)) & (hBOOL(hAPP(nat,bool,even_odd_even(nat),X_1)) -> hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),X_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),X_1))) = hAPP(nat,real,cos_coeff,X_1)))) # label(fact_2535_cos__coeff__def) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3727 (all X_a (real_algebra_1(X_a) & real_normed_vector(X_a) -> (exists K_2 ((all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(real,X_a,of_real(X_a),X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),X_1)),K_2)))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2)))))) # label(fact_2826_of__real_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3728 (all A_3 all P_4 (hBOOL(hAPP(int,bool,zprime,P_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),P_4)) -> hAPP(int,nat,nat_1,A_3) = hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,A_3),P_4))))) # label(fact_3229_Bnor__prime) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3729 (all N all A_1 all B all P_5 (hBOOL(hAPP(nat,bool,prime,P_5)) -> (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),B)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),N)),A_1)))))) # label(fact_3094_prime__power__dvd__cancel__right) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3730 (all X_a all X_b (comm_monoid_mult(X_b) -> (all F all F_1 ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),F_1)) -> hAPP(X_a,X_b,F,X_1) = one_one(X_b))) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),F_1) = one_one(X_b))))) # label(fact_4797_setprod__1_H) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3731 (all M all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),M) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)) # label(fact_600_nat__mult__commute) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3732 (all X_a (division_ring(X_a) -> (all A_1 all B hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B)))) # label(fact_1941_minus__divide__left) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3733 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,K)),hAPP(int,int,bit1,L))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)))) # label(fact_218_rel__simps_I34_J) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3734 (all A_2 (hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2)) -> hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2)) = hAPP(fun(int,bool),nat,finite_card(int),A_2))) # label(fact_3690_Nat__Transfer_Otransfer__int__nat__set__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3735 (all U_1 all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),U_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) # label(fact_2039_realpow__square__minus__le) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3736 (all X_a all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,finite_finite_1(fun(X_a,bool)),hAPP(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool),collect(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),combc(fun(X_a,bool),fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool))),A_2)))))) # label(fact_3845_finite__Collect__subsets) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3737 (all N all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) -> hAPP(real,real,hAPP(nat,fun(real,real),root,hAPP(nat,nat,suc,N)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(nat,nat,suc,N))) = X)) # label(fact_2707_real__root__pos) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3738 (all X_a (linordered_field(X_a) -> (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),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),zero_zero(X_a))))))) # label(fact_2590_negative__imp__inverse__negative) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3739 (all X_a (comm_monoid_add(X_a) & topological_space(X_a) -> (all F all L (hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),L)) -> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)))))) # label(fact_4118_sums__summable) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3740 (all A_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_1),hAPP(nat,nat,suc,zero_zero(nat))))) # label(fact_2887_coprime__Suc0) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3741 (all N all F_4 all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),F_4),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),F_4),N)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),F_4),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M),N)))))) # label(fact_1323_zdvd__zmod) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3742 (all X_a (dvd(X_a) & comm_ring(X_a) -> (all Ta all D_1 all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),D)) -> (all X_1 all K_1 (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),K_1),D))),Ta))) <-> -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),Ta))))))))) # label(fact_718_inf__period_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3743 (all X_a (semiring_div(X_a) -> (all A_1 zero_zero(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),A_1)))) # label(fact_1280_mod__self) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3744 (all X_a (real_div_algebra(X_a) -> (all X all A_1 (zero_zero(real) != A_1 -> (zero_zero(X_a) != ti(X_a,X) -> hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,inverse_inverse(real),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),X)) = hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X))))))) # label(fact_4768_nonzero__inverse__scaleR__distrib) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3745 (all M all I_1 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),J_2))))) # label(fact_414_trans__le__add2) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3746 (all X_a (real_vector(X_a) -> (all X all Xa_2 hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,uminus_uminus(real),X)),Xa_2) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),X),Xa_2))))) # label(fact_4739_scaleR__left_Ominus) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3747 (all X_a (division_ring(X_a) -> (all A_1 all B all C_1 (ti(X_a,C_1) != zero_zero(X_a) -> (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1) = ti(X_a,B) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C_1) = ti(X_a,A_1)))))) # label(fact_1854_eq__divide__imp) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3748 (all Na all P_1 (hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) -> ((all N_1 (hBOOL(hAPP(nat,bool,P_1,N_1)) -> hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,suc,N_1))))) -> hBOOL(hAPP(nat,bool,P_1,Na))))) # label(fact_2832_nat__induct) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3749 (all X_a (real_normed_vector(X_a) -> (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),Xa)),zero_zero(real))) <-> zero_zero(X_a) = ti(X_a,Xa))))) # label(fact_1997_norm__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3750 (all U (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),U)) -> hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),hAPP(nat,fun(nat,bool),ord_lessThan(nat),hAPP(int,nat,nat_1,U))) = hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),zero_zero(int)),U))) # label(fact_4350_image__atLeastZeroLessThan__int) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3751 (all M all I_1 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),M))))) # label(fact_413_trans__le__add1) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3752 (all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),M),zero_zero(int))) -> zero_zero(int) = hAPP(int,int,fact_fact(int),M))) # label(fact_3213_fact__neg__int) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3753 (all P_1 all Na all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K)) -> ((all I_2 all J_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J_1)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),J_1),K)) & hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),I_2)),J_1) = ti(int,Na) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,I_2),J_1)))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Na),K)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Na),K)))))) # label(fact_1499_split__pos__lemma) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3754 (all X_a all G all A_2 hAPP(fun(X_a,bool),real,hAPP(fun(X_a,real),fun(fun(X_a,bool),real),big_co1399186613setsum(X_a,real),hAPP(fun(X_a,complex),fun(X_a,real),hAPP(fun(complex,real),fun(fun(X_a,complex),fun(X_a,real)),combb(complex,real,X_a),re),G)),A_2) = hAPP(complex,real,re,hAPP(fun(X_a,bool),complex,hAPP(fun(X_a,complex),fun(fun(X_a,bool),complex),big_co1399186613setsum(X_a,complex),G),A_2))) # label(fact_4273_Re_Osetsum) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3755 (all X_a (semiring_div(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)))) # label(fact_1297_mod__add__self2) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3756 (all X_a all F all Xa all A_2 ((all I_2 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I_2),A_2)) -> one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(X_a,int,F,I_2)),Xa))) -> one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co302546470etprod(X_a,int),F),A_2)),Xa))) # label(fact_4441_setprod__coprime__int) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3757 (all X_a (real_normed_vector(X_a) -> (all G all F all L_2 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),L_2),sequentially)) -> (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),F)),G)),zero_zero(X_a)),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),G),L_2),sequentially))))))) # label(fact_4511_LIMSEQ__diff__approach__zero2) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3758 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 hAPP(X_a,X_a,abs_abs(X_a),A_1) = hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1))))) # label(fact_1943_abs__minus__cancel) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3759 (all N -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,real_1(nat),N)),zero_zero(real)))) # label(fact_2382_not__real__of__nat__less__zero) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3760 (all A_1 all B all C_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C_1)) -> hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),C_1))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C_1)))) # label(fact_1479_zmod__zmult2__eq) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3761 (all X all N hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(nat,int,semiring_1_of_nat(int),X)),N) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),N))) # label(fact_178_Nat__Transfer_Otransfer__int__nat__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3762 (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),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Y),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),N_1),X))))),X))))) # label(fact_1064_best__division__abs) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3763 (all X_a (linord20386208strict(X_a) -> (all C_1 all D_3 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D_3)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D_3)))))))))) # label(fact_483_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3764 (all X_a (real_normed_vector(X_a) -> (all R_3 all A_1 all B hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),R_3),A_1)),B)))) # label(fact_4710_scaleR_OscaleR__left) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3765 (all X_a fequal(X_a) = ti(fun(X_a,fun(X_a,bool)),fequal(X_a))) # label(tsy_c_fequal_res) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3766 (all X_a (preorder(X_a) -> (all Z_1 all X all 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),Z_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Z_1))))))) # label(fact_518_order__less__trans) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3767 (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),ord_less(int),hAPP(int,int,fact_fact(int),M)),hAPP(int,int,fact_fact(int),N)))))) # label(fact_3214_fact__less__mono__int) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3768 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real))) -> hAPP(real,nat,natceiling,X) = zero_zero(nat))) # label(fact_1673_natceiling__neg) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3769 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),K)),hAPP(int,int,number_number_of(int),L))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)))) # label(fact_225_less__eq__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3770 (all P_1 all A0 all A1 (hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),posDivAlg_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))) -> ((all A_4 all B_4 (hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),posDivAlg_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_4),B_4))) -> ((-(hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_4),B_4)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_4),zero_zero(int)))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B_4)))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A_4),B_4))))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A0),A1))))) # label(fact_1681_posDivAlg_Opinduct) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3771 (all Xa (hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,suc,Xa))) <-> -hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa)))) # label(fact_2472_even__Suc) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3772 (all X_a all X_b (real_normed_algebra(X_b) & comm_ring_1(X_b) -> (all X_2 all L_2 all S ((all N_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),N_1),S)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_b,fun(filter(nat),bool),hAPP(fun(nat,X_b),fun(X_b,fun(filter(nat),bool)),tendsto(nat,X_b),hAPP(X_a,fun(nat,X_b),X_2,N_1)),hAPP(X_a,X_b,L_2,N_1)),sequentially)))) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_b,fun(filter(nat),bool),hAPP(fun(nat,X_b),fun(X_b,fun(filter(nat),bool)),tendsto(nat,X_b),hAPP(fun(X_a,bool),fun(nat,X_b),hAPP(fun(nat,fun(fun(X_a,bool),X_b)),fun(fun(X_a,bool),fun(nat,X_b)),combc(nat,fun(X_a,bool),X_b),hAPP(fun(nat,fun(X_a,X_b)),fun(nat,fun(fun(X_a,bool),X_b)),hAPP(fun(fun(X_a,X_b),fun(fun(X_a,bool),X_b)),fun(fun(nat,fun(X_a,X_b)),fun(nat,fun(fun(X_a,bool),X_b))),combb(fun(X_a,X_b),fun(fun(X_a,bool),X_b),nat),big_co302546470etprod(X_a,X_b)),hAPP(fun(X_a,fun(nat,X_b)),fun(nat,fun(X_a,X_b)),combc(X_a,nat,X_b),X_2))),S)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),L_2),S)),sequentially)))))) # label(fact_4682_LIMSEQ__setprod) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3773 (all X_a all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)),hAPP(fun(X_a,bool),nat,finite_card(X_a),B_1)))))) # label(fact_2748_card__mono) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3774 (all N hAPP(nat,nat,suc,N) != N) # label(fact_1278_n__not__Suc__n) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3775 (all A_3 all B_2 all P_4 (hBOOL(hAPP(nat,bool,prime,P_4)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),A_3)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),B_2)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_3),B_2)))))) # label(fact_3077_prime__divprod__eq) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3776 (all X_2 ((all J_1 exists M_3 all M_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),M_2)) -> (all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(nat,real,X_2,M_2)),hAPP(nat,real,X_2,N_1)))),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,J_1))))))))) <-> hBOOL(hAPP(fun(nat,real),bool,cauchy(real),X_2)))) # label(fact_3339_Cauchy__iff2) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3777 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,ln,Xa)),zero_zero(real))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),one_one(real)))))) # label(fact_1721_ln__less__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3778 (all A_1 all B hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B))) # label(fact_2070_zmod__zminus__zminus) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3779 (all A_1 hAPP(complex,real,norm_norm(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,cos,A_1)),hAPP(real,real,sin,A_1))) = one_one(real)) # label(fact_2767_cmod__unit__one) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3780 (all Ta all B_1 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Ta),one_one(int))),B_1)) -> (all X_1 ((all Xa_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D))) -> (all Xb (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1)) -> hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa_1) != ti(int,X_1))))) -> (ti(int,Ta) = ti(int,X_1) -> ti(int,Ta) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),D))))))) # label(fact_4660_bset_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3781 (all X_a (number_ring(X_a) -> hAPP(int,X_a,number_number_of(X_a),pls) = zero_zero(X_a))) # label(fact_85_number__of__Pls) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3782 (all B_2 all A_3 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),A_3),zOdd)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),zOdd)) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_3),B_2)),zEven))))) # label(fact_1712_IntNatAux_Oodd__plus__odd) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3783 (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_353_le__antisym) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3784 (all X_a (semiring_1(X_a) -> (all M all N hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N))))) # label(fact_372_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3785 (all X all Y hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,X)),hAPP(real,real,cos,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,sin,Y))) = hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y))) # label(fact_2259_sin__add) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3786 (all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),N) = N) # label(fact_159_plus__nat_Oadd__0) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3787 (all X_a (comm_ring_1(X_a) -> (all X ((-hBOOL(hAPP(nat,bool,even_odd_even(nat),X)) -> hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),X)) & (hBOOL(hAPP(nat,bool,even_odd_even(nat),X)) -> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),X) = one_one(X_a)))))) # label(fact_2494_minus__one__even__odd__power) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3788 (all Xa all Ya (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Xa),Ya)) | -hBOOL(hAPP(nat,bool,even_odd_even(nat),Ya)) & -hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa)) | hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa)) & hBOOL(hAPP(nat,bool,even_odd_even(nat),Ya)) <-> hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Xa),Ya))))) # label(fact_2482_even__difference__nat) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3789 (all X_a (archim1727834104eiling(X_a) -> (all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),hAPP(X_a,int,archim2085082626_floor(X_a),Xa))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),Xa)))))) # label(fact_2910_one__le__floor) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3790 (all X_a (linord219039673up_add(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),A_3)),zero_zero(X_a))))))) # label(fact_198_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3791 (all X_a all A_2 all B_1 hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)),A_2))) # label(fact_2213_Diff__subset) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3792 (all X_a (linord1117847801e_zero(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),C_1)))))))) # label(fact_1836_divide__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3793 (all X_a ti(fun(product_prod(int,int),X_a),abs_Frac(X_a)) = abs_Frac(X_a)) # label(tsy_c_Nitpick_OAbs__Frac_res) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3794 (all R_3 all P_5 all Q_3 (hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_5),Q_3) = hAPP(rat,product_prod(int,int),quotient_of,R_3) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Q_3)))) # label(fact_3740_quotient__of__denom__pos) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3795 (all X all Y hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(complex,real,im,X)),hAPP(complex,real,im,Y)) = hAPP(complex,real,im,hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),X),Y))) # label(fact_3003_Im_Oadd) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3796 (all A_1 all 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)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B))) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,number_number_of(int),min)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),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)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)))) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),B) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B)))))) # label(fact_1638_negDivAlg_Osimps) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3797 (all X_a (metric_space(X_a) -> (all X_2 all L_2 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),L_2),sequentially)) <-> (all R_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1)) -> (exists No ((all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),No),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),hAPP(nat,X_a,X_2,N_1)),L_2)),R_1)))) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),No)))))))))) # label(fact_4542_LIMSEQ__iff__nz) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3798 (all B_2 all A_3 all Ma (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,A_3),Ma))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2)))) # label(fact_3234_Bnor__mem__zg) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3799 (all X_a (ab_group_add(X_a) -> (all A_3 all B_2 (ti(X_a,B_2) = ti(X_a,A_3) <-> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2) = zero_zero(X_a))))) # label(fact_734_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3800 (all P (hBOOL(hAPP(bool,bool,fNot,P)) | hBOOL(P))) # label(help_fNot_2_1_U) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3801 (all Xa all Ya ((hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)) <-> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ya),zEven))) <-> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xa),Ya)),zEven)))) # label(fact_1699_even__diff) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3802 (all X_a all X_b (comm_monoid_mult(X_b) -> (all F all S all T_3 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),T_3)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),S),T_3)) -> ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),T_3),S))) -> one_one(X_b) = hAPP(X_a,X_b,F,X_1))) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),T_3) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),S))))))) # label(fact_3987_setprod__mono__one__right) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3803 (all D_3 all I_1 (zero_zero(int) != ti(int,I_1) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_3),I_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,abs_abs(int),D_3)),hAPP(int,int,abs_abs(int),I_1)))))) # label(fact_1027_dvd__imp__le__int) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3804 (all X_a all X_b all P_1 hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),the(product_prod(X_a,X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),P_1)) = hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),the(product_prod(X_a,X_b)),hAPP(fun(product_prod(X_a,X_b),X_b),fun(product_prod(X_a,X_b),bool),hAPP(fun(product_prod(X_a,X_b),fun(X_b,bool)),fun(fun(product_prod(X_a,X_b),X_b),fun(product_prod(X_a,X_b),bool)),combs(product_prod(X_a,X_b),X_b,bool),hAPP(fun(product_prod(X_a,X_b),X_a),fun(product_prod(X_a,X_b),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,bool)),fun(fun(product_prod(X_a,X_b),X_a),fun(product_prod(X_a,X_b),fun(X_b,bool))),combb(X_a,fun(X_b,bool),product_prod(X_a,X_b)),P_1),product_fst(X_a,X_b))),product_snd(X_a,X_b)))) # label(fact_4172_The__split) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3805 (all X_a (real_n2089651433ebra_1(X_a) -> (all X all N hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(X_a,real,norm_norm(X_a),X)),N)))))) # label(fact_2011_norm__power__ineq) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3806 (all L_4 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_4)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),L_4),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),L_4),M)))))) # label(fact_809_diff__less__mono2) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3807 (all X_a (linordered_idom(X_a) -> (all Xa all Wa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),hAPP(int,nat,number_number_of(nat),Wa))),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),zero_zero(X_a))) & -hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,number_number_of(nat),Wa))))))) # label(fact_2511_power__less__zero__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3808 (all A_3 all B_2 all Na (Na != zero_zero(nat) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A_3),Na)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B_2),Na))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_3),B_2))))) # label(fact_1236_pow__divides__eq__nat) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3809 (all Xa hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),sin_coeff)),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(real,real,sin,Xa)))) # label(fact_4123_sin__converges) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3810 (all A_3 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),Ma)) -> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,A_3),Ma))),Ma) = one_one(int))) # label(fact_3958_Bnor__prod__zgcd) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3811 (all Y all N all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),hAPP(nat,int,hAPP(int,fun(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_5),Y)))))) # label(fact_923_zpower__zdvd__prop2) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3812 (all Z (hAPP(int,int,number_number_of(int),min) = ti(int,Z) | one_one(int) = ti(int,Z) <-> hAPP(int,int,abs_abs(int),Z) = one_one(int))) # label(fact_1007_abs__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3813 (all X_a (field(X_a) -> (all X all Y all Z_1 (zero_zero(X_a) != ti(X_a,Z_1) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Z_1)),Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),Y))),Z_1))))) # label(fact_1878_divide__add__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3814 (all Xa -(hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)) & hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd)))) # label(fact_1704_even__odd__conj) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3815 (all X_a (semiring_div(X_a) -> (all A_1 all B (zero_zero(X_a) != ti(X_a,B) -> ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),B))))) # label(fact_1088_div__mult__self2__is__id) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3816 (all A_1 all B all Q_3 all R_3 (hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_3),R_3))) -> (zero_zero(int) != ti(int,B) -> ti(int,Q_3) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)))) # label(fact_1618_divmod__int__rel__div) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3817 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C_1)))) # label(fact_296_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3818 (all B all A_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A_1)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(real,nat,natfloor,A_1)),hAPP(real,nat,natfloor,B))),hAPP(real,nat,natfloor,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_1),B))))))) # label(fact_1676_le__mult__natfloor) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3819 (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),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Y),hAPP(nat,nat,suc,N)))))) # label(fact_1407_divides__rexp) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3820 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X))))))) # label(fact_1740_ln__one__plus__pos__lower__bound) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3821 (all X_4 all N_5 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_5)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_4)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(nat,fun(real,real),root,N_5),X_4)))))) # label(fact_2708_real__root__pos__pos) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3822 (all X_a (real_vector(X_a) -> (all A_1 all X all Y hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),Y))))) # label(fact_4701_scaleR__right_Odiff) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3823 (all X_a all X_b all X_c ti(fun(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c)),product_prod_case(X_a,X_b,X_c)) = product_prod_case(X_a,X_b,X_c)) # label(tsy_c_Product__Type_Oprod_Oprod__case_res) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3824 (all N N = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),one_one(nat))) # label(fact_620_nat__mult__1__right) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3825 (all X_a all X_b (real_normed_vector(X_b) -> (all F all A_2 (hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),zero_zero(real)),A_2)) <-> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),A_2)))))) # label(fact_4473_tendsto__norm__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3826 (all X_1 hAPP(real,real,sin,X_1) = hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),sin_coeff)),hAPP(real,fun(nat,real),power_power(real),X_1)))) # label(fact_3997_lemma__sin__ext) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3827 (all X_a (number_ring(X_a) -> (all W hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),hAPP(int,X_a,number_number_of(X_a),W))),hAPP(int,X_a,number_number_of(X_a),W)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,W))))) # label(fact_103_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3828 (all X_a (cancel_semigroup_add(X_a) -> (all B all A_1 all C_1 (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),A_1) -> ti(X_a,B) = ti(X_a,C_1))))) # label(fact_290_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3829 (all A_1 hAPP(real,complex,cis,hAPP(real,real,uminus_uminus(real),A_1)) = hAPP(complex,complex,inverse_inverse(complex),hAPP(real,complex,cis,A_1))) # label(fact_3055_cis__inverse) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3830 (all X_a (comple44059174attice(X_a) -> (all Va all U all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),U),A_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Va),U)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Va),hAPP(fun(X_a,bool),X_a,complete_Sup_Sup(X_a),A_2)))))))) # label(fact_1817_complete__lattice__class_OSup__upper2) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3831 (all X_a (linorder(X_a) -> (all Z all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Xa),Ya))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),Ya)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),Xa)))))) # label(fact_4984_less__max__iff__disj) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3832 (all X_a all S all T_3 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),T_3)) -> (-hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S)) -> -hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),S),T_3)))))) # label(fact_3061_Diff__infinite__finite) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3833 (all Xa all Ya 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_eq(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Xa),Z)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Ya),Z))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya))))) # label(fact_636_real__mult__le__cancel__iff1) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3834 (all Wa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,Wa)),zero_zero(int))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),zero_zero(int))))) # label(fact_58_bin__less__0__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3835 (all A_1 all B hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,cis,A_1)),hAPP(real,complex,cis,B)) = hAPP(real,complex,cis,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_1),B))) # label(fact_3053_cis__mult) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3836 (all X_a (real_vector(X_a) -> (all X zero_zero(X_a) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),zero_zero(real)),X)))) # label(fact_4737_scaleR__left_Ozero) # label(axiom) # label(non_clause). [assumption]. 7.85/7.82 3837 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),one_one(real))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,sqrt,Xa)),one_one(real))))) # label(fact_2150_real__sqrt__lt__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3838 (all Z_1 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_1)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z_1),X)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Z_1))))) # label(fact_937_dvd_Oless__le__trans) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3839 (all X_a (boolean_algebra(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)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Y)),hAPP(X_a,X_a,uminus_uminus(X_a),X))))))) # label(fact_2983_compl__mono) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3840 (all M all N all K_4 hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)),K_4)) # label(fact_611_add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3841 (all X_a all X_b (comm_monoid_add(X_b) -> (all F all G all S all T_3 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),T_3)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),S),T_3)) -> ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),T_3),S))) -> zero_zero(X_b) = hAPP(X_a,X_b,G,X_1))) -> ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),S)) -> hAPP(X_a,X_b,F,X_1) = hAPP(X_a,X_b,G,X_1))) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),T_3) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),S)))))))) # label(fact_4442_setsum__mono__zero__cong__left) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3842 (all X_a all F all A_2 hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),A_2) = hAPP(int,nat,nat_1,hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),hAPP(fun(X_a,nat),fun(X_a,int),hAPP(fun(nat,int),fun(fun(X_a,nat),fun(X_a,int)),combb(nat,int,X_a),semiring_1_of_nat(int)),F)),A_2))) # label(fact_4277_transfer__nat__int__sum__prod2_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3843 (all Xa all P_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),standardRes,P_4),Xa)),hAPP(int,fun(int,bool),sRStar,P_4))) <-> -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Xa),zero_zero(int)),P_4))))) # label(fact_1565_StandardRes__SRStar__prop1) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3844 (all X_a (sgn_if(X_a) -> sgn_sgn(X_a) = ti(fun(X_a,X_a),sgn_sgn(X_a)))) # label(tsy_c_Groups_Osgn__class_Osgn_1_res) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3845 (all Xa all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (one_one(real) = Xa <-> one_one(real) = hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)))) # label(fact_2690_real__root__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3846 (all A_1 all B hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A_1)),B))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) # label(fact_789_zdiff__power2) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3847 (all X_a (linorder(X_a) -> (all T_5 exists Z_2 all X_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Z_2)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),T_5)))))) # label(fact_2853_minf_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3848 (all Ra hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),times_times(real),Ra)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),plus_plus(real),one_one(real))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),uminus_uminus(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),suc)))))),Ra),sequentially))) # label(fact_4523_LIMSEQ__inverse__real__of__nat__add__minus__mult) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3849 (all X_a (linord1117847801e_zero(X_a) -> (all X all Y ((all Z_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Z_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_2),one_one(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_2),X)),Y))))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)))))) # label(fact_1894_field__le__mult__one__interval) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3850 (all M hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),hAPP(nat,nat,suc,zero_zero(nat))) = zero_zero(nat)) # label(fact_1531_mod__1) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3851 (all N all M hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),N),M) = hAPP(int,quickcheck_code_int,quickcheck_of_int,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(quickcheck_code_int,int,quickcheck_int_of,N)),hAPP(quickcheck_code_int,int,quickcheck_int_of,M)))) # label(fact_3565_times__code__int__def) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3852 (all Y all X (-hBOOL(hAPP(int,bool,even_odd_even(int),X)) -> (hBOOL(hAPP(int,bool,even_odd_even(int),Y)) -> -hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),Y)))))) # label(fact_1595_odd__plus__even) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3853 (all N all M_4 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,N)),M_4)) -> (exists M_2 M_4 = hAPP(nat,nat,suc,M_2)))) # label(fact_2728_Suc__le__D) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3854 (all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,nat,suc,hAPP(nat,nat,suc,N))) # label(fact_1486_add__2__eq__Suc_H) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3855 (all Na all Ma (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_eq(nat),hAPP(nat,nat,suc,Na)),Ma)))) # label(fact_1392_less__eq__Suc__le) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3856 (all X_a (real_n2089651433ebra_1(X_a) -> (all R_3 hAPP(X_a,real,norm_norm(X_a),hAPP(real,X_a,of_real(X_a),R_3)) = hAPP(real,real,abs_abs(real),R_3)))) # label(fact_2808_norm__of__real) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3857 (all X_a (real_normed_field(X_a) & banach(X_a) -> (all X all Y hAPP(X_a,X_a,exp(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,exp(X_a),X)),hAPP(X_a,X_a,exp(X_a),Y))))) # label(fact_1775_exp__add) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3858 (all X_a (linordered_field(X_a) -> (all C_1 all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),C_1),B))))))))) # label(fact_1875_divide__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3859 (all X all Y all K_4 all P_5 (hBOOL(hAPP(nat,bool,prime,P_5)) -> (hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),K_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),X),Y) -> (exists I_2 exists J_1 (X = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),I_2) & Y = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),J_1)))))) # label(fact_3451_prime__power__mult) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3860 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) -> hAPP(real,real,hAPP(real,fun(real,real),log,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,ln,hAPP(real,real,exp(real),one_one(real)))),hAPP(real,real,ln,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))),hAPP(real,real,ln,X)))) # label(fact_1898_log__base__10__eq1) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3861 (all X_a (real_n2089651433ebra_1(X_a) -> hAPP(X_a,X_a,sgn_sgn(X_a),one_one(X_a)) = one_one(X_a))) # label(fact_2266_sgn__one) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3862 (all X_2 all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),X_2),A_3),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),im),X_2)),hAPP(complex,real,im,A_3)),sequentially)))) # label(fact_4537_Im_OLIMSEQ) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3863 (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_265_transfer__int__nat__quantifiers_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3864 (all X_a (linordered_semidom(X_a) -> (all N all R_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),R_3)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),R_3),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,hAPP(X_a,fun(nat,X_a),power_power(X_a),R_3),hAPP(nat,nat,suc,N))),R_3))))))) # label(fact_1456_realpow__Suc__le__self) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3865 (all A_3 all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),Xa)) -> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),inverse_divide(real),A_3)),hAPP(real,fun(nat,real),power_power(real),Xa))),zero_zero(real)),sequentially)))) # label(fact_4461_LIMSEQ__divide__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3866 (all X_2 (hBOOL(hAPP(fun(nat,complex),bool,cauchy(complex),X_2)) -> hBOOL(hAPP(fun(nat,real),bool,cauchy(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),im),X_2))))) # label(fact_3878_Im_OCauchy) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3867 (all Xa all X_2 (hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),Xa),X_2)) -> ((exists Z_2 all X_1 (hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_1),X_2)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),Z_2)))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),hAPP(fun(real,bool),real,complete_Sup_Sup(real),X_2)))))) # label(fact_3721_Sup__upper__EX) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3868 (all X_a (number_ring(X_a) -> (all A_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(int,X_a,number_number_of(X_a),pls)) = ti(X_a,A_1)))) # label(fact_88_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3869 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,exp(real),Xa)),one_one(real))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),zero_zero(real))))) # label(fact_1784_exp__le__one__iff) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3870 (all A_3 all B_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2)) -> (hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),A_3))))) # label(fact_3815_one__le__Fract__iff) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3871 (all X_a (comm_semiring_1(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C_1))))))) # label(fact_712_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3872 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),one_one(real))) -> hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(real,real,sin,hAPP(real,real,arccos,X))))) # label(fact_2350_sin__arccos) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3873 (all X_a (linordered_field(X_a) -> (all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))))))) # label(fact_2609_inverse__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3874 (all X_a (real_normed_field(X_a) -> (all Db all F all G all Xa all Da (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),hAPP(X_a,X_a,G,Xa)),Da)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),Xa),Db)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),F),G)),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Da),Db)))))))) # label(fact_3885_DERIV__chain2) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3875 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),zero_zero(int)))))) # label(fact_1108_div__nonneg__neg__le0) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3876 (all X_a (number_semiring(X_a) -> (all V (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V)) -> hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,succ,V)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),hAPP(int,X_a,number_number_of(X_a),V)))))) # label(fact_271_semiring__1__add__number__of) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3877 (all N all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)) = M)) # label(fact_847_le__add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3878 (all L all U hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),L),U)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),U),L)),one_one(int)))) # label(fact_4630_card__atLeastAtMost__int) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3879 (all X_a (real_normed_algebra(X_a) -> (all C all F all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),A_3)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),F)),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C))))))) # label(fact_4108_sums__mult2) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3880 (all A_1 all B hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) = hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,A_1),B))) # label(fact_3132_div__int__def) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3881 (all Na all K all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),Ma)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),Na)) -> (Ma = Na <-> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),K) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Ma),K))))) # label(fact_817_eq__diff__iff) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3882 (all P_1 all Na ((exists M_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_2),Na)) & hBOOL(hAPP(nat,bool,P_1,M_2)))) <-> (exists X_1 (hBOOL(hAPP(nat,bool,P_1,X_1)) & hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_1),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)),Na))))))) # label(fact_4560_ex__nat__less) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3883 (all L all K ((zero_zero(int) != ti(int,K) -> (ti(int,L) = zero_zero(int) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,K),L)) & (ti(int,L) != zero_zero(int) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,K),L) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(fun(int,int),fun(product_prod(int,int),product_prod(int,int)),product_apsnd(int,int,int),hAPP(int,fun(int,int),times_times(int),hAPP(int,int,sgn_sgn(int),L))),hAPP(product_prod(int,int),product_prod(int,int),hAPP(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),hAPP(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),if(product_prod(int,int)),hAPP(int,bool,hAPP(int,fun(int,bool),fequal(int),hAPP(int,int,sgn_sgn(int),K)),hAPP(int,int,sgn_sgn(int),L))),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),pdivmod,K),L)),hAPP(product_prod(int,int),product_prod(int,int),hAPP(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_prod_case(int,int,product_prod(int,int)),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int)))),combs(int,fun(int,product_prod(int,int)),fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),hAPP(fun(fun(int,fun(product_prod(int,int),product_prod(int,int))),fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int))))),combb(fun(int,fun(product_prod(int,int),product_prod(int,int))),fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int))),int),combs(int,product_prod(int,int),product_prod(int,int))),hAPP(fun(int,product_prod(int,int)),fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),hAPP(fun(product_prod(int,int),fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(fun(int,product_prod(int,int)),fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int))))),combb(product_prod(int,int),fun(int,fun(product_prod(int,int),product_prod(int,int))),int),hAPP(fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),fun(product_prod(int,int),fun(int,fun(product_prod(int,int),product_prod(int,int)))),combc(int,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),hAPP(fun(int,bool),fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),hAPP(fun(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),fun(fun(int,bool),fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))),combb(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),int),if(product_prod(int,int))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),fequal(int)),zero_zero(int))))),hAPP(int,fun(int,product_prod(int,int)),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int))),combc(int,int,product_prod(int,int)),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int)),uminus_uminus(int))),zero_zero(int))))),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(fun(int,int),fun(int,product_prod(int,int)))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combc(int,fun(int,int),fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(fun(int,int),fun(int,product_prod(int,int)))),hAPP(fun(fun(int,product_prod(int,int)),fun(fun(int,int),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(fun(int,int),fun(int,product_prod(int,int))))),combb(fun(int,product_prod(int,int)),fun(fun(int,int),fun(int,product_prod(int,int))),int),combb(int,product_prod(int,int),int)),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int)),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),hAPP(fun(int,int),fun(int,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(int,int),fun(int,fun(int,int))),combb(int,fun(int,int),int),minus_minus(int)),uminus_uminus(int))),one_one(int))))),hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,abs_abs(int),L))))),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),pdivmod,K),L)))))) & (ti(int,K) = zero_zero(int) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),zero_zero(int)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,K),L)))) # label(fact_4096_divmod__int__code) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3884 (all K_4 all I_1 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),I_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),J_2))))) # label(fact_616_mult__le__mono2) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3885 (all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,sin,X)),one_one(real)))) # label(fact_2273_sin__le__one) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3886 (all Z_1 hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(complex,real,re,Z_1)),hAPP(real,real,uminus_uminus(real),hAPP(complex,real,im,Z_1))) = hAPP(complex,complex,cnj,Z_1)) # label(fact_3037_cnj__def) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3887 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),ln),Xa),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),Xa))))) # label(fact_2653_DERIV__ln__divide) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3888 (all A_3 all B_2 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_3),Ma)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B_2),Ma)),Ma)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),B_2),Ma)))) # label(fact_1321_zcong__zmod) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3889 (all Xa all Ya (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ya),zEven)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Xa),Ya)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Xa),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Ya),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) # label(fact_1735_even__div__2__l) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3890 (all X all Y hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,X),Y)))) # label(fact_3473_zgcd__geq__zero) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3891 (all X_a (real_normed_vector(X_a) -> (all A_1 all R_3 all B hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),B))))) # label(fact_4712_scaleR_OscaleR__right) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3892 (all X_a (real_n2089651433ebra_1(X_a) & banach(X_a) -> (all Xa hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))))))) # label(fact_4773_summable__exp__generic) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3893 (all X_a (real_normed_algebra(X_a) -> (all A_1 all B hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B))))) # label(fact_1909_mult_Ominus__right) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3894 (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_949_dvd_Oantisym) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3895 (all Na hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less_eq(int)),Na)))))) # label(fact_3853_bdd__int__set__le__finite) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3896 (all Ya all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),Xa),Ya))))) # label(fact_4562_SetInterval_Otransfer__nat__int__set__function__closures) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3897 (all B_2 all P_4 all A_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_3)) -> (-hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_3),one_one(int))),P_4))) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,A_3),P_4))) -> ti(int,B_2) = hAPP(int,int,hAPP(int,fun(int,int),inv,P_4),A_3) | ti(int,B_2) = ti(int,A_3))))) # label(fact_1208_wset__mem__imp__or) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3898 (all X_a (comm_semiring_1(X_a) -> (all A_1 zero_zero(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),zero_zero(X_a)),A_1)))) # label(fact_378_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3899 (all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N) = hAPP(nat,nat,suc,hAPP(nat,nat,suc,N))) # label(fact_1485_add__2__eq__Suc) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3900 (all Xa all K (hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),K)),one_one(int)) = ti(int,Xa) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd)))) # label(fact_1230_zOddI) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3901 (all X all Y hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),Y),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),Y))) # label(fact_4399_gcd__red__int) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3902 (all X_a (ring_1(X_a) -> (all A_1 all N hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),N)))) # label(fact_1999_power__minus) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3903 (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_941_dvd_Oless__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3904 (all P_5 (hBOOL(hAPP(nat,bool,prime,P_5)) -> -hBOOL(hAPP(nat,bool,even_odd_even(nat),P_5)) | hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) = P_5)) # label(fact_3104_prime__odd) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3905 (all Z_1 all W hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Z_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,W))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Z_1),hAPP(int,nat,number_number_of(nat),W))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Z_1),hAPP(int,nat,number_number_of(nat),W)))) # label(fact_3880_zpower__number__of__even) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3906 (all X_a (real_normed_vector(X_a) -> (all Xa all X_2 (hBOOL(hAPP(fun(nat,real),bool,cauchy(real),X_2)) -> hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),X_2)),Xa))))))) # label(fact_4748_scaleR__left_OCauchy) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3907 (all N all D_3 (zero_zero(int) != ti(int,D_3) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_3),N)) -> hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),N),D_3)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,real_1(int),N)),hAPP(int,real,real_1(int),D_3))))) # label(fact_3420_real__of__int__div) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3908 (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_255_zero__le__imp__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3909 (all X_a power_1(X_a) = ti(fun(X_a,fun(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)))),power_1(X_a))) # label(tsy_c_Power_Opower_Opower_res) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3910 (all L all F ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,F,N_1)),hAPP(nat,real,F,hAPP(nat,nat,suc,N_1))))) -> ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,F,N_1)),L))) -> ((all E (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E)) -> (exists N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),L),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,F,N_1)),E)))))) -> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),F),L),sequentially)))))) # label(fact_4898_increasing__LIMSEQ) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3911 (all X_a (ring_div(X_a) -> (all A_1 all B all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C_1))),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),C_1)))) # label(fact_1301_mod__diff__right__eq) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3912 (all Z_1 all W hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z_1),W) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z_1),hAPP(int,int,uminus_uminus(int),W))) # label(fact_2076_diff__int__def__symmetric) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3913 (all X_a all X_b (comm_semiring_1(X_b) & no_zero_divisors(X_b) -> (all F all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> (zero_zero(X_b) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2) <-> (exists X_1 (zero_zero(X_b) = hAPP(X_a,X_b,F,X_1) & hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))))))))) # label(fact_3977_setprod__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3914 (all K all Na all F_3 hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F_3),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),K)))),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F_3),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K))) = hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),F_3),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))) # label(fact_4204_sumr__offset2) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3915 (all X all Y hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,X),Y) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y)) # label(fact_3746_gcd__eq__nitpick__gcd) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3916 (all X_a (ring_char_0(X_a) -> (all A_3 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),ring_1_Ints(X_a))) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),A_3)),A_3) != zero_zero(X_a))))) # label(fact_3836_Ints__odd__nonzero) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3917 (all X_a (linordered_idom(X_a) -> (all A_3 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),ring_1_Ints(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),A_3)),A_3)),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a)))))))) # label(fact_3822_Ints__odd__less__0) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3918 (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)))))))) # label(fact_493_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3919 (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),zero_zero(X_a)))))) # label(fact_99_power2__less__0) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3920 (all X_a all X_b all X_c all X_d all Xa all P_1 ((all A_4 all B_4 all C_4 all D_2 hBOOL(hAPP(product_prod(X_a,product_prod(X_b,product_prod(X_c,X_d))),bool,P_1,hAPP(product_prod(X_b,product_prod(X_c,X_d)),product_prod(X_a,product_prod(X_b,product_prod(X_c,X_d))),hAPP(X_a,fun(product_prod(X_b,product_prod(X_c,X_d)),product_prod(X_a,product_prod(X_b,product_prod(X_c,X_d)))),product_Pair(X_a,product_prod(X_b,product_prod(X_c,X_d))),A_4),hAPP(product_prod(X_c,X_d),product_prod(X_b,product_prod(X_c,X_d)),hAPP(X_b,fun(product_prod(X_c,X_d),product_prod(X_b,product_prod(X_c,X_d))),product_Pair(X_b,product_prod(X_c,X_d)),B_4),hAPP(X_d,product_prod(X_c,X_d),hAPP(X_c,fun(X_d,product_prod(X_c,X_d)),product_Pair(X_c,X_d),C_4),D_2)))))) -> hBOOL(hAPP(product_prod(X_a,product_prod(X_b,product_prod(X_c,X_d))),bool,P_1,Xa)))) # label(fact_3177_prod__induct4) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3921 (all A_3 all B_2 all P_1 ((all A_4 all B_4 all C_4 (hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),B_4),C_4))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B_4),C_4)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_4),B_4)) & hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_4),B_4))) -> hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_4),C_4))))) -> (-hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_3),B_2))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2)) -> (all N_1 -hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),hAPP(product_prod(real,real),real,product_fst(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),N_1))),hAPP(product_prod(real,real),real,product_snd(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),N_1)))))))))) # label(fact_3170_not__P__Bolzano__bisect_H) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3922 (all P_1 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,P_1,hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),Xa),Ya))))) -> ((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),Ya),zero_zero(int))) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),Xa),hAPP(int,int,uminus_uminus(int),Ya)))))) -> ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Ya)) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,uminus_uminus(int),Xa)),Ya))))) -> ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Ya),zero_zero(int))) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,uminus_uminus(int),Xa)),hAPP(int,int,uminus_uminus(int),Ya)))))) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),Xa),Ya)))))))) # label(fact_4427_gcd__cases__int) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3923 (all Xa all A_2 (hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),A_2)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa))))) # label(fact_3685_Nat__Transfer_Otransfer__nat__int__set__function__closures_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3924 (all M all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N))) # label(fact_3272_gcd__add2__nat) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3925 (all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),N) = hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,suc,N)))) # label(fact_1462_Suc3__eq__add__3) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3926 (all Z_1 all W (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Z_1),W)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),W),Z_1)) -> Z_1 = W))) # label(fact_596_real__le__antisym) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3927 (all X all Y hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),hAPP(int,int,uminus_uminus(int),Y))) # label(fact_4408_gcd__neg2__int) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3928 (all P_5 all M ((zero_zero(nat) = M -> one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),M)) & (zero_zero(nat) != M -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),P_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_5),M)))) # label(fact_881_power__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3929 (all A_3 all 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(int),A_3),zero_zero(int))) <-> hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_2)),zero_zero(rat)))))) # label(fact_3805_Fract__less__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3930 (all X_a all X_b (comm_monoid_add(X_b) -> (all F all A_2 (-hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2) = zero_zero(X_b))))) # label(fact_4234_setsum__infinite) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3931 (all X_a (ord(X_a) -> ord_atLeastLessThan(X_a) = ti(fun(X_a,fun(X_a,fun(X_a,bool))),ord_atLeastLessThan(X_a)))) # label(tsy_c_SetInterval_Oord__class_OatLeastLessThan_res) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3932 (all X_a (semiring_1(X_a) -> (all Inc all I ti(X_a,I) = hAPP(X_a,X_a,hAPP(nat,fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(nat,fun(X_a,X_a)),semiri532925092at_aux(X_a),Inc),zero_zero(nat)),I)))) # label(fact_4187_of__nat__aux_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3933 (all X all Y hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y))) # label(fact_4417_transfer__int__nat__gcd_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3934 (all X_a (monoid_add(X_a) -> (all A_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),zero_zero(X_a)),A_1) = ti(X_a,A_1)))) # label(fact_324_add__0__left) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3935 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X))),X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))))) # label(fact_1896_abs__ln__one__plus__x__minus__x__bound__nonpos) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3936 (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) # label(fact_315_zero__le__power2) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3937 (all Z_5 (hBOOL(hAPP(int,bool,nat_neg,Z_5)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_5),zero_zero(int))))) # label(fact_1603_neg__def) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3938 (all X X = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),zero_zero(nat))) # label(fact_3280_gcd__0__nat) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3939 (all X_a all X_b all R_2 all S (hBOOL(hAPP(fun(X_a,fun(X_b,bool)),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),bool),ord_less_eq(fun(X_a,fun(X_b,bool))),hAPP(fun(product_prod(X_a,X_b),bool),fun(X_a,fun(X_b,bool)),hAPP(fun(X_a,fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool))),fun(fun(product_prod(X_a,X_b),bool),fun(X_a,fun(X_b,bool))),combc(X_a,fun(product_prod(X_a,X_b),bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),fun(X_a,fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool))),hAPP(fun(fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool)),fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool))),fun(fun(X_a,fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),fun(X_a,fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool)))),combb(fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool)),fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool)),X_a),combc(X_b,fun(product_prod(X_a,X_b),bool),bool)),hAPP(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),hAPP(fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),fun(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool)))),combb(fun(X_b,product_prod(X_a,X_b)),fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool)),X_a),hAPP(fun(product_prod(X_a,X_b),fun(fun(product_prod(X_a,X_b),bool),bool)),fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),combb(product_prod(X_a,X_b),fun(fun(product_prod(X_a,X_b),bool),bool),X_b),member(product_prod(X_a,X_b)))),product_Pair(X_a,X_b)))),R_2)),hAPP(fun(product_prod(X_a,X_b),bool),fun(X_a,fun(X_b,bool)),hAPP(fun(X_a,fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool))),fun(fun(product_prod(X_a,X_b),bool),fun(X_a,fun(X_b,bool))),combc(X_a,fun(product_prod(X_a,X_b),bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),fun(X_a,fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool))),hAPP(fun(fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool)),fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool))),fun(fun(X_a,fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),fun(X_a,fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool)))),combb(fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool)),fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool)),X_a),combc(X_b,fun(product_prod(X_a,X_b),bool),bool)),hAPP(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),hAPP(fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),fun(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool)))),combb(fun(X_b,product_prod(X_a,X_b)),fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool)),X_a),hAPP(fun(product_prod(X_a,X_b),fun(fun(product_prod(X_a,X_b),bool),bool)),fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),combb(product_prod(X_a,X_b),fun(fun(product_prod(X_a,X_b),bool),bool),X_b),member(product_prod(X_a,X_b)))),product_Pair(X_a,X_b)))),S))) <-> hBOOL(hAPP(fun(product_prod(X_a,X_b),bool),bool,hAPP(fun(product_prod(X_a,X_b),bool),fun(fun(product_prod(X_a,X_b),bool),bool),ord_less_eq(fun(product_prod(X_a,X_b),bool)),R_2),S)))) # label(fact_4362_pred__subset__eq2) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3940 (all X_a all C all X_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),X_2)) -> hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),C)),hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(X_a,bool),nat,finite_card(X_a),X_2))) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),hAPP(nat,fun(X_a,nat),combk(nat,X_a),C)),X_2)))) # label(fact_4290_setsum__const2) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3941 (all M all I_1 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_2)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),J_2))))) # label(fact_136_trans__less__add2) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3942 (all X all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),X)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))))),one_one(int)),P_5))))) # label(fact_1049_Little__Fermat) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3943 (all Z all Z_3 ((hBOOL(hAPP(int,bool,nat_neg,Z_3)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,nat_1,Z)),hAPP(int,nat,nat_1,Z_3)) = hAPP(int,nat,nat_1,Z)) & (-hBOOL(hAPP(int,bool,nat_neg,Z_3)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,nat_1,Z)),hAPP(int,nat,nat_1,Z_3)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),hAPP(int,bool,nat_neg,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z),Z_3))),zero_zero(nat)),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z),Z_3)))))) # label(fact_3860_diff__nat__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3944 (all Ta all B_1 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D)) -> (all X_1 ((all Xa_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D))) -> (all Xb (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1)) -> hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa_1) != ti(int,X_1))))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X_1),Ta)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),D)),Ta))))))) # label(fact_4658_bset_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3945 (all N all X hAPP(real,real,exp(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),X)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,exp(real),X)),N)) # label(fact_2401_exp__real__of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3946 (all X_a all X_b all B_1 all F all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2))) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_2)) -> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))))) # label(fact_3678_finite__surjI) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3947 (all Ya all Xa (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ya),zEven)) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xa),Ya)),zOdd))))) # label(fact_1714_odd__minus__even) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3948 (all X_a (order(X_a) -> (all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya)) <-> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),ord_atMost(X_a),Xa)),hAPP(X_a,fun(X_a,bool),ord_atMost(X_a),Ya))))))) # label(fact_4962_atMost__subset__iff) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3949 (all X_b all X_a all F1 all A_3 all B_2 (hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),F1,A_3),B_2)) -> hBOOL(hAPP(product_prod(X_a,X_b),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),F1),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_3),B_2))))) # label(fact_4067_prod__caseI) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3950 (all X_a (metric_space(X_a) -> (all Xa all Ya (ti(X_a,Ya) = ti(X_a,Xa) <-> hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Xa),Ya) = zero_zero(real))))) # label(fact_3542_dist__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3951 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))) -> hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)))) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)))) # label(fact_3146_mod__neg__neg) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3952 (all J_2 all K_4 all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),J_2)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),K_4)),M)) -> hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),K_4) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),J_2)))) # label(fact_790_neg__one__power__eq__mod__m) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3953 (all X_a (archim1727834104eiling(X_a) -> (all Z_1 all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)),one_one(X_a))),X)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),Z_1))) -> ti(int,Z_1) = hAPP(X_a,int,archimedean_ceiling(X_a),X)))))) # label(fact_3631_ceiling__unique) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3954 (all X_b all X_a (topological_space(X_a) & real_normed_algebra(X_b) -> (all G all M_1 all F all L_2 all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),M_1),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),L_2),M_1)),hAPP(X_a,filter(X_a),at(X_a),A_3)))))))) # label(fact_4591_LIM__mult) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3955 (all X_a (linord1117847801e_zero(X_a) & number(X_a) -> (all Wa all B_2 all 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(X_a),C),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),C)))) & (-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_eq(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),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)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),C)),B_2))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),C))))))) # label(fact_1803_le__divide__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3956 (all R_3 all X hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),R_3),X) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),hAPP(complex,real,re,X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),hAPP(complex,real,im,X)))) # label(fact_4769_complex__scaleR__def) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3957 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) -> (N != hAPP(nat,nat,suc,M) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,M)),N))))) # label(fact_1328_Suc__lessI) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3958 (all A_2 all Ma (hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),A_2),hAPP(int,fun(fun(int,bool),bool),rsetR,Ma))) -> hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),A_2)))) # label(fact_3665_RsetR__fin) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3959 (all X_a all X_b (metric_space(X_b) -> (all G all Ma all F all L all A_2 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),A_2)) -> (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),Ma),A_2)) -> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_a,fun(X_b,real)),fun(fun(X_a,X_b),fun(X_a,real)),combs(X_a,X_b,real),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,real)),hAPP(fun(X_b,fun(X_b,real)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,real))),combb(X_b,fun(X_b,real),X_a),dist_dist(X_b)),F)),G)),hAPP(X_b,real,hAPP(X_b,fun(X_b,real),dist_dist(X_b),L),Ma)),A_2))))))) # label(fact_4485_tendsto__dist) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3960 (all Y hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,arctan,Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) # label(fact_2223_arctan__ubound) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3961 (all A_1 all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5)) -> (hBOOL(hAPP(int,bool,zprime,P_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),P_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,zero_zero(int)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),P_5)))))) # label(fact_1134_Euler__part2) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3962 (all X_a all A_2 hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),A_2)))) # label(fact_2846_Pow__top) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3963 (all P_1 all A0 all A1 (hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),norm_frac_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))) -> ((all A_4 all B_4 (hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),norm_frac_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_4),B_4))) -> ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_4),zero_zero(int))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,hAPP(int,int,uminus_uminus(int),A_4)),hAPP(int,int,uminus_uminus(int),B_4)))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A_4),B_4))))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A0),A1))))) # label(fact_3578_norm__frac_Opinduct) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3964 (all Na all N_6 all F ((all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,N_1)),hAPP(nat,nat,F,hAPP(nat,nat,suc,N_1)))) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,hAPP(nat,nat,suc,N_1))),hAPP(nat,nat,F,N_1))))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Na),N_6)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,N_6)),hAPP(nat,nat,F,Na))) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,Na)),hAPP(nat,nat,F,N_6)))))) # label(fact_1570_dvd_Olift__Suc__mono__less) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3965 (all L all U hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),L),U)))) # label(fact_4270_finite__atLeastLessThan__int) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3966 (all Xa hAPP(real,int,archim2085082626_floor(real),Xa) = hAPP(fun(int,bool),int,the(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(real,fun(int,bool),hAPP(fun(int,fun(real,bool)),fun(real,fun(int,bool)),combc(int,real,bool),hAPP(fun(int,real),fun(int,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(int,real),fun(int,fun(real,bool))),combb(real,fun(real,bool),int),ord_less_eq(real)),ring_1_of_int(real))),Xa))),hAPP(fun(int,real),fun(int,bool),hAPP(fun(real,bool),fun(fun(int,real),fun(int,bool)),combb(real,bool,int),hAPP(real,fun(real,bool),ord_less(real),Xa)),hAPP(fun(int,int),fun(int,real),hAPP(fun(int,real),fun(fun(int,int),fun(int,real)),combb(int,real,int),ring_1_of_int(real)),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),plus_plus(int)),one_one(int))))))) # label(fact_4173_floor__real__def) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3967 (all Xa hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,Xa)) = hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),uminus_uminus(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),sin_coeff)),hAPP(real,fun(nat,real),power_power(real),Xa))))) # label(fact_4001_lemma__sin__minus) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3968 (all X all Y hAPP(real,real,hAPP(real,fun(real,real),dist_dist(real),X),Y) = hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y))) # label(fact_3544_dist__real__def) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3969 (all X_b all X_a (topological_space(X_a) & real_normed_vector(X_b) -> (all F all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3)) -> (zero_zero(X_b) != hAPP(X_a,X_b,F,A_3) -> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),sgn_sgn(X_b)),F)),A_3))))))) # label(fact_4909_isCont__sgn) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3970 (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N))))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)))))) # label(fact_1511_odd__0__le__power__imp__0__le) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3971 (all Wa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,Wa)),zero_zero(int))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),zero_zero(int))))) # label(fact_56_bin__less__0__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3972 (all K all L (hAPP(quickcheck_code_int,int,quickcheck_int_of,L) = hAPP(quickcheck_code_int,int,quickcheck_int_of,K) <-> K = L)) # label(fact_3187_Quickcheck__Narrowing_Oint__of__inject) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3973 (all X all Y all Z_1 hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),Y)),Z_1) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Y),Z_1))) # label(fact_625_zpower__zpower) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3974 (all X_b all X_a (metric_space(X_a) & real_normed_algebra(X_b) -> (all G all F all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),F)),G)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3)))))))) # label(fact_4598_LIM__mult__zero) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3975 (all X hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),one_one(nat)) = one_one(nat)) # label(fact_3276_gcd__lcm__complete__lattice__nat_Oinf__bot__right) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3976 (all X_a all X_b all C all P_4 (hBOOL(hAPP(product_prod(X_a,X_b),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),C),P_4)) -> -(all X_1 all Y_1 (ti(product_prod(X_a,X_b),P_4) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X_1),Y_1) -> -hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),C,X_1),Y_1)))))) # label(fact_4088_splitE) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3977 (all X_a all X_b (ord(X_b) -> (all F all G (hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less(fun(X_a,X_b)),F),G)) <-> -hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),G),F)) & hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G)))))) # label(fact_595_less__fun__def) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3978 (all M all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M)),hAPP(nat,nat,suc,N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)) # label(fact_1347_diff__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3979 (all X_a (zero(X_a) -> zero_zero(X_a) = ti(X_a,zero_zero(X_a)))) # label(tsy_c_Groups_Ozero__class_Ozero_res) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3980 (all Z_1 all X all 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),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z_1))))) # label(fact_3065_termination__basic__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3981 (all K all I all J (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J)) -> (hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),I) = J <-> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),I) = K))) # label(fact_851_le__imp__diff__is__add) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3982 (all N all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),M),N)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M),hAPP(int,int,fact_fact(int),N)))))) # label(fact_3222_dvd__fact__int) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3983 (all X_a (real_normed_field(X_a) & banach(X_a) -> (all Xa hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_a),fun(X_a,bool),isCont(X_a,X_a),exp(X_a)),Xa))))) # label(fact_4929_isCont__exp) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3984 (all X_a (division_ring(X_a) -> (all B all A_1 all C_1 (ti(X_a,C_1) != zero_zero(X_a) -> (ti(X_a,B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1) -> ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C_1)))))) # label(fact_1853_divide__eq__imp) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3985 (all X_a (topological_space(X_a) -> (all F all L (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),L),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),suc)),L),sequentially)))))) # label(fact_4494_LIMSEQ__Suc) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3986 (all X_a all C all F all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),hAPP(fun(X_a,int),fun(X_a,int),hAPP(fun(int,int),fun(fun(X_a,int),fun(X_a,int)),combb(int,int,X_a),hAPP(int,fun(int,int),times_times(int),C)),F)),A_2) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),F),A_2)))) # label(fact_4276_setsum__const__mult) # label(axiom) # label(non_clause). [assumption]. 7.85/7.83 3987 (all Na all Xa (zero_zero(real) != Xa -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (exists T_4 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,abs_abs(real),T_4))) & hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,exp(real),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),Na))) = hAPP(real,real,exp(real),Xa) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),T_4)),hAPP(real,real,abs_abs(real),Xa)))))))) # label(fact_4296_Maclaurin__exp__lt) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 3988 (all B_2 all P_4 all A_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_3)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_3),one_one(int))),P_4))) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,A_3),P_4)))))) # label(fact_1205_wset__subset) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 3989 (all N all J_2 all K_4 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J_2),K_4)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),N)),K_4)))) # label(fact_808_less__imp__diff__less) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 3990 (all I_1 all M hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),M))))) # label(fact_1385_less__add__Suc1) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 3991 (all Y all X (-hBOOL(hAPP(int,bool,even_odd_even(int),X)) -> (-hBOOL(hAPP(int,bool,even_odd_even(int),Y)) -> -hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y)))))) # label(fact_1590_Parity_Oodd__times__odd) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 3992 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx all Ry hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ly)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),Ry)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ly)),Ry))))) # label(fact_365_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 3993 (all X_a (archim1804426504_field(X_a) -> (all X exists X_1 ((all Y_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Y_1),one_one(int))))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Y_1)),X)) -> ti(int,X_1) = ti(int,Y_1))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),one_one(int))))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),X_1)),X)))))) # label(fact_3641_floor__exists1) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 3994 (all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),one_one(real))),hAPP(nat,real,real_1(nat),hAPP(real,nat,natfloor,X))))) # label(fact_2414_real__natfloor__gt__diff__one) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 3995 -(all S_1 all W_1 -(hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,abs_abs(int),W_1))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))) & hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),y),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),S_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))) = ti(int,W_1))) # label(fact_926__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]. 7.85/7.84 3996 (all X_a (number_ring(X_a) -> (all X (hBOOL(hAPP(nat,bool,even_odd_even(nat),X)) -> one_one(X_a) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,number_number_of(X_a),min)),X))))) # label(fact_2513_Parity_Oneg__one__even__power) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 3997 (all X_a (linord1117847801e_zero(X_a) -> (all A_3 all B_2 all C (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),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),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C)),B_2))) & (-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),C),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),zero_zero(X_a)))) & (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C))))))))) # label(fact_1821_le__divide__eq) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 3998 (all Xa hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),sin)),sin)),Xa),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,cos,Xa))),hAPP(real,real,sin,Xa))))) # label(fact_3932_DERIV__sin__sin__mult2) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 3999 (all X (-hBOOL(hAPP(nat,bool,even_odd_even(nat),X)) -> hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),X),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,suc,X)),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))))) # label(fact_2500_odd__nat__plus__one__div__two) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4000 (all A_3 all B_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_2),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_3)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_3),B_2)),zero_zero(int)))))) # label(fact_1079_neg__imp__zdiv__neg__iff) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4001 (all Ma all Wa ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa)) -> ti(int,Wa) = hAPP(nat,int,semiring_1_of_nat(int),Ma)) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Wa)) -> Ma = zero_zero(nat)) <-> Ma = hAPP(int,nat,nat_1,Wa))) # label(fact_1038_nat__eq__iff2) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4002 (exists K_2 ((all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(complex,real,re,X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_1)),K_2)))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2)))) # label(fact_3046_Re_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4003 (all Z_1 hAPP(real,complex,of_real(complex),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(complex,real,re,Z_1))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),Z_1),hAPP(complex,complex,cnj,Z_1))) # label(fact_3042_complex__add__cnj) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4004 (all K_4 all A_1 all J_2 all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5)) -> (hBOOL(hAPP(int,bool,zprime,P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),zero_zero(int)),P_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),J_2)),J_2)),K_4)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),J_2)),A_1)),P_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,K_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),J_2))),P_5))))))) # label(fact_1189_aux______4) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4005 (all X_a (real_n1866405975lgebra(X_a) -> (all X all N hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(X_a,real,norm_norm(X_a),X)),N) = hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N))))) # label(fact_1982_norm__power) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4006 (all X_a all X_b (real_normed_vector(X_b) -> (all X_2 all L_2 all S ((all N_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),N_1),S)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_b,fun(filter(nat),bool),hAPP(fun(nat,X_b),fun(X_b,fun(filter(nat),bool)),tendsto(nat,X_b),hAPP(X_a,fun(nat,X_b),X_2,N_1)),hAPP(X_a,X_b,L_2,N_1)),sequentially)))) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_b,fun(filter(nat),bool),hAPP(fun(nat,X_b),fun(X_b,fun(filter(nat),bool)),tendsto(nat,X_b),hAPP(fun(X_a,bool),fun(nat,X_b),hAPP(fun(nat,fun(fun(X_a,bool),X_b)),fun(fun(X_a,bool),fun(nat,X_b)),combc(nat,fun(X_a,bool),X_b),hAPP(fun(nat,fun(X_a,X_b)),fun(nat,fun(fun(X_a,bool),X_b)),hAPP(fun(fun(X_a,X_b),fun(fun(X_a,bool),X_b)),fun(fun(nat,fun(X_a,X_b)),fun(nat,fun(fun(X_a,bool),X_b))),combb(fun(X_a,X_b),fun(fun(X_a,bool),X_b),nat),big_co1399186613setsum(X_a,X_b)),hAPP(fun(X_a,fun(nat,X_b)),fun(nat,fun(X_a,X_b)),combc(X_a,nat,X_b),X_2))),S)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),L_2),S)),sequentially)))))) # label(fact_4683_LIMSEQ__setsum) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4007 (all P_1 all Xa all Ya ((hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Xa),Ya)) -> hBOOL(hAPP(int,bool,P_1,zero_zero(int)))) & (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ya),Xa)) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,semiring_1_of_nat(int),Xa)),hAPP(nat,int,semiring_1_of_nat(int),Ya))))) <-> hBOOL(hAPP(int,bool,P_1,hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Xa),Ya)))))) # label(fact_884_zdiff__int__split) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4008 (all A_1 all B all C_1 (ti(int,C_1) != zero_zero(int) -> hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C_1),A_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C_1),B)))) # label(fact_3761_mult__rat__cancel) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4009 (all X_a (linorder(X_a) -> (all B all A_1 all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),X)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)),X)))))) # label(fact_4847_min__max_Ole__infI1) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4010 (all X_a (ord(X_a) -> (all C_1 all A_1 all B (ti(X_a,A_1) = ti(X_a,B) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),C_1))))))) # label(fact_522_ord__eq__less__trans) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4011 (all Ya all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Ya),Xa)) -> hAPP(fun(real,bool),real,complete_Sup_Sup(real),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_gr676655987ssThan(real),Ya),Xa)) = Xa)) # label(fact_3720_Sup__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4012 (all X_b all X_a all R_2 all K all Ta all Sb (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),Sb)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),Ta)) -> ((all X_1 (hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),Ta)) -> hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),Sb))),hAPP(X_b,fun(X_a,bool),hAPP(fun(X_a,fun(X_b,bool)),fun(X_b,fun(X_a,bool)),combc(X_a,X_b,bool),R_2),X_1)))) = hAPP(X_b,nat,K,X_1))) -> hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,nat),hAPP(fun(fun(X_b,bool),nat),fun(fun(X_a,fun(X_b,bool)),fun(X_a,nat)),combb(fun(X_b,bool),nat,X_a),finite_card(X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_a),collect(X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_a),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),Ta)))),R_2)))),Sb) = hAPP(fun(X_b,bool),nat,hAPP(fun(X_b,nat),fun(fun(X_b,bool),nat),big_co1399186613setsum(X_b,nat),K),Ta))))) # label(fact_4330_setsum__multicount__gen) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4013 (all Q_3 all B all R_3 all C_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_3),zero_zero(int))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C_1)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Q_3),C_1))),R_3))))))) # label(fact_1476_zmult2__lemma__aux1) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4014 (all X_2 (hBOOL(hAPP(fun(nat,rat),bool,vanishes,X_2)) <-> (all R_1 (hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R_1)) -> (exists K_1 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N_1)) -> hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,abs_abs(rat),hAPP(nat,rat,X_2,N_1))),R_1)))))))) # label(fact_4889_vanishes__def) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4015 (all X_a (number(X_a) & semiring_div(X_a) -> (all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(int,X_a,number_number_of(X_a),Xa)),hAPP(int,X_a,number_number_of(X_a),Ya))) <-> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(int,X_a,number_number_of(X_a),Ya)),hAPP(int,X_a,number_number_of(X_a),Xa)) = zero_zero(X_a))))) # label(fact_1409_dvd__eq__mod__eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4016 (all X_a all C_2 all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),C_2)) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),C_2))))) # label(fact_2200_psubset__subset__trans) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4017 (all X_a (real_normed_field(X_a) -> (all Ya all A_3 hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),inverse_divide(X_a)),Ya)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_3),Ya)),hAPP(X_a,filter(X_a),at(X_a),A_3)))))) # label(fact_4578_divide_Ocont) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4018 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> (hAPP(real,real,sin,X) = zero_zero(real) -> (exists N_1 (X = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) & hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1))))))) # label(fact_2530_sin__zero__lemma) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4019 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)))))) # label(fact_729_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4020 (all X_a (order(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1)) -> (ti(X_a,B) != ti(X_a,A_1) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1))))))) # label(fact_556_xt1_I11_J) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4021 (all X all A_1 all B (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),X),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),X),B)) -> hAPP(int,int,abs_abs(int),X) = one_one(int))))) # label(fact_4429_coprime__common__divisor__int) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4022 (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),hAPP(nat,nat,suc,M)),hAPP(nat,nat,suc,N))))) # label(fact_1269_Suc__mono) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4023 (all X_a (metric_space(X_a) -> (all X_2 all L_2 all A_3 (hBOOL(hAPP(filter(real),bool,hAPP(X_a,fun(filter(real),bool),hAPP(fun(real,X_a),fun(X_a,fun(filter(real),bool)),tendsto(real,X_a),X_2),L_2),hAPP(real,filter(real),at(real),A_3))) <-> (all S_2 (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),S_2),A_3),sequentially)) & (all N_1 hAPP(nat,real,S_2,N_1) != A_3) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,real),fun(nat,X_a),hAPP(fun(real,X_a),fun(fun(nat,real),fun(nat,X_a)),combb(real,X_a,nat),X_2),S_2)),L_2),sequentially)))))))) # label(fact_4659_LIMSEQ__SEQ__conv) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4024 (all A_3 hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),d22set,A_3)))) # label(fact_2730_d22set__fin) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4025 (all X_a (ordered_ring(X_a) -> (all A_3 all E_2 all C all B_2 all D_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2)),E_2)),C)),D_1)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),E_2)),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),E_2)),D_1))))))) # label(fact_775_less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4026 (all X_a (comm_monoid_add(X_a) & topological_space(X_a) -> ti(fun(fun(X_a,bool),X_a),the(X_a)) = the(X_a))) # label(tsy_c_HOL_OThe_3_res) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4027 (all X_a (semiring_1(X_a) -> (all M all N hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),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_173_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4028 (all X_b all X_a (metric_space(X_a) & metric_space(X_b) -> (all X_2 all L_2 all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),X_2),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> (all S_2 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),S_2),A_3),sequentially)) & (all N_1 hAPP(nat,X_a,S_2,N_1) != ti(X_a,A_3)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_b,fun(filter(nat),bool),hAPP(fun(nat,X_b),fun(X_b,fun(filter(nat),bool)),tendsto(nat,X_b),hAPP(fun(nat,X_a),fun(nat,X_b),hAPP(fun(X_a,X_b),fun(fun(nat,X_a),fun(nat,X_b)),combb(X_a,X_b,nat),X_2),S_2)),L_2),sequentially)))))))) # label(fact_4680_LIMSEQ__SEQ__conv1) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4029 (all X_a (real_normed_algebra(X_a) -> (all X hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),zero_zero(X_a)) = zero_zero(X_a)))) # label(fact_1242_mult__right_Ozero) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4030 (all X_a (topological_space(X_a) -> (all A_3 hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_a),fun(X_a,bool),isCont(X_a,X_a),combi(X_a)),A_3))))) # label(fact_4920_isCont__ident) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4031 (all Xa all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(rat,real,ratreal,Xa)),hAPP(rat,real,ratreal,Ya))) <-> hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),Xa),Ya)))) # label(fact_3788_real__less__eq__code) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4032 (all X_a (linord219039673up_add(X_a) -> (all A_3 (hAPP(X_a,X_a,uminus_uminus(X_a),A_3) = ti(X_a,A_3) <-> zero_zero(X_a) = ti(X_a,A_3))))) # label(fact_1933_neg__equal__zero) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4033 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all N hAPP(X_a,X_a,abs_abs(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,number_number_of(X_a),min)),N)) = one_one(X_a)))) # label(fact_1029_abs__power__minus__one) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4034 (all X_a (monoid_mult(X_a) -> (all N one_one(X_a) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),one_one(X_a)),N)))) # label(fact_169_power__one) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4035 (all X_a (number_ring(X_a) -> (all V all W all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),W))),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,number_number_of(X_a),W)),C_1))))) # label(fact_777_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4036 (all X (-hBOOL(hAPP(nat,bool,even_odd_even(nat),X)) -> hAPP(nat,nat,suc,zero_zero(nat)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))))) # label(fact_2502_odd__nat__mod__two__eq__one) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4037 (all X all Y hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,uminus_uminus(int),X)),Y) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y)) # label(fact_4409_gcd__neg1__int) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4038 (all N hAPP(real,nat,natfloor,hAPP(int,real,number_number_of(real),N)) = hAPP(int,nat,number_number_of(nat),N)) # label(fact_1665_natfloor__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4039 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) -> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) = hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),B))))) # label(fact_3134_div__pos__pos) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4040 (all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,exp(real),X)))) # label(fact_1767_exp__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4041 (all X_a (field(X_a) -> (all X all W all Z_1 all Y (zero_zero(X_a) != ti(X_a,Y) -> (zero_zero(X_a) != ti(X_a,Z_1) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Z_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),W),Y))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),W),Z_1))))))) # label(fact_1877_add__frac__eq) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4042 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),X)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,ln,X))))) # label(fact_1724_ln__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4043 (all X hAPP(real,real,sqrt,hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(real,real,abs_abs(real),X)) # label(fact_2166_real__sqrt__abs) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4044 (all P_4 hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),resSet,P_4),hAPP(int,fun(int,bool),sRStar,P_4)))) # label(fact_2942_ResSet__SRStar__prop) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4045 (all X_a finite_finite_1(X_a) = ti(fun(fun(X_a,bool),bool),finite_finite_1(X_a))) # label(tsy_c_Finite__Set_Ofinite_res) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4046 (all X_a (number(X_a) & field_inverse_zero(X_a) -> (all B_2 all Wa all A_3 (ti(X_a,A_3) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),hAPP(int,X_a,number_number_of(X_a),Wa)) <-> (zero_zero(X_a) = hAPP(int,X_a,number_number_of(X_a),Wa) -> zero_zero(X_a) = ti(X_a,A_3)) & (zero_zero(X_a) != hAPP(int,X_a,number_number_of(X_a),Wa) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),hAPP(int,X_a,number_number_of(X_a),Wa)) = ti(X_a,B_2)))))) # label(fact_1779_divide__eq__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4047 (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_153_zless__int) # label(axiom) # label(non_clause). [assumption]. 7.85/7.84 4048 (all X_a (metric_space(X_a) -> (all X_2 all L_2 ((all R_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1)) -> (exists No all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),No),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),hAPP(nat,X_a,X_2,N_1)),L_2)),R_1)))))) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),L_2),sequentially)))))) # label(fact_4687_metric__LIMSEQ__I) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4049 (all Xa all Ya (hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Xa),Xa)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Ya),Ya)) = zero_zero(real) <-> Xa = zero_zero(real) & Ya = zero_zero(real))) # label(fact_638_real__two__squares__add__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4050 (all X all Y hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) # label(fact_2178_real__sqrt__sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4051 (all X_a (archim1727834104eiling(X_a) -> hAPP(X_a,int,archimedean_ceiling(X_a),zero_zero(X_a)) = zero_zero(int))) # label(fact_3304_ceiling__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4052 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))) -> hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)))) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)))) # label(fact_3137_div__neg__neg) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4053 (all X_a (real_normed_algebra(X_a) -> (all C all F (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) -> hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),F)),C)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),F)),C))))) # label(fact_4015_suminf__mult2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4054 (all X_a (real_vector(X_a) -> (all A_3 all Xa all Ya (hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_3),Ya) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_3),Xa) <-> zero_zero(real) = A_3 | ti(X_a,Ya) = ti(X_a,Xa))))) # label(fact_4728_scaleR__cancel__left) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4055 (all M all N all I_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),I_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),I_1),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),I_1),N))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))))) # label(fact_144_nat__power__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4056 (all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,nat_1,P_5)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) # label(fact_1062_Int2_Oaux__1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4057 (all P_1 all P_2 all D_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_1)) -> ((all X_1 all K_1 (hBOOL(hAPP(int,bool,P_2,X_1)) <-> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_1),D_1)))))) -> ((exists Z_2 all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_2),X_1)) -> (hBOOL(hAPP(int,bool,P_2,X_1)) <-> hBOOL(hAPP(int,bool,P_1,X_1))))) -> ((exists X1 hBOOL(hAPP(int,bool,P_2,X1))) -> (exists X1 hBOOL(hAPP(int,bool,P_1,X1)))))))) # label(fact_2870_plusinfinity) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4058 (all A_1 all B all C_1 all Q_3 all R_3 (hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,B),C_1),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Q_3),R_3))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C_1)) -> hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)),C_1),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),Q_3)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),R_3)),C_1))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),R_3)),C_1))))))) # label(fact_3121_divmod__nat__rel__mult1__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4059 (all X_b all X_a (semiring_1(X_a) & comm_monoid_mult(X_a) -> (all F all A_2 hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),hAPP(fun(X_b,nat),fun(X_b,X_a),hAPP(fun(nat,X_a),fun(fun(X_b,nat),fun(X_b,X_a)),combb(nat,X_a,X_b),semiring_1_of_nat(X_a)),F)),A_2) = hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(fun(X_b,bool),nat,hAPP(fun(X_b,nat),fun(fun(X_b,bool),nat),big_co302546470etprod(X_b,nat),F),A_2))))) # label(fact_3955_of__nat__setprod) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4060 (all A_1 ((-hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_1),zero_zero(real))) -> hAPP(real,real,abs_abs(real),A_1) = A_1) & (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_1),zero_zero(real))) -> hAPP(real,real,uminus_uminus(real),A_1) = hAPP(real,real,abs_abs(real),A_1)))) # label(fact_2024_abs__real__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4061 (all X_a (ordered_ab_group_add(X_a) -> (all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),B_2)),A_3)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)),B_2)))))) # label(fact_1931_minus__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4062 (all X_b all X_a all F all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),F),B_1))))) # label(fact_3664_image__mono) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4063 (all X_a (dense_linorder(X_a) -> (all A_3 all B_2 all C all D_1 ((hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),D_1)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),A_3))) <-> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_gr676655987ssThan(X_a),A_3),B_2)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),C),D_1))))))) # label(fact_4555_greaterThanLessThan__subseteq__atLeastAtMost__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4064 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,exp(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X)))))) # label(fact_1745_aux4) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4065 (all X_a all P_1 (hBOOL(hAPP(X_a,bool,P_1,hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1))) <-> (exists X1 hBOOL(hAPP(X_a,bool,P_1,X1))))) # label(fact_4812_some__eq__ex) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4066 (all X_a (linord626643107strict(X_a) -> (all V 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)) -> (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),U_1),V) = one_one(X_a) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),U_1),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),V),Y))),A_1)))))))))) # label(fact_542_convex__bound__lt) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4067 (all X_a (real_algebra_1(X_a) -> (all R_3 all X hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(real,X_a,of_real(X_a),R_3)),X)))) # label(fact_4715_scaleR__conv__of__real) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4068 (all R_3 all P_5 all Q_3 (hAPP(product_prod(int,int),product_prod(int,int),normalize,R_3) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_5),Q_3) -> one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),P_5),Q_3))) # label(fact_4431_normalize__coprime) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4069 (all A_3 all B_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_2)) -> (hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_3))))) # label(fact_3804_zero__less__Fract__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4070 (all Wa (hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,Wa)))) <-> hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),Wa))))) # label(fact_1607_neg__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4071 (all Xa all Ya (hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Xa),Ya))) <-> hBOOL(hAPP(nat,bool,even_odd_even(nat),Xa)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Ya)))) # label(fact_2506_even__power__nat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4072 (all X_a all X_b (comm_monoid_add(X_b) & ordere779506340up_add(X_b) -> (all F all A_2 ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_1)),zero_zero(X_b))))) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2)),zero_zero(X_b))))))) # label(fact_4335_setsum__nonpos) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4073 (all Ma all Va (hAPP(int,int,number_number_of(int),Va) = hAPP(nat,int,semiring_1_of_nat(int),Ma) <-> Ma = hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),Va)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),Va))))) # label(fact_1045_int__eq__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4074 (all X_a (division_ring(X_a) -> (all B all A_1 (zero_zero(X_a) != ti(X_a,A_1) -> (zero_zero(X_a) != ti(X_a,B) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),B)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)) = hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))))))) # label(fact_2595_nonzero__inverse__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4075 (all X_b all X_a (topological_space(X_a) & real_normed_field(X_b) -> (all Ya all G all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),inverse_divide(X_b)),G)),Ya)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4611_divide_OLIM__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4076 (all Y all X (hBOOL(hAPP(int,bool,even_odd_even(int),X)) -> (hBOOL(hAPP(int,bool,even_odd_even(int),Y)) -> hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),Y)))))) # label(fact_1597_Parity_Oeven__plus__even) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4077 (all A_3 hBOOL(hAPP(filter(complex),bool,hAPP(real,fun(filter(complex),bool),hAPP(fun(complex,real),fun(real,fun(filter(complex),bool)),tendsto(complex,real),im),hAPP(complex,real,im,A_3)),hAPP(complex,filter(complex),at(complex),A_3)))) # label(fact_4572_Im_Ocont) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4078 (all X_a (real_normed_field(X_a) -> (all Na all F all Xa all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),Xa),D)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(nat,fun(X_a,X_a),hAPP(fun(X_a,fun(nat,X_a)),fun(nat,fun(X_a,X_a)),combc(X_a,nat,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(nat,X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(nat,X_a))),combb(X_a,fun(nat,X_a),X_a),power_power(X_a)),F)),hAPP(nat,nat,suc,Na))),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),Na))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),D),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,F,Xa)),Na))))))))) # label(fact_3916_DERIV__power__Suc) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4079 (all X_a ti(fun(int,X_a),number_of_frac(X_a)) = number_of_frac(X_a)) # label(tsy_c_Nitpick_Onumber__of__frac_res) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4080 (all X_a (comm_semiring_1(X_a) -> (all X all P_5 all Q_3 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),P_5)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),Q_3)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),P_5),Q_3))))) # label(fact_466_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4081 (all X_a (archim1727834104eiling(X_a) -> (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(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(X_a,int,archim2085082626_floor(X_a),Xa))))))) # label(fact_2906_zero__le__floor) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4082 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ly)),Rx) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ly),Rx))))) # label(fact_362_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4083 (all R_3 all X hAPP(complex,real,re,hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),R_3),X)) = hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),R_3),hAPP(complex,real,re,X))) # label(fact_4706_Re_OscaleR) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4084 (all F all B_2 all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B_2),Xa)) -> ((all X_1 (hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_1),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_gr676655987ssThan(real),B_2),Xa))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,F,X_1))))) -> (hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),F),Xa)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,F,Xa))))))) # label(fact_4945_LIM__less__bound) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4085 (all X_a all X_b all X_c all X_d all Y -(all A_4 all B_4 all C_4 all D_2 hAPP(product_prod(X_b,product_prod(X_c,X_d)),product_prod(X_a,product_prod(X_b,product_prod(X_c,X_d))),hAPP(X_a,fun(product_prod(X_b,product_prod(X_c,X_d)),product_prod(X_a,product_prod(X_b,product_prod(X_c,X_d)))),product_Pair(X_a,product_prod(X_b,product_prod(X_c,X_d))),A_4),hAPP(product_prod(X_c,X_d),product_prod(X_b,product_prod(X_c,X_d)),hAPP(X_b,fun(product_prod(X_c,X_d),product_prod(X_b,product_prod(X_c,X_d))),product_Pair(X_b,product_prod(X_c,X_d)),B_4),hAPP(X_d,product_prod(X_c,X_d),hAPP(X_c,fun(X_d,product_prod(X_c,X_d)),product_Pair(X_c,X_d),C_4),D_2))) != ti(product_prod(X_a,product_prod(X_b,product_prod(X_c,X_d))),Y))) # label(fact_3176_prod__cases4) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4086 (all I_1 all J_2 hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,J_2),I_1) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I_1),J_2)) # label(fact_3465_zgcd__commute) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4087 (all X_b all X_a (real_normed_field(X_a) & real_normed_vector(X_b) -> (all F all K_3 all K (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K)) -> ((all H_1 (zero_zero(X_a) != ti(X_a,H_1) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),H_1)),K)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_a,X_b,F,H_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),K_3),hAPP(X_a,real,norm_norm(X_a),H_1))))))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a))))))))) # label(fact_4651_lemma__termdiff4) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4088 (all W hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),W)),hAPP(int,nat,number_number_of(nat),W)) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,W))) # label(fact_3862_nat__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4089 (all R_3 exists X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(product_prod(int,int),int,product_snd(int,int),X_1))) & hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(product_prod(int,int),int,product_fst(int,int),X_1)),hAPP(product_prod(int,int),int,product_snd(int,int),X_1)) = one_one(int) & (all Y_1 (hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(product_prod(int,int),int,product_fst(int,int),Y_1)),hAPP(product_prod(int,int),int,product_snd(int,int),Y_1)) = R_3 & one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(product_prod(int,int),int,product_fst(int,int),Y_1)),hAPP(product_prod(int,int),int,product_snd(int,int),Y_1)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(product_prod(int,int),int,product_snd(int,int),Y_1))) -> X_1 = Y_1)) & hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(product_prod(int,int),int,product_fst(int,int),X_1)),hAPP(product_prod(int,int),int,product_snd(int,int),X_1)) = R_3)) # label(fact_4437_quotient__of__unique) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4090 (all X_a (real_normed_field(X_a) -> (all Y_2 all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2)) -> (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),Y_2)) -> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),plus_plus(X_a)),X_2)),Y_2)))))))) # label(fact_4039_summable__add) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4091 (all Y all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Y),X)) -> hAPP(int,int,hAPP(int,fun(int,int),nat_tsub,X),Y) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),Y))) # label(fact_1480_tsub__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4092 (all K_4 all L_4 hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,bit1,K_4)),hAPP(int,int,bit1,L_4)) = hAPP(int,int,bit0,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K_4),L_4))) # label(fact_765_diff__bin__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4093 (all Ya all Xa (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Xa),Ya)),zEven)))) # label(fact_1696_even__times__either) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4094 (all X_b (number(X_b) & monoid_mult(X_b) -> (all V all W ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W))) -> one_one(X_b) = hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,W)))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W))) -> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(int,nat,number_number_of(nat),W)))),hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(int,nat,number_number_of(nat),W))) = hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,W)))))))) # label(fact_3934_power__number__of__odd__number__of) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4095 (all N one_one(real) = hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(nat,real,real_1(nat),N))),pi))) # label(fact_2445_cos__2npi) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4096 (all K (hBOOL(hAPP(rat,bool,iszero(rat),hAPP(int,rat,number_number_of(rat),K))) <-> hBOOL(hAPP(int,bool,iszero(int),hAPP(int,int,number_number_of(int),K))))) # label(fact_3706_iszero__rat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4097 (all X_a (real_normed_vector(X_a) -> (all A_1 all A_6 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_6),B)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A_1),A_6)),B)))) # label(fact_4697_scaleR_Odiff__left) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4098 (all X_a all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),B_1)),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2))) -> ti(fun(X_a,bool),B_1) = ti(fun(X_a,bool),A_2))))) # label(fact_2747_card__seteq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4099 (all X_a (linord581940658strict(X_a) -> (all A_3 all B_2 all C (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_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B_2))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),A_3))))))) # label(fact_484_mult__le__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4100 (all X_a trivial_limit(X_a) = ti(fun(filter(X_a),bool),trivial_limit(X_a))) # label(tsy_c_Limits_Otrivial__limit_res) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4101 (all A_1 all B hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)),B) = zero_zero(int)) # label(fact_1365_zmod__zdiv__trivial) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4102 (all X_a (real_n1866405975lgebra(X_a) -> (all X_2 all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially)) -> (zero_zero(X_a) != ti(X_a,A_3) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),inverse_inverse(X_a)),X_2)),hAPP(X_a,X_a,inverse_inverse(X_a),A_3)),sequentially))))))) # label(fact_4471_LIMSEQ__inverse) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4103 (all Y all X all A_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1)) -> (A_1 != one_one(real) -> (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)) -> hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),log,A_1),X)),hAPP(real,real,hAPP(real,fun(real,real),log,A_1),Y)) = hAPP(real,real,hAPP(real,fun(real,real),log,A_1),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y))))))) # label(fact_2042_log__mult) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4104 (all N all M all R_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M),R_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N),R_3)) -> (hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),N) = one_one(int) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),N)),R_3)))))) # label(fact_4376_divides__mult__int) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4105 (all Ya all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Ya)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,ln,Xa)),hAPP(real,real,ln,Ya))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya)))))) # label(fact_1705_ln__less__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4106 (all X_a all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) & ti(fun(X_a,bool),A_2) != ti(fun(X_a,bool),B_1) <-> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1)))) # label(fact_2197_psubset__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4107 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),Xa)),one_one(real))) -> hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),inverse_divide(real),one_one(real))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat)))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),Xa)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat))))))))) # label(fact_4006_summable__arctan__series) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4108 (all X_a all X_b (comm_monoid_add(X_b) -> (all F all A_2 ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2)) -> hAPP(X_a,X_b,F,X_1) = zero_zero(X_b))) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2) = zero_zero(X_b))))) # label(fact_4800_setsum__0_H) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4109 (all R_3 all A_1 hAPP(complex,real,re,hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R_3),A_1)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),hAPP(real,real,cos,A_1))) # label(fact_3194_Re__rcis) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4110 (all X_b all X_a all F all A_2 (-hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),F),A_2))) -> (exists X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2)) & -hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),A_2))),hAPP(X_b,fun(X_a,bool),hAPP(fun(X_a,fun(X_b,bool)),fun(X_b,fun(X_a,bool)),combc(X_a,X_b,bool),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,bool)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,bool))),combb(X_b,fun(X_b,bool),X_a),fequal(X_b)),F)),hAPP(X_a,X_b,F,X_1))))))))))) # label(fact_4358_pigeonhole__infinite) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4111 (all Z all X_2 (hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),Z),X_2)) -> ((all X_1 (hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_1),X_2)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),Z)))) -> Z = hAPP(fun(real,bool),real,complete_Sup_Sup(real),X_2)))) # label(fact_3731_Sup__eq__maximum) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4112 (all M all N hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N)),N))) # label(fact_3239_gcd__dvd2__nat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4113 (all X all A_1 all B (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_1),B)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),B)))))) # label(fact_3375_powr__mono) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4114 (all X_a (real_normed_vector(X_a) -> (all Xa (ti(X_a,Xa) = zero_zero(X_a) <-> hAPP(X_a,X_a,sgn_sgn(X_a),Xa) = zero_zero(X_a))))) # label(fact_2263_sgn__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4115 (all X_a (archim1727834104eiling(X_a) -> (all Xa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),zero_zero(X_a))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archim2085082626_floor(X_a),Xa)),zero_zero(int))))))) # label(fact_2905_floor__less__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4116 (all K_4 hAPP(int,int,pred,hAPP(int,int,bit0,K_4)) = hAPP(int,int,bit1,hAPP(int,int,pred,K_4))) # label(fact_2925_pred__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4117 (all X_a (linord581940658strict(X_a) -> (all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),zero_zero(X_a))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_2)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_2)),zero_zero(X_a))))))) # label(fact_425_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4118 (all X_a (comm_semiring_1(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),one_one(X_a)),A_1)))) # label(fact_388_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4119 (all X_b all X_a (ord(X_a) & comm_monoid_add(X_b) -> (all F all G all B_2 all D_1 all A_3 all C (ti(X_a,A_3) = ti(X_a,C) -> (ti(X_a,D_1) = ti(X_a,B_2) -> ((all X_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),X_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),D_1)) -> hAPP(X_a,X_b,G,X_1) = hAPP(X_a,X_b,F,X_1)))) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),A_3),B_2)) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),C),D_1)))))))) # label(fact_4329_setsum__ivl__cong) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4120 (all X_a all X_b (metric_space(X_a) & zero(X_a) & real_n2089651433ebra_1(X_b) -> (all A_3 all K (zero_zero(X_a) != ti(X_a,K) -> -hBOOL(hAPP(filter(X_b),bool,hAPP(X_a,fun(filter(X_b),bool),hAPP(fun(X_b,X_a),fun(X_a,fun(filter(X_b),bool)),tendsto(X_b,X_a),hAPP(X_a,fun(X_b,X_a),combk(X_a,X_b),K)),zero_zero(X_a)),hAPP(X_b,filter(X_b),at(X_b),A_3))))))) # label(fact_4592_LIM__not__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4121 (all Q_5 all R_4 all A_1 all B all Q_3 all R_3 (hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_3),R_3))) -> (hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_5),R_4))) -> (ti(int,B) != zero_zero(int) -> ti(int,R_3) = ti(int,R_4))))) # label(fact_1612_unique__remainder) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4122 (all X_a (comm_monoid_mult(X_a) & comm_monoid_add(X_a) -> (all F all G all A_2 all B_1 (A_2 = B_1 -> (hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,B_1)) -> ((all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) -> hAPP(int,X_a,G,X_1) = hAPP(int,X_a,F,X_1))) -> hAPP(fun(int,bool),X_a,hAPP(fun(int,X_a),fun(fun(int,bool),X_a),big_co302546470etprod(int,X_a),F),A_2) = hAPP(fun(int,bool),X_a,hAPP(fun(int,X_a),fun(fun(int,bool),X_a),big_co302546470etprod(int,X_a),G),B_1))))))) # label(fact_3989_transfer__nat__int__sum__prod__cong_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4123 (all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) -> zero_zero(nat) = B | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A_1),B)))) # label(fact_839_divides__ge) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4124 (all X_a (field_char_0(X_a) -> (all B_2 all A_3 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),field_char_0_Rats(X_a))) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),field_char_0_Rats(X_a))) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_2)),field_char_0_Rats(X_a)))))))) # label(fact_3708_Rats__mult) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4125 (all Ma all K all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ma),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Na),K))))) # label(fact_628_mult__less__cancel2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4126 (all Ya all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Ya),Xa)) -> Xa = hAPP(fun(real,bool),real,complete_Sup_Sup(real),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastLessThan(real),Ya),Xa)))) # label(fact_4280_Sup__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4127 (all K_4 all L_4 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,K_4)),hAPP(int,int,bit0,L_4)) = hAPP(int,int,bit1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),L_4))) # label(fact_93_add__Bit1__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4128 (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),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,N))),one_one(X_a)))))))) # label(fact_1457_power__Suc__less__one) # label(axiom) # label(non_clause). [assumption]. 7.88/7.84 4129 (all N all V ((-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V))) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,succ,V))),N) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),N))) & (hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V))) -> hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),N)))) # label(fact_1635_Suc__nat__number__of__add) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4130 (all M M = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),hAPP(nat,nat,suc,zero_zero(nat)))) # label(fact_1403_div__1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4131 (all X_a (banach(X_a) -> (all F all N_3 all C (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),one_one(real))) -> ((all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_3),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,F,hAPP(nat,nat,suc,N_1)))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,F,N_1))))))) -> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))))))) # label(fact_4081_ratio__test) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4132 (all M all N hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),N) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),M),N))) # label(fact_4386_gcd__add2__int) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4133 (all X hAPP(real,real,hAPP(real,fun(real,real),log,hAPP(real,real,exp(real),one_one(real))),X) = hAPP(real,real,ln,X)) # label(fact_2034_log__ln) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4134 (all X_a (ord(X_a) -> (all I all K (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I),hAPP(X_a,fun(X_a,bool),ord_atMost(X_a),K))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),I),K)))))) # label(fact_4961_atMost__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4135 (all K_4 all I_1 all J_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_1),J_2)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),I_1)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),J_2))))) # label(fact_224_zadd__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4136 (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),hAPP(X_a,X_a,abs_abs(X_a),Ma)),K)))))) # label(fact_980_abs__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4137 (all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,sqrt,Ya))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Ya)))) # label(fact_2141_real__sqrt__gt__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4138 (all X_a (real_normed_field(X_a) -> (all C all F (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) -> hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),F)),C)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),F)),C))))) # label(fact_4012_suminf__divide) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4139 (all X_a (real_field(X_a) -> (all X all Y (zero_zero(real) != Y -> hAPP(real,X_a,of_real(X_a),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(real,X_a,of_real(X_a),X)),hAPP(real,X_a,of_real(X_a),Y)))))) # label(fact_2814_nonzero__of__real__divide) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4140 (all X_a (sgn_div_norm(X_a) -> (all X hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,inverse_inverse(real),hAPP(X_a,real,norm_norm(X_a),X))),X) = hAPP(X_a,X_a,sgn_sgn(X_a),X)))) # label(fact_4750_sgn__div__norm) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4141 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) -> M = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N))) # label(fact_1526_mod__less) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4142 (all X all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)) = hAPP(real,real,hAPP(nat,fun(real,real),root,N),hAPP(real,real,inverse_inverse(real),X)))) # label(fact_2694_real__root__inverse) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4143 (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(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)) <-> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ya),zEven))) -> hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),hAPP(int,nat,nat_1,Xa)) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),hAPP(int,nat,nat_1,Ya)))))) # label(fact_1739_neg__one__power__parity) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4144 (all K_4 all L_4 hAPP(int,int,bit1,L_4) != hAPP(int,int,bit0,K_4)) # label(fact_69_rel__simps_I49_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4145 (all A_3 all 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(int),A_3),B_2)) <-> hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_2)),one_one(rat)))))) # label(fact_3810_Fract__less__one__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4146 (all X_a (linordered_idom(X_a) -> (all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,ring_1_of_int(X_a),Z))))))) # label(fact_3625_of__int__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4147 (all X_a (ring_1(X_a) -> hAPP(int,X_a,ring_1_of_int(X_a),one_one(int)) = one_one(X_a))) # label(fact_3603_of__int__1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4148 (all X_a (ab_group_add(X_a) -> (all F all K all Na hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Na),K)))),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))))) # label(fact_4203_sumr__offset) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4149 (all X_a (topological_space(X_a) -> (all G all F all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> ((all X_1 (ti(X_a,X_1) != ti(X_a,A_3) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(X_a,real,G,X_1))))) -> ((all X_1 (ti(X_a,A_3) != ti(X_a,X_1) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,G,X_1)),hAPP(X_a,real,F,X_1))))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),G),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))))) # label(fact_4801_real__LIM__sandwich__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4150 (all K_4 hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),one_one(rat)),hAPP(int,rat,number_number_of(rat),K_4)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,one_one(int)),hAPP(int,int,number_number_of(int),K_4))) # label(fact_3755_Fract__1__number__of) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4151 (all A_1 one_one(real) = hAPP(real,real,hAPP(real,fun(real,real),powr,one_one(real)),A_1)) # label(fact_3350_powr__one__eq__one) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4152 (all X (hBOOL(hAPP(int,bool,even_odd_even(int),X)) -> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) # label(fact_1642_even__plus__one__div__two) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4153 (all I_1 hAPP(quickcheck_code_int,nat,quickcheck_nat_of,I_1) = hAPP(int,nat,nat_1,hAPP(quickcheck_code_int,int,quickcheck_int_of,I_1))) # label(fact_3201_nat__of__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4154 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,pred,K)),L)))) # label(fact_2927_le__iff__pred__less) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4155 (all A_3 all B_2 all Ma (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,B_2),Ma) -> (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),A_3)) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,A_3),Ma))))))) # label(fact_3498_Bnor__mem__if) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4156 (all M all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),M) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)) # label(fact_125_nat__add__commute) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4157 -(all S1 -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),S1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,int,number_number_of(int),min)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))))) # label(fact_705__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]. 7.88/7.85 4158 (all Xa all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,exp(real),Xa)),hAPP(real,real,exp(real),Ya))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),Ya)))) # label(fact_1756_exp__less__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4159 (all X_a (power(X_a) -> power_power(X_a) = hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),hAPP(X_a,fun(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a))),power_1(X_a),one_one(X_a)),times_times(X_a)))) # label(fact_2863_power__power__power) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4160 (all X_a (real_normed_field(X_a) & banach(X_a) -> (all X hAPP(real,X_a,of_real(X_a),hAPP(real,real,exp(real),X)) = hAPP(X_a,X_a,exp(X_a),hAPP(real,X_a,of_real(X_a),X))))) # label(fact_2810_exp__of__real) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4161 (all X_a zero_frac(X_a) = ti(X_a,zero_frac(X_a))) # label(tsy_c_Nitpick_Ozero__frac_res) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4162 (all N all M hAPP(int,quickcheck_code_int,quickcheck_of_int,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(quickcheck_code_int,int,quickcheck_int_of,N)),hAPP(quickcheck_code_int,int,quickcheck_int_of,M))) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),N),M)) # label(fact_3566_plus__code__int__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4163 (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),M)))))))) # label(fact_749_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4164 (all F all Xa all L (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),Xa),L)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),L),zero_zero(real))) -> (exists D_2 ((all H_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H_1)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),H_1),D_2)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,Xa)),hAPP(real,real,F,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Xa),H_1))))))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_2))))))) # label(fact_2663_DERIV__neg__dec__left) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4165 (all Ma all Na (Ma != 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),Na),Ma)))) # label(fact_124_nat__neq__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4166 (all Z_1 hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z_1),one_one(int)) = ti(int,Z_1)) # label(fact_406_zmult__1__right) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4167 (all M all N hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),M)),hAPP(nat,int,semiring_1_of_nat(int),N))) # label(fact_639_int__mult) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4168 (all X_a (divisi14063676e_zero(X_a) -> (all A_1 all N hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),N)))) # label(fact_2563_power__inverse) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4169 (all X_b all X_c all X_a product_prod_rec(X_b,X_c,X_a) = ti(fun(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a)),product_prod_rec(X_b,X_c,X_a))) # label(tsy_c_Product__Type_Oprod_Oprod__rec_res) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4170 (all V ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),V),pls)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,number_number_of(nat),V)),one_one(nat)) = zero_zero(nat)) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),V),pls)) -> hAPP(int,nat,number_number_of(nat),hAPP(int,int,pred,V)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,number_number_of(nat),V)),one_one(nat))))) # label(fact_2923_nat__number__of__diff__1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4171 (all X_a (real_normed_vector(X_a) -> (all Ra all X_2 all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),X_2),A_3)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra)),X_2)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra),A_3))))))) # label(fact_4746_scaleR__right_Osums) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4172 (all Na all A_3 all B_2 all P_1 ((all A_4 all B_4 all C_4 (hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_4),B_4))) -> (hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),B_4),C_4))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_4),B_4)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B_4),C_4)) -> hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_4),C_4)))))))) -> (-hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_3),B_2))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2)) -> -hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),hAPP(product_prod(real,real),real,product_fst(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),Na))),hAPP(product_prod(real,real),real,product_snd(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),Na))))))))) # label(fact_3169_not__P__Bolzano__bisect) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4173 (all R_3 all A_1 all N hAPP(real,complex,hAPP(real,fun(real,complex),rcis,hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),R_3),N)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),A_1)) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R_3),A_1)),N)) # label(fact_3183_DeMoivre2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4174 (all X_a all G all A_2 hAPP(fun(X_a,bool),complex,hAPP(fun(X_a,complex),fun(fun(X_a,bool),complex),big_co1399186613setsum(X_a,complex),hAPP(fun(X_a,complex),fun(X_a,complex),hAPP(fun(complex,complex),fun(fun(X_a,complex),fun(X_a,complex)),combb(complex,complex,X_a),cnj),G)),A_2) = hAPP(complex,complex,cnj,hAPP(fun(X_a,bool),complex,hAPP(fun(X_a,complex),fun(fun(X_a,bool),complex),big_co1399186613setsum(X_a,complex),G),A_2))) # label(fact_4275_cnj_Osetsum) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4175 (all X_a (semiring_div(X_a) -> (all N all K_4 all M (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_4),M)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_4),N)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_4),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),M),N)))))))) # label(fact_1303_dvd__mod) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4176 (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(int,int,fact_fact(int),X))))) # label(fact_3217_transfer__nat__int__factorial__closure) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4177 (all K_4 all L_4 hAPP(int,int,bit0,L_4) != hAPP(int,int,bit1,K_4)) # label(fact_68_rel__simps_I50_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4178 (all C_1 all A_1 all B all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),C_1)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),C_1)),M)))) # label(fact_900_zcong__shift) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4179 (all X_b all X_a (ord(X_a) & real_normed_vector(X_b) -> (all X_2 all M_1 ((all N_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),M_1),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_a,X_b,X_2,M_1)),hAPP(X_a,X_b,X_2,N_1)))),one_one(real))))) -> (all N_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),M_1),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_a,X_b,X_2,N_1))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(X_b,real,norm_norm(X_b),hAPP(X_a,X_b,X_2,M_1))))))))))) # label(fact_2115_lemmaCauchy) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4180 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),N)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),M)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),M)))))) # label(fact_1439_n__less__n__mult__m) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4181 (all X_a (cancel_semigroup_add(X_a) -> (all A_3 all B_2 all C (ti(X_a,C) = ti(X_a,B_2) <-> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),C))))) # label(fact_295_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4182 (all X_b all X_a (real_normed_vector(X_a) & topological_space(X_b) -> (all F all A_3 all L_2 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_a),fun(X_a,X_b),hAPP(fun(X_a,X_b),fun(fun(X_a,X_a),fun(X_a,X_b)),combb(X_a,X_b,X_a),F),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3))),L_2),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a)))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4606_LIM__offset__zero__cancel) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4183 (all X_a (linordered_idom(X_a) -> (all Z (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(int,X_a,ring_1_of_int(X_a),Z))))))) # label(fact_3627_of__int__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4184 (all Na (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Na)) -> hAPP(int,nat,nat_1,Na) = hAPP(fun(int,bool),nat,finite_card(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),Na)))))) # label(fact_3854_card__bdd__int__set__l) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4185 (all N hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N))) # label(fact_350_le0) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4186 (all X_a (real_normed_algebra(X_a) -> (all X all Y hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),Y)))) # label(fact_1912_mult__left_Ominus) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4187 (all F all A_3 all B_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),B_2)) -> ((all X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),B_2)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),X_1)) -> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),F),X_1)))) -> ((all X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),B_2)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),X_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),zero_zero(real))))) -> (all X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),X_1)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),B_2)) -> hAPP(real,real,F,X_1) = hAPP(real,real,F,A_3))))))) # label(fact_4951_DERIV__isconst1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4188 (all X hAPP(complex,complex,sgn_sgn(complex),X) = hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),hAPP(real,real,inverse_inverse(real),hAPP(complex,real,norm_norm(complex),X))),X)) # label(fact_4767_complex__sgn__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4189 (all W (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),W)),zero_zero(int))) -> hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,hAPP(int,int,uminus_uminus(int),one_one(int))),hAPP(int,int,uminus_uminus(int),hAPP(int,int,number_number_of(int),W))))) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),one_one(int)),hAPP(int,int,number_number_of(int),W)))) # label(fact_3131_div__pos__neg__1__number__of) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4190 (all N all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,real_1(int),N)),hAPP(int,real,real_1(int),X))),hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),N),X)))))) # label(fact_3434_real__of__int__div2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4191 (all K_4 hAPP(int,rat,hAPP(int,fun(int,rat),fract,K_4),one_one(int)) = hAPP(int,rat,ring_1_of_int(rat),K_4)) # label(fact_3770_Fract__of__int__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4192 (all X_a (linordered_idom(X_a) -> (all A_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),A_1))))) # label(fact_973_abs__mult__self) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4193 (all X hAPP(rat,real,ratreal,hAPP(rat,rat,uminus_uminus(rat),X)) = hAPP(real,real,uminus_uminus(real),hAPP(rat,real,ratreal,X))) # label(fact_3791_real__uminus__code) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4194 (all Z_1 hAPP(complex,complex,cnj,hAPP(complex,complex,cnj,Z_1)) = Z_1) # label(fact_2959_complex__cnj__cnj) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4195 (all X_a (linord219039673up_add(X_a) -> (all X all Y hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),hAPP(X_a,X_a,uminus_uminus(X_a),Y))))) # label(fact_4972_minus__max__eq__min) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4196 (all X_a (comm_semiring_1(X_a) -> (all A_1 all M hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),one_one(X_a))),M) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),M)),M)))) # label(fact_464_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4197 (all X_a (number(X_a) & linord1117847801e_zero(X_a) -> (all B_2 all Wa all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),hAPP(int,X_a,number_number_of(X_a),Wa))),A_3)) <-> (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Wa))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),hAPP(int,X_a,number_number_of(X_a),Wa))),B_2))) & (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_3)))) & (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Wa))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),hAPP(int,X_a,number_number_of(X_a),Wa))))))))) # label(fact_1795_divide__less__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4198 (all A_1 zero_zero(int) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),hAPP(int,int,number_number_of(int),min))) # label(fact_1428_zmod__minus1__right) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4199 (all C_1 all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_1),B)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C_1),A_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A_1),C_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),B),C_1)))))) # label(fact_840_diff__less__mono) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4200 (all X_b all X_a (topological_space(X_a) & real_normed_algebra(X_b) -> (all G all F all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3)) -> (hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),G),A_3)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),F)),G)),A_3))))))) # label(fact_4911_mult_OisCont) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4201 (all X_a (monoid_mult(X_a) -> (all A_1 hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),A_1)),A_1)))) # label(fact_540_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4202 (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M)),hAPP(nat,X_a,hAPP(X_a,fun(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_238_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4203 (all X_a (linord1117847801e_zero(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),A_3))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_3)))))) # label(fact_2594_inverse__positive__iff__positive) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4204 (all Ta all A_2 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Ta),one_one(int))),A_2)) -> (all X_1 ((all Xa_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D))) -> (all Xb (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2)) -> hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa_1) != ti(int,X_1))))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X_1),Ta)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),D)),Ta)))))))) # label(fact_4656_aset_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4205 (all A_1 hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),A_1) = zero_zero(int)) # label(fact_1306_zmod__self) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4206 (all A_1 all B (zero_zero(nat) != B -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),B)))) # label(fact_3285_gcd__le2__nat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4207 (all D_1 all F all Xa all L (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),Xa),L)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_1)) -> ((all Y_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Xa),Y_1))),D_1)) -> hAPP(real,real,F,Xa) = hAPP(real,real,F,Y_1))) -> zero_zero(real) = L)))) # label(fact_2661_DERIV__local__const) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4208 (all A_1 all B all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M),B)) -> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)),M) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),M)))) # label(fact_1427_zmod__zdvd__zmod) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4209 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,exp(real),Xa)),one_one(real))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),zero_zero(real))))) # label(fact_1787_exp__less__one__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4210 (all X_a (archim1727834104eiling(X_a) -> (all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archim2085082626_floor(X_a),Xa)),one_one(int))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),one_one(X_a))))))) # label(fact_2909_floor__less__one) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4211 (all X_a (archim1804426504_field(X_a) -> (all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X)) -> (exists N_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,suc,N_1)))),X))))))) # label(fact_2662_ex__inverse__of__nat__Suc__less) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4212 (all K_4 all L_4 hAPP(int,int,bit1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,pred,K_4)),L_4)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,bit0,K_4)),hAPP(int,int,bit1,L_4))) # label(fact_2934_diff__bin__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4213 (all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Ma),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) <-> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Ma),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = one_one(nat))) # label(fact_1563_mod2__gr__0) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4214 (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)),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y)))))) # label(fact_4426_transfer__nat__int__gcd__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4215 (all X_a (order(X_a) -> (all Xa all Ya all F (hBOOL(hAPP(fun(nat,X_a),bool,hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,X_a),bool),mono(nat,X_a),dvd_dvd(nat)),F)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,F,Xa)),hAPP(nat,X_a,F,Ya)))))))) # label(fact_4818_dvd_OmonoD) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4216 (all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),Z_1))))) # label(fact_307_le__imp__0__less) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4217 (all X all N hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),pi))) = hAPP(real,real,tan,X)) # label(fact_2377_tan__periodic__nat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4218 (all K_4 all I_1 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),K_4))))) # label(fact_415_add__le__mono1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4219 (all X_a (real_vector(X_a) -> (all A_3 all Xa all B_2 (hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_3),Xa) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),B_2),Xa) <-> B_2 = A_3 | zero_zero(X_a) = ti(X_a,Xa))))) # label(fact_4732_scaleR__cancel__right) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4220 (all M all K_4 all N (hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K_4),N) = one_one(int) -> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M)),N))) # label(fact_3456_zgcd__zmult__cancel) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4221 (all Z_1 hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Z_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Z_1),Z_1)) # label(fact_646_nat__mult__2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4222 (all X_b all X_a (real_normed_vector(X_b) & topological_space(X_a) -> (all F all L all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),minus_minus(X_b)),F)),L)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4609_LIM__zero__cancel) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4223 (all Na (one_one(nat) = Na <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,Na),Na)))) # label(fact_2884_coprime__refl) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4224 (all X_a all X_b (comm_monoid_mult(X_b) -> (all G all H all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2)) -> hAPP(X_a,X_b,H,X_1) = hAPP(X_a,X_b,G,X_1))) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),A_2) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),H),A_2)))))) # label(fact_4794_setprod_Ocong) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4225 (all X_a (linordered_idom(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)) = hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))))) # label(fact_974_abs__mult) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4226 (all X hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,X),zero_zero(nat)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int))) # label(fact_3060_bezw__0) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4227 (all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) <-> hBOOL(hAPP(int,bool,nat_is_nat,Xa)))) # label(fact_4865_is__nat__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4228 (all X_a (linordered_idom(X_a) -> (all Xa all Na (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),Na))) <-> zero_zero(nat) = Na | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Xa)) & -hBOOL(hAPP(nat,bool,even_odd_even(nat),Na)) | hBOOL(hAPP(nat,bool,even_odd_even(nat),Na)) & zero_zero(X_a) != ti(X_a,Xa))))) # label(fact_2509_zero__less__power__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4229 (all X all Y all Z_1 hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),Y)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),Z_1)) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z_1))) # label(fact_472_zpower__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4230 (all Na hAPP(nat,nat,suc,Na) = hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less_eq(nat)),Na)))) # label(fact_3868_card__Collect__le__nat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4231 (all Y all A_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1)) -> (A_1 != one_one(real) -> Y = hAPP(real,real,hAPP(real,fun(real,real),log,A_1),hAPP(real,real,hAPP(real,fun(real,real),powr,A_1),Y))))) # label(fact_3432_log__powr__cancel) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4232 (all X_a (metric_space(X_a) -> (all X2_1 all X1_1 all Y all E_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X1_1),Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),E_1),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X2_1),Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),E_1),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X1_1),X2_1)),E_1))))))) # label(fact_3527_dist__triangle__half__l) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4233 (all Xa (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)) -> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Xa),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = zero_zero(int))) # label(fact_1734_even__div__2__prop1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4234 (all X_a (comm_monoid_add(X_a) & t2_space(X_a) -> zero_zero(X_a) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(X_a,fun(nat,X_a),combk(X_a,nat),zero_zero(X_a))))) # label(fact_3995_suminf__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4235 (all Xa all Na (-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),Na))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),hAPP(int,real,number_number_of(real),Na))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(real,nat,natceiling,Xa)),hAPP(int,nat,number_number_of(nat),Na))))))) # label(fact_1659_natceiling__le__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4236 (all X_a (ordered_ab_group_add(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)),zero_zero(X_a))))))) # label(fact_1949_neg__le__0__iff__le) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4237 (all X_a (real_normed_field(X_a) & banach(X_a) -> (all X hAPP(X_a,X_a,exp(X_a),X) != zero_zero(X_a)))) # label(fact_1763_exp__not__eq__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4238 (all Ta all B_1 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D)) -> (all X_1 ((all Xa_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D))) -> (all Xb (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1)) -> ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa_1))))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),Ta)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),D)),Ta))))))) # label(fact_4663_bset_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4239 (all X_a (linord20386208strict(X_a) -> (all C_1 all D_3 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D_3)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D_3)))))))))) # label(fact_480_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4240 (all X_a all X_c all X_b all F all Xa all Ya hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),Xa),hAPP(X_c,X_b,F,Ya)) = hAPP(product_prod(X_a,X_c),product_prod(X_a,X_b),hAPP(fun(X_c,X_b),fun(product_prod(X_a,X_c),product_prod(X_a,X_b)),product_apsnd(X_c,X_b,X_a),F),hAPP(X_c,product_prod(X_a,X_c),hAPP(X_a,fun(X_c,product_prod(X_a,X_c)),product_Pair(X_a,X_c),Xa),Ya))) # label(fact_4094_apsnd__conv) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4241 (all X_a (order(X_a) -> (all X_2 ((all M_2 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_2),N_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,X_2,M_2)),hAPP(nat,X_a,X_2,N_1))))) -> hBOOL(hAPP(fun(nat,X_a),bool,monoseq(X_a),X_2)))))) # label(fact_4158_monoI1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4242 (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),dvd_dvd(nat),M),N)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))))) # label(fact_1237_dvd__pos__nat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4243 (all X_a (dense_linorder(X_a) -> ti(fun(X_a,fun(X_a,fun(X_a,bool))),ord_gr676655987ssThan(X_a)) = ord_gr676655987ssThan(X_a))) # label(tsy_c_SetInterval_Oord__class_OgreaterThanLessThan_0_res) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4244 (all Z_1 -(all M_2 all N_1 hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,semiring_1_of_nat(int),M_2)),hAPP(nat,int,semiring_1_of_nat(int),N_1)) != ti(int,Z_1))) # label(fact_1685_int__diff__cases) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4245 (all M all N (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N))) # label(fact_1535_mod__geq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4246 (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N))))))) # label(fact_252_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4247 (all L all U hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),L),U) = hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),L),hAPP(nat,nat,suc,U))) # label(fact_4564_atLeastLessThanSuc__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4248 (all A_3 all C all D_1 all B_2 (ti(int,B_2) != zero_zero(int) -> (zero_zero(int) != ti(int,D_1) -> (hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_2) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,C),D_1) <-> hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_3),D_1) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),B_2))))) # label(fact_3760_eq__rat_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4249 (all A_1 all B (hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)),hAPP(int,rat,ring_1_of_int(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),one_one(int))))) & hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,ring_1_of_int(rat),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B))),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B))))) # label(fact_3814_rat__floor__lemma) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4250 (all N all M all R_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),R_3)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N),R_3)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,M),N)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)),R_3)))))) # label(fact_2894_divides__mul) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4251 (all X hAPP(real,real,sgn_sgn(real),X) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),hAPP(real,real,abs_abs(real),X))) # label(fact_2281_real__sgn__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4252 (all X_a all X_b all X_c all F all Xa hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),hAPP(product_prod(X_a,X_c),product_prod(X_a,X_b),hAPP(fun(X_c,X_b),fun(product_prod(X_a,X_c),product_prod(X_a,X_b)),product_apsnd(X_c,X_b,X_a),F),Xa)) = hAPP(product_prod(X_a,X_c),X_a,product_fst(X_a,X_c),Xa)) # label(fact_4093_fst__apsnd) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4253 (all X_a (linorder(X_a) -> (all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y) = ti(X_a,Y))))) # label(fact_4850_min__max_Oinf__absorb2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4254 (all X_a (ordere236663937imp_le(X_a) -> (all A_3 all C all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B_2),C))))))) # label(fact_215_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4255 (all N all M all R_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),R_3)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N),R_3)) -> (hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N) = one_one(nat) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)),R_3)))))) # label(fact_3313_divides__mult__nat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4256 (all X_a (ordere1490568538miring(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B)))))))) # label(fact_433_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4257 (all X_a (monoid_add(X_a) -> plus_plus(X_a) = ti(fun(X_a,fun(X_a,X_a)),plus_plus(X_a)))) # label(tsy_c_Groups_Oplus__class_Oplus_1_res) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4258 (all K_4 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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N),K_4))))) # label(fact_1156_div__le__mono) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4259 (all X_a (field(X_a) -> (all A_1 all N all B (ti(X_a,B) != zero_zero(X_a) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)),N))))) # label(fact_1764_nonzero__power__divide) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4260 (all X_b all X_a (real_normed_vector(X_b) & topological_space(X_a) -> (all F all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3))) <-> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4636_LIM__norm__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4261 (all Xa all Ya (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Xa),Ya)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),Xa)),hAPP(nat,int,semiring_1_of_nat(int),Ya))))) # label(fact_470_Nat__Transfer_Otransfer__int__nat__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4262 (all X hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),hAPP(real,nat,natfloor,X)))) # label(fact_1663_zero__le__natfloor) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4263 (all A_3 all P_4 hAPP(fun(int,bool),fun(fun(int,bool),bool),hAPP(fun(int,fun(int,bool)),fun(fun(int,bool),fun(fun(int,bool),bool)),image(int,fun(int,bool)),hAPP(int,fun(int,fun(int,bool)),hAPP(int,fun(int,fun(int,fun(int,bool))),multInvPair,A_3),P_4)),hAPP(int,fun(int,bool),sRStar,P_4)) = hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,A_3),P_4)) # label(fact_3677_SetS__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4264 (all X_a (number(X_a) & linord1117847801e_zero(X_a) -> (all A_3 all B_2 all Wa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B_2),hAPP(int,X_a,number_number_of(X_a),Wa)))) <-> (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Wa))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),hAPP(int,X_a,number_number_of(X_a),Wa))))) & (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),zero_zero(X_a))))) & (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Wa))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),hAPP(int,X_a,number_number_of(X_a),Wa))),B_2))))))) # label(fact_1804_le__divide__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4265 (all X_a (topological_space(X_a) -> (all F all L all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,real),fun(X_a,real),hAPP(fun(real,real),fun(fun(X_a,real),fun(X_a,real)),combb(real,real,X_a),abs_abs(real)),F)),hAPP(real,real,abs_abs(real),L)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4577_LIM__rabs) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4266 (all N (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N)) -> hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N),one_one(int))),hAPP(int,int,fact_fact(int),N)) = hAPP(int,int,fact_fact(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N),one_one(int))))) # label(fact_3226_fact__plus__one__int) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4267 (all X_a (archim1727834104eiling(X_a) -> (all N hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)) = hAPP(nat,int,semiring_1_of_nat(int),N)))) # label(fact_3310_ceiling__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4268 (all X all Y ((Y = zero_zero(nat) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) = hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,X),Y)) & (Y != zero_zero(nat) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)))),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)))),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),X),Y))))) = hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,X),Y)))) # label(fact_3138_bezw_Osimps) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4269 (all X_a (archim1804426504_field(X_a) -> (all X exists N_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1)))))) # label(fact_2836_ex__le__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4270 (all X_a (order(X_a) -> (all C_1 all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) -> (ti(X_a,C_1) = ti(X_a,B) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),A_1))))))) # label(fact_519_xt1_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4271 (all X_a hAPP(product_prod(int,int),X_a,abs_Frac(X_a),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))) = one_frac(X_a)) # label(fact_3819_Nitpick_Oone__frac__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4272 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1)) -> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A_1)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),A_1))) # label(fact_1130_pos__zdiv__mult__2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4273 (all Xa hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),cos)),cos)),Xa),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,min))),hAPP(real,real,cos,Xa))),hAPP(real,real,sin,Xa))))) # label(fact_3921_DERIV__cos__cos__mult2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4274 (all X all Y all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> hAPP(real,real,hAPP(nat,fun(real,real),root,N),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),Y)))) # label(fact_2686_real__root__mult) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4275 (all P_1 all Na all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Na),K)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Na),K))) <-> (all I_2 all J_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),J_1),zero_zero(int))) & ti(int,Na) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),I_2)),J_1) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),J_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,I_2),J_1))))))) # label(fact_1500_split__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4276 (all X (-hBOOL(hAPP(int,bool,even_odd_even(int),X)) -> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),one_one(int)))) # label(fact_1641_odd__plus__one__div__two) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4277 (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),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),X)),X)))))))) # label(fact_491_mult__left__le__one__le) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4278 (all A_1 zero_zero(real) = hAPP(real,real,hAPP(real,fun(real,real),log,A_1),one_one(real))) # label(fact_1995_log__one) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4279 (all Xa all A_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archim2085082626_floor(real),Xa)),A_3)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),A_3)),one_one(real)))))) # label(fact_3439_floor__le__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4280 (all Xa (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)) | hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd)))) # label(fact_1702_even__odd__disj) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4281 (all Ta all A_2 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ta),A_2)) -> (all X_1 ((all Xa_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D))) -> (all Xb (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2)) -> ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa_1))))) -> (ti(int,Ta) != ti(int,X_1) -> hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),D) != ti(int,Ta))))))) # label(fact_4666_aset_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4282 (all D_1 all F all Xa all L (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),Xa),L)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_1)) -> ((all Y_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Xa),Y_1))),D_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,F,Y_1)),hAPP(real,real,F,Xa))))) -> zero_zero(real) = L)))) # label(fact_2659_DERIV__local__max) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4283 (all A_3 all Ra (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Ra)) -> (hBOOL(hAPP(fun(nat,real),bool,summable(real),A_3)) -> (exists N_2 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),A_3),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),N_1))))),Ra))))))) # label(fact_4083_suminf__exist__split) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4284 (all Xa hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),Xa))))) # label(fact_4056_summable__exp) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4285 (all X_a (real_normed_algebra(X_a) -> (all A_1 all A_6 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_6),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),A_6)),B)))) # label(fact_1245_mult_Oadd__left) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4286 (all B_2 all A_3 all P_4 (hBOOL(hAPP(int,bool,zprime,P_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),P_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int)))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),B_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_2),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int)))) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),inv,P_4),B_2)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,A_3),P_4))) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,A_3),P_4)))))))))) # label(fact_1199_wset__inv__mem__mem) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4287 (all W all V (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,number_number_of(int),V))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),W)),hAPP(int,int,number_number_of(int),V))) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,hAPP(int,int,number_number_of(int),W)),hAPP(int,int,number_number_of(int),V)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),hAPP(int,int,number_number_of(int),W))) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),W)),hAPP(int,int,number_number_of(int),V))) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,hAPP(int,int,number_number_of(int),W)),hAPP(int,int,number_number_of(int),V)) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,hAPP(int,int,number_number_of(int),V)),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,hAPP(int,int,number_number_of(int),W)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,number_number_of(int),V))))))) # label(fact_1649_posDivAlg__eqn__number__of) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4288 (all X_a (linorder(X_a) -> (all X hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),X) = ti(X_a,X)))) # label(fact_4842_Min_Oidem) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4289 (all X_a (real_normed_vector(X_a) -> (all A_1 all B hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(X_a,real,norm_norm(X_a),A_1)),hAPP(X_a,real,norm_norm(X_a),B))),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B))))))) # label(fact_2015_norm__triangle__ineq2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4290 (all X_a all C_2 all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),C_2)) -> ti(fun(X_a,bool),A_2) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),B_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),C_2),A_2))))) # label(fact_2211_double__diff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4291 (all X_a all A_3 all F all Va all Na hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(X_a,fun(fun(nat,X_a),fun(nat,X_a)),nat_case(X_a),A_3),F),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),Va)),Na)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),hAPP(bool,fun(X_a,fun(X_a,X_a)),if(X_a),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(X_a,fun(fun(nat,X_a),fun(nat,X_a)),nat_case(X_a),A_3),F),Na)),hAPP(nat,X_a,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),Na)))) # label(fact_4135_nat__case__add__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.88/7.85 4292 (all X_a (linorder(X_a) -> (all A_3 all B_2 all C all D_1 (hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),C),D_1) = hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),A_3),B_2) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_1)) -> ti(X_a,D_1) = ti(X_a,B_2))))))) # label(fact_4222_atLeastLessThan__inj_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4293 (all X_a (real_normed_algebra(X_a) -> (all R_3 all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),X)),Y) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y))))) # label(fact_4723_mult__left_OscaleR) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4294 (all B all A_1 (zero_zero(int) != ti(int,A_1) | ti(int,B) != zero_zero(int) -> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B))) = one_one(int))) # label(fact_4379_div__gcd__coprime__int) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4295 (all X_a (semiring_div(X_a) -> (all A_1 all C_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),C_1),B)) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C_1))))) # label(fact_1304_mod__mod__cancel) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4296 (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_249_not__one__le__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4297 (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_885_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4298 (all X all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),pi)),Y)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,cos,Y)),hAPP(real,real,cos,X))))))) # label(fact_2308_cos__monotone__minus__pi__0_H) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4299 (all Xa ((exists A_4 exists B_4 ti(int,Xa) = hAPP(product_prod(int,int),int,twoSqu1094008024sum2sq,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_4),B_4))) <-> hBOOL(hAPP(int,bool,twoSqu1091635170sum2sq,Xa)))) # label(fact_1067_is__sum2sq__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4300 (all Z_3 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_3)) -> (hAPP(int,nat,nat_1,Z_3) = hAPP(int,nat,nat_1,Z) <-> ti(int,Z_3) = ti(int,Z))))) # label(fact_1000_eq__nat__nat__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4301 (all X_a (field_inverse_zero(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(X_a,X_a,uminus_uminus(X_a),B))))) # label(fact_1942_minus__divide__divide) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4302 (all X all Y 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_eq(real),zero_zero(real)),Y)) -> (hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),N) = X -> Y = hAPP(real,real,hAPP(nat,fun(real,real),root,N),X))))) # label(fact_2714_real__root__pos__unique) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4303 (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),Na),hAPP(nat,nat,suc,Ma))))) # label(fact_1334_not__less__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4304 (all V_2 hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),one_one(int)),hAPP(int,int,number_number_of(int),V_2))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(int,nat,number_number_of(nat),V_2))) # label(fact_1510_one__div__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4305 (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_eq(nat),N),M)) -> hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N)))) # label(fact_1497_le__div__geq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4306 (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)) -> hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),Y)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,nat_1,X)),hAPP(int,nat,nat_1,Y))))) # label(fact_1043_Nat__Transfer_Otransfer__nat__int__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4307 (all X all Y hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) # label(fact_2172_real__sqrt__sum__squares__ge1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4308 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),Xa)),one_one(real))) -> hBOOL(hAPP(fun(nat,real),bool,monoseq(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),inverse_divide(real),one_one(real))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat)))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),Xa)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat)))))))) # label(fact_3947_monoseq__arctan__series) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4309 (all X_b all X_a (topological_space(X_a) & real_normed_algebra(X_b) -> (all Xa all G all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),hAPP(X_b,fun(X_b,X_b),times_times(X_b),Xa)),G)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4601_mult__right_OLIM__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4310 (all X hAPP(complex,complex,cnj,hAPP(real,complex,of_real(complex),X)) = hAPP(real,complex,of_real(complex),X)) # label(fact_2954_complex__cnj__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4311 (all Va all Na hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),hAPP(int,nat,number_number_of(nat),Va)),hAPP(nat,nat,suc,Na)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),hAPP(nat,nat,suc,Na)),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),Na)))) # label(fact_4970_max__Suc__number__of) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4312 (all X all N (-hBOOL(hAPP(nat,bool,even_odd_even(nat),N)) -> X = hAPP(real,real,hAPP(nat,fun(real,real),root,N),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N)))) # label(fact_2680_odd__real__root__power__cancel) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4313 (all X_b all X_a (topological_space(X_a) & real_normed_algebra(X_b) -> (all Ya all G all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),G)),Ya)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4599_mult__left_OLIM__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4314 (all Z all P_1 ((all N_1 hBOOL(hAPP(int,bool,P_1,hAPP(nat,int,semiring_1_of_nat(int),N_1)))) -> ((all N_1 hBOOL(hAPP(int,bool,P_1,hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N_1)))))) -> hBOOL(hAPP(int,bool,P_1,Z))))) # label(fact_3554_int__of__nat__induct) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4315 (all Ya all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Ya)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Ya)))))) # label(fact_2695_real__root__ge__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4316 (all K_4 hAPP(int,rat,ring_1_of_int(rat),K_4) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,K_4),one_one(int))) # label(fact_3771_of__int__rat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4317 (all Na all Ma (hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),Na),Ma)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(quickcheck_code_int,int,quickcheck_int_of,Na)),hAPP(quickcheck_code_int,int,quickcheck_int_of,Ma))))) # label(fact_3184_less__eq__code__int__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4318 (all X_a all Q_1 all P_1 (hBOOL(hAPP(X_a,bool,P_1,hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1))) -> (hBOOL(hAPP(X_a,bool,Q_1,hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),Q_1))) -> (hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1) = hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),Q_1) -> (exists A_4 (hBOOL(hAPP(X_a,bool,P_1,A_4)) & hBOOL(hAPP(X_a,bool,Q_1,A_4)))))))) # label(fact_4891_EulerFermat_Oaux) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4319 (all X_a (real_normed_field(X_a) -> (all X all Y all Ya_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Ya_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Y),Ya_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y)),Ya_1)))) # label(fact_1752_divide_Odiff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4320 (all N all V ((hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V))) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,number_number_of(nat),V)),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),zero_zero(nat)),N)) & (-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V))) -> hAPP(int,nat,nat_1,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),V)),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,number_number_of(nat),V)),N)))) # label(fact_1631_power__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4321 (all Z_1 all W (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),W)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),Z_1)) -> ti(int,W) = ti(int,Z_1)))) # label(fact_203_zle__antisym) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4322 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),one_one(real))) -> hBOOL(hAPP(fun(nat,real),bool,monoseq(real),hAPP(real,fun(nat,real),power_power(real),Xa)))))) # label(fact_3951_monoseq__realpow) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4323 (all M hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,M),zero_zero(int)),M))) # label(fact_889_zcong__id) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4324 (all X all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M)) -> (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,X),M) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(int,nat,phi,M))),one_one(int)),M))))) # label(fact_3452_Euler__Fermat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4325 (all F all F_2 all A_3 all B_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),B_2)) -> ((all X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),B_2)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),X_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),hAPP(real,real,F_2,X_1))))) -> (exists Z_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_3),Z_2)) & hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,F,B_2)),hAPP(real,real,F,A_3)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B_2),A_3)),hAPP(real,real,F_2,Z_2)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Z_2),B_2))))))) # label(fact_2673_MVT2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4326 (all X_a (metric_space(X_a) -> (all K all L (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(X_a,fun(nat,X_a),combk(X_a,nat),K)),L),sequentially)) <-> ti(X_a,K) = ti(X_a,L))))) # label(fact_4491_LIMSEQ__const__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4327 (all X_a (field_inverse_zero(X_a) -> (all A_1 all N hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),one_one(X_a)),A_1)),N) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),one_one(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N))))) # label(fact_1765_power__one__over) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4328 (all X_a (group_add(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B))))) # label(fact_1967_Groups_Odiff__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4329 (all M all N all K_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K_4)) -> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),N)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N)))) # label(fact_3480_zgcd__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4330 (all K_4 zero_zero(rat) = hAPP(product_prod(int,int),rat,frct,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K_4))) # label(fact_3727_Frct__code__post_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4331 (all X_a (ab_group_add(X_a) -> (all A_1 all C_1 all B all D_3 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),C_1),D_3)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D_3))))) # label(fact_741_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4332 (all X_a (linordered_idom(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_3)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,sgn_sgn(X_a),A_3))))))) # label(fact_2280_sgn__greater) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4333 (all Z all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(int,nat,nat_1,Z)),Ma)) <-> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) -> Ma = zero_zero(nat)) & (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),dvd_dvd(int),Z),hAPP(nat,int,semiring_1_of_nat(int),Ma)))))) # label(fact_1056_nat__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4334 (exists K_2 all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(complex,real,re,X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_1)),K_2)))) # label(fact_3047_Re_Obounded) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4335 (all X_a (linorder(X_a) -> (all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya)) -> (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Ya)) <-> ti(X_a,Ya) = ti(X_a,Xa)))))) # label(fact_559_linorder__antisym__conv2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4336 (all R1 all A_1 all R2 all B hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R1),A_1)),hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R2),B)) = hAPP(real,complex,hAPP(real,fun(real,complex),rcis,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),R1),R2)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A_1),B))) # label(fact_3197_rcis__divide) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4337 (all A_1 all C_1 hAPP(int,rat,hAPP(int,fun(int,rat),fract,zero_zero(int)),A_1) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,zero_zero(int)),C_1)) # label(fact_3762_eq__rat_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4338 (all X_a all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) -> hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),B_1))))) # label(fact_2842_PowI) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4339 (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,hAPP(real,fun(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)) -> Y_1 = X_1)) & hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_1),N) = A_1))))) # label(fact_340_realpow__pos__nth__unique) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4340 (all P_5 all Q_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Q_3),zero_zero(int))) -> hAPP(product_prod(int,int),product_prod(int,int),normalize,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_5),Q_3)) = hAPP(product_prod(int,int),product_prod(int,int),normalize,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,uminus_uminus(int),P_5)),hAPP(int,int,uminus_uminus(int),Q_3))))) # label(fact_3745_normalize__negative) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4341 (all Na all Ma hAPP(nat,nat,hAPP(fun(nat,nat),fun(nat,nat),hAPP(nat,fun(fun(nat,nat),fun(nat,nat)),nat_case(nat),zero_zero(nat)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),suc),hAPP(nat,fun(nat,nat),ord_min(nat),Na))),Ma) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(nat,nat,suc,Na)),Ma)) # label(fact_4858_min__Suc1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4342 (all X_a (real_normed_field(X_a) -> (all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2)) -> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),X_2))))))) # label(fact_4025_summable__minus) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4343 (all I_1 all J_2 all K_4 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),J_2)),K_4)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),K_4)))) # label(fact_140_add__lessD1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4344 (all K_4 hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(nat,int,semiring_1_of_nat(int),K_4)),one_one(int)) = hAPP(nat,rat,semiring_1_of_nat(rat),K_4)) # label(fact_3775_of__nat__rat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4345 (all X (zero_zero(real) != hAPP(real,real,cos,X) -> (hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X)) != zero_zero(real) -> hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,tan,X))),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,tan,X)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) # label(fact_2365_tan__double) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4346 (all Xa all Ya all Xaa (hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Xa),Ya) = hAPP(real,complex,of_real(complex),Xaa) <-> Xa = Xaa & Ya = zero_zero(real))) # label(fact_2801_complex__eq__cancel__iff2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4347 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),sqrt),Xa),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,inverse_inverse(real),hAPP(real,real,sqrt,Xa))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) # label(fact_2654_DERIV__real__sqrt) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4348 (all Na (-hBOOL(hAPP(nat,bool,even_odd_even(nat),Na)) <-> (exists M_2 hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),M_2)) = Na))) # label(fact_2524_odd__Suc__mult__two__ex) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4349 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 all B hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B))))))) # label(fact_1013_abs__triangle__ineq4) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4350 (all X_a (real_normed_algebra(X_a) -> (all Xa_2 all X hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_2),X)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_2),hAPP(X_a,X_a,uminus_uminus(X_a),X))))) # label(fact_1910_mult__right_Ominus) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4351 (all X_a (metric_space(X_a) -> (all X hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),X) = zero_zero(real)))) # label(fact_3541_dist__self) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4352 (all P_2 all P_1 all Xa ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> (hBOOL(P_2) <-> hBOOL(P_1))) -> ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> hBOOL(P_1)) <-> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> hBOOL(P_2))))) # label(fact_574_imp__le__cong) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4353 (all X_a (linordered_field(X_a) -> (all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),B)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)))))))) # label(fact_2588_less__imp__inverse__less__neg) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4354 (all P_1 all A_3 all B_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_2)) -> (all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(product_prod(real,real),real,product_snd(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),hAPP(nat,nat,suc,N_1)))),hAPP(product_prod(real,real),real,product_snd(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),A_3),B_2),N_1))))))) # label(fact_3168_Bolzano__bisect__Suc__le__snd) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4355 (all A_1 all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int)))) -> hAPP(int,int,hAPP(int,fun(int,int),inv,P_5),A_1) != zero_zero(int))))) # label(fact_1182_inv__not__0) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4356 (all X one_one(real) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,cos,X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,X)),hAPP(real,real,sin,X)))) # label(fact_2267_sin__cos__squared__add3) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4357 (all X_b all X_a (topological_space(X_b) & real_normed_vector(X_a) -> (all K all F all L_2 all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_a),fun(X_a,X_b),hAPP(fun(X_a,X_b),fun(fun(X_a,X_a),fun(X_a,X_b)),combb(X_a,X_b,X_a),F),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),plus_plus(X_a)),K))),L_2),hAPP(X_a,filter(X_a),at(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),K)))))))) # label(fact_4610_LIM__offset) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4358 (all N all M hAPP(int,quickcheck_code_int,quickcheck_of_int,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(quickcheck_code_int,int,quickcheck_int_of,N)),hAPP(quickcheck_code_int,int,quickcheck_int_of,M))) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),minus_minus(quickcheck_code_int),N),M)) # label(fact_3567_minus__code__int__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4359 (all Ma (hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,zero_zero(int)),Ma) = one_one(int) <-> one_one(int) = hAPP(int,int,abs_abs(int),Ma))) # label(fact_3489_zgcd__0__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4360 (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)) -> (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),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_229_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4361 (all X_a all X_b all X_c all Xa all P_1 ((all A_4 all B_4 all C_4 hBOOL(hAPP(product_prod(X_a,product_prod(X_b,X_c)),bool,P_1,hAPP(product_prod(X_b,X_c),product_prod(X_a,product_prod(X_b,X_c)),hAPP(X_a,fun(product_prod(X_b,X_c),product_prod(X_a,product_prod(X_b,X_c))),product_Pair(X_a,product_prod(X_b,X_c)),A_4),hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),B_4),C_4))))) -> hBOOL(hAPP(product_prod(X_a,product_prod(X_b,X_c)),bool,P_1,Xa)))) # label(fact_3178_prod__induct3) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4362 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,K)),hAPP(int,int,bit1,L))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)))) # label(fact_31_rel__simps_I17_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4363 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),min)),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),arccos),Xa))))) # label(fact_4942_isCont__arccos) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4364 (all V (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(int,nat,number_number_of(nat),V))) -> hAPP(int,nat,number_number_of(nat),V) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,number_number_of(nat),V)),one_one(nat))))) # label(fact_1490_expand__Suc) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4365 (all D_3 all A_1 all B (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_3),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)) -> one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_3),B))) # label(fact_4416_coprime__rmult__int) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4366 (all X_a (comm_ring_1(X_a) -> (all X all N (-hBOOL(hAPP(nat,bool,even_odd_even(nat),N)) -> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),N) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N)))))) # label(fact_2478_power__minus__odd) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4367 (all X_a (semiring_0(X_a) & power(X_a) -> (all N ((zero_zero(nat) != N -> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),zero_zero(X_a)),N) = zero_zero(X_a)) & (N = zero_zero(nat) -> one_one(X_a) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),zero_zero(X_a)),N)))))) # label(fact_112_power__0__left) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4368 (all D_1 all Ta (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),hAPP(int,int,uminus_uminus(int),Ta))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),Ta)))) # label(fact_2066_uminus__dvd__conv_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4369 (all X_a (ordered_ab_group_add(X_a) -> (all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),hAPP(X_a,X_a,uminus_uminus(X_a),A_3))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),hAPP(X_a,X_a,uminus_uminus(X_a),B_2))))))) # label(fact_1902_le__minus__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4370 (all X_a (ring_1(X_a) -> (all Na all A_3 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),ring_1_Ints(X_a))) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_3),Na)),ring_1_Ints(X_a))))))) # label(fact_3828_Ints__power) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4371 (all X_a (group_add(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),B)),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B))))) # label(fact_1915_minus__add) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4372 (all X_a (banach(X_a) -> (all F all G ((exists N_2 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,F,N_1))),hAPP(nat,real,G,N_1))))) -> (hBOOL(hAPP(fun(nat,real),bool,summable(real),G)) -> hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),F)))))))) # label(fact_4044_summable__norm__comparison__test) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4373 (all J_2 all A_1 all K_4 all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5)) -> (hBOOL(hAPP(int,bool,zprime,P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,K_4),zero_zero(int)),P_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),J_2),K_4)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),K_4))),K_4)),P_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),J_2),K_4)),A_1),P_5))))))) # label(fact_1190_aux______2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4374 (all X hAPP(real,real,arctan,X) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,arctan,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))))) # label(fact_2161_arctan__half) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4375 (all X all A_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),X)),A_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X),hAPP(real,nat,natfloor,A_1))))) # label(fact_2400_le__natfloor) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4376 (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)))))) # label(fact_106_of__nat__less__two__power) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4377 (all X_a (number_semiring(X_a) -> (all V_2 all V (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_2)) -> hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),V_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),V_2))))))) # label(fact_305_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4378 (all X_a (real_normed_vector(X_a) -> (all X -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),X)),zero_zero(real)))))) # label(fact_1975_norm__not__less__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4379 (all Y_2 all X_2 (hBOOL(hAPP(fun(nat,rat),bool,vanishes,X_2)) -> (hBOOL(hAPP(fun(nat,rat),bool,vanishes,Y_2)) -> hBOOL(hAPP(fun(nat,rat),bool,vanishes,hAPP(fun(nat,rat),fun(nat,rat),hAPP(fun(nat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,rat)),combs(nat,rat,rat),hAPP(fun(nat,rat),fun(nat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,fun(rat,rat))),combb(rat,fun(rat,rat),nat),minus_minus(rat)),X_2)),Y_2)))))) # label(fact_4878_vanishes__diff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4380 (all X_a (archim1727834104eiling(X_a) -> (all Z all Xa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)),Xa)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z),hAPP(X_a,int,archimedean_ceiling(X_a),Xa))))))) # label(fact_3620_less__ceiling__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4381 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> hAPP(real,nat,natceiling,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),one_one(real))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(real,nat,natceiling,X)),one_one(nat)))) # label(fact_1678_natceiling__add__one) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4382 (all X_a (archim1727834104eiling(X_a) -> (all X hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(X_a,int,archimedean_ceiling(X_a),X))))))) # label(fact_3607_le__of__int__ceiling) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4383 (all I_1 all J_2 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I_1),J_2)),I_1))) # label(fact_3454_zgcd__zdvd1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4384 (all Xa (hAPP(real,real,sin,Xa) = zero_zero(real) <-> (exists N_1 (hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1)) & Xa = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) | (exists N_1 (hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1)) & hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = Xa)))) # label(fact_2466_sin__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4385 (all Ma all Na hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Ma),hAPP(nat,nat,suc,Na)) = hAPP(nat,nat,hAPP(fun(nat,nat),fun(nat,nat),hAPP(nat,fun(fun(nat,nat),fun(nat,nat)),nat_case(nat),zero_zero(nat)),combi(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Ma),Na))) # label(fact_4150_diff__Suc) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4386 (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)))))))) # label(fact_111_one__less__power) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4387 (all X_a all X_b all X_c all Z all C all P_4 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Z),hAPP(product_prod(X_b,X_c),fun(X_a,bool),hAPP(fun(X_b,fun(X_c,fun(X_a,bool))),fun(product_prod(X_b,X_c),fun(X_a,bool)),product_prod_case(X_b,X_c,fun(X_a,bool)),C),P_4))) -> -(all X_1 all Y_1 (hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),X_1),Y_1) = ti(product_prod(X_b,X_c),P_4) -> -hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Z),hAPP(X_c,fun(X_a,bool),hAPP(X_b,fun(X_c,fun(X_a,bool)),C,X_1),Y_1))))))) # label(fact_4087_mem__splitE) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4388 (all X_c all X_a all X_b all C all P_4 all Z (hBOOL(hAPP(X_c,bool,hAPP(product_prod(X_a,X_b),fun(X_c,bool),hAPP(fun(X_a,fun(X_b,fun(X_c,bool))),fun(product_prod(X_a,X_b),fun(X_c,bool)),product_prod_case(X_a,X_b,fun(X_c,bool)),C),P_4),Z)) -> -(all X_1 all Y_1 (ti(product_prod(X_a,X_b),P_4) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X_1),Y_1) -> -hBOOL(hAPP(X_c,bool,hAPP(X_b,fun(X_c,bool),hAPP(X_a,fun(X_b,fun(X_c,bool)),C,X_1),Y_1),Z)))))) # label(fact_4084_splitE_H) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4389 (all W (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W))) -> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),one_one(int)),hAPP(int,int,number_number_of(int),W)) = hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,one_one(int)),hAPP(int,int,number_number_of(int),W))))) # label(fact_3144_mod__pos__pos__1__number__of) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4390 (all Ya all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Xa)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd)) -> (exists N_1 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Ya),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),N_1),Xa))))),Xa)))))) # label(fact_1232_best__odd__division__abs) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4391 (all X_a all Q_1 all P_1 all Xa (hBOOL(hAPP(X_a,bool,P_1,Xa)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),P_1),Q_1)) -> hBOOL(hAPP(X_a,bool,Q_1,Xa))))) # label(fact_4364_rev__predicate1D) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4392 (all Z_1 hAPP(complex,real,norm_norm(complex),Z_1) = hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,cnj,Z_1))) # label(fact_2963_complex__mod__cnj) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4393 (all X_2 all A_3 (hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),X_2),A_3)) -> hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(complex,complex),fun(fun(nat,complex),fun(nat,complex)),combb(complex,complex,nat),cnj),X_2)),hAPP(complex,complex,cnj,A_3))))) # label(fact_4136_cnj_Osums) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4394 (all K_4 all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),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_4),N)))))) # label(fact_803_dvd__diffD1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4395 (all X_a (linordered_field(X_a) -> (all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),B)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)))))))) # label(fact_2589_less__imp__inverse__less) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4396 (all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> hAPP(int,int,number_number_of(int),min) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,number_number_of(int),min)),B))) # label(fact_1122_div__eq__minus1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4397 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),zero_zero(real))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,sqrt,Xa)),zero_zero(real))))) # label(fact_2139_real__sqrt__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4398 (all Ta all B_1 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Ta),one_one(int))),B_1)) -> (all X_1 ((all Xa_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D))) -> (all Xb (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1)) -> hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa_1) != ti(int,X_1))))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Ta),X_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Ta),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),D))))))))) # label(fact_4657_bset_I8_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4399 (all A_1 all N all B hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B)),N) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),N)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),N))) # label(fact_4400_gcd__exp__int) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4400 (all X_a (number_ring(X_a) -> (all V hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit1,pls)))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),one_one(X_a))))) # label(fact_2103_diff__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4401 (all A_1 all B hAPP(int,rat,ring_1_of_int(rat),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,sgn_sgn(int),A_1)),hAPP(int,int,sgn_sgn(int),B))) = hAPP(rat,rat,sgn_sgn(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B))) # label(fact_3803_sgn__rat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4402 (all J_2 all K_4 all A_1 all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5)) -> (hBOOL(hAPP(int,bool,zprime,P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,K_4),zero_zero(int)),P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),zero_zero(int)),P_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),J_2))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_5),K_4))),P_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),K_4),P_5))))))))) # label(fact_1192_MultInv__zcong__prop3) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4403 (all X_a (linordered_idom(X_a) -> (all X all N (hBOOL(hAPP(nat,bool,even_odd_even(nat),N)) -> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,abs_abs(X_a),X)),N))))) # label(fact_2481_power__even__abs) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4404 (all X_a (real_normed_field(X_a) -> (all Y_2 all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2)) -> (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),Y_2)) -> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),X_2)),Y_2)))))))) # label(fact_4038_summable__diff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4405 (all X_a (number_ring(X_a) -> (all X (-hBOOL(hAPP(nat,bool,even_odd_even(nat),X)) -> hAPP(int,X_a,number_number_of(X_a),min) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,number_number_of(X_a),min)),X))))) # label(fact_2498_Parity_Oneg__one__odd__power) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4406 (all X_a all A_2 all B_1 hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1) = hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),A_2))),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(bool,bool),fun(fun(X_a,bool),fun(X_a,bool)),combb(bool,bool,X_a),fNot),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),B_1))))) # label(fact_2245_set__diff__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4407 (all A_3 all P_4 (hBOOL(hAPP(int,bool,zprime,P_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),zero_zero(int)),P_4)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_4),A_3)) -> hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(fun(int,bool),bool),nat,finite_card(fun(int,bool)),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,A_3),P_4))) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) # label(fact_1689_SetS__card) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4408 (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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K)),Ma)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K)),Na))))) # label(fact_1393_Suc__mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4409 (all X_a (division_ring(X_a) -> (all A_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),one_one(X_a)) = ti(X_a,A_1)))) # label(fact_1830_divide__1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4410 (all X all Y hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y))) # label(fact_3247_gcd__semilattice__nat_Oinf__left__idem) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4411 (all M all N all Q_3 all R_3 (hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M),N),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Q_3),R_3))) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N) = Q_3)) # label(fact_3126_div__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4412 (all X_b all X_a (topological_space(X_a) & topological_space(X_b) -> (all F all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),hAPP(X_a,X_b,F,A_3)),hAPP(X_a,filter(X_a),at(X_a),A_3))) <-> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),A_3)))))) # label(fact_4902_isCont__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4413 (all X_a (semiring_div(X_a) -> (all A_1 all B ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B)),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B))))) # label(fact_1412_mod__div__equality) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4414 (all B 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_eq(real),zero_zero(real)),Y)) -> hAPP(real,real,hAPP(real,fun(real,real),log,B),hAPP(real,real,hAPP(real,fun(real,real),powr,X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Y),hAPP(real,real,hAPP(real,fun(real,real),log,B),X))))) # label(fact_3443_log__powr) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4415 (all X_a (linordered_field(X_a) -> (all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))))))) # label(fact_2610_inverse__le__imp__le__neg) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4416 (all Z_4 all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),zero_zero(int))) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,nat_1,hAPP(int,int,uminus_uminus(int),Z_1))),hAPP(int,nat,nat_1,hAPP(int,int,uminus_uminus(int),Z_4))) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z_1),Z_4)))) # label(fact_2097_nat__mult__distrib__neg) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4417 (all X_a (number_ring(X_a) -> (all W all Z_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,uminus_uminus(int),W))),Z_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(int,X_a,number_number_of(X_a),W))),Z_1)))) # label(fact_1926_minus__number__of__mult) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4418 (all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),zero_zero(nat)),N) = zero_zero(nat)) # label(fact_807_diff__0__eq__0) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4419 (all X_a (metric_space(X_a) -> (all X_2 all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially)) -> hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_2)))))) # label(fact_4459_LIMSEQ__imp__Cauchy) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4420 (all D_1 all F all Xa all L (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),Xa),L)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_1)) -> ((all Y_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Xa),Y_1))),D_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,F,Xa)),hAPP(real,real,F,Y_1))))) -> L = zero_zero(real))))) # label(fact_2660_DERIV__local__min) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4421 (all X_a (mult_zero(X_a) -> (all A_1 zero_zero(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),zero_zero(X_a))))) # label(fact_374_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4422 (all X_a (linordered_field(X_a) -> (all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)))))))) # label(fact_1842_divide__pos__pos) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4423 (all A_1 all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int)))) -> hAPP(int,int,hAPP(int,fun(int,int),inv,P_5),A_1) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int)))))) # label(fact_1177_inv__not__p__minus__1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4424 (all X all Y (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),X),Y)) -> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y) = hAPP(int,int,abs_abs(int),X))) # label(fact_4387_gcd__proj1__if__dvd__int) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4425 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> (zero_zero(real) = hAPP(real,real,sqrt,X) -> X = zero_zero(real)))) # label(fact_2137_real__sqrt__eq__zero__cancel) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4426 (all X_a all Xa all Ya all F (hBOOL(hAPP(fun(product_prod(X_a,X_a),bool),bool,hAPP(product_prod(X_a,X_a),fun(fun(product_prod(X_a,X_a),bool),bool),member(product_prod(X_a,X_a)),hAPP(X_a,product_prod(X_a,X_a),hAPP(X_a,fun(X_a,product_prod(X_a,X_a)),product_Pair(X_a,X_a),Xa),Ya)),hAPP(fun(X_a,nat),fun(product_prod(X_a,X_a),bool),measure(X_a),F))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(X_a,nat,F,Xa)),hAPP(X_a,nat,F,Ya))))) # label(fact_3344_in__measure) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4427 (all Z_1 ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1)) -> ti(int,Z_1) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,nat_1,Z_1))) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1)) -> zero_zero(int) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,nat_1,Z_1))))) # label(fact_1024_int__nat__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4428 (all X_a (sgn_div_norm(X_a) -> ti(fun(X_a,real),norm_norm(X_a)) = norm_norm(X_a))) # label(tsy_c_RealVector_Onorm__class_Onorm_0_res) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4429 (all A_1 all B hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(nat,int,semiring_1_of_nat(int),A_1)),hAPP(nat,int,semiring_1_of_nat(int),B)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A_1),B))) # label(fact_1541_zmod__int) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4430 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B)) -> ti(int,A_1) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)))) # label(fact_1420_mod__pos__pos__trivial) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4431 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,Ma)),hAPP(nat,nat,suc,Na))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)))) # label(fact_1332_Suc__less__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4432 (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),hAPP(int,int,bit0,K))))) # label(fact_689_rel__simps_I25_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4433 (all X_a (comm_monoid_add(X_a) -> (all K all F (zero_zero(X_a) = hAPP(nat,X_a,F,zero_zero(nat)) -> hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K)) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),hAPP(nat,nat,suc,zero_zero(nat))),K)))))) # label(fact_4221_setsum__shift__lb__Suc0__0__upt) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4434 (all K_4 pls != hAPP(int,int,bit1,K_4)) # label(fact_66_rel__simps_I46_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4435 (all Xa (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(real,nat,natceiling,Xa)),one_one(nat))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),one_one(real))))) # label(fact_1675_natceiling__le__eq__one) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4436 (all M all N hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),M)),hAPP(nat,real,real_1(nat),N)) = hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N))) # label(fact_2385_real__of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4437 (all V ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls)) -> hAPP(int,nat,number_number_of(nat),hAPP(int,int,succ,V)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),hAPP(int,nat,number_number_of(nat),V))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls)) -> one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),hAPP(int,nat,number_number_of(nat),V))))) # label(fact_202_nat__1__add__number__of) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4438 (all X_a (ordered_ab_group_add(X_a) -> (all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),B)),hAPP(X_a,X_a,uminus_uminus(X_a),A_1))))))) # label(fact_1899_le__imp__neg__le) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4439 (all I_1 all M all N (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),I_1),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),N))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),I_1),M)))) # label(fact_4402_dvd__gcd__D1__int) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4440 (all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> (hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) != ti(int,P_5) -> (hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))) != ti(int,P_5) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),P_5)))))) # label(fact_704_prime__g__5) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4441 (all Ya all Xa (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ya),zOdd)) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xa),Ya)),zEven))))) # label(fact_1715_odd__minus__odd) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4442 (all P_1 all D_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_1)) -> ((all X_1 all K_1 (hBOOL(hAPP(int,bool,P_1,X_1)) <-> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_1),D_1)))))) -> ((exists X1 hBOOL(hAPP(int,bool,P_1,X1))) <-> (exists X_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_1))) & hBOOL(hAPP(int,bool,P_1,X_1)))))))) # label(fact_4668_periodic__finite__ex) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4443 (all X all Y hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,Y)),hAPP(real,real,cos,X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,Y)),hAPP(real,real,sin,X))) = hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y))) # label(fact_2269_cos__diff2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4444 (all X all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),pi)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,cos,X)),hAPP(real,real,cos,Y))))))) # label(fact_2307_cos__monotone__0__pi) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4445 (all X_a (monoid_mult(X_a) -> (all A_1 all N hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) # label(fact_645_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4446 (all K_4 hAPP(nat,rat,semiring_1_of_nat(rat),K_4) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(nat,int,semiring_1_of_nat(int),K_4)),one_one(int))) # label(fact_3776_Fract__of__nat__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4447 (all N all A_1 hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,cis,A_1)),hAPP(real,complex,cis,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),A_1))) = hAPP(real,complex,cis,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N))),A_1))) # label(fact_3059_cis__real__of__nat__Suc__mult) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4448 (all Y all X all P_5 (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),X),P_5)) -> -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),Y))))) # label(fact_791_Euler_Oaux____1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4449 (all X_a (number_ring(X_a) -> hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,number_number_of(int),min)) = hAPP(int,X_a,number_number_of(X_a),min))) # label(fact_3616_of__int__m1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4450 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,number_number_of(X_a),Xa)),hAPP(int,X_a,number_number_of(X_a),Ya))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),Ya)))))) # label(fact_207_le__number__of) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4451 (all Z all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Z),hAPP(nat,int,semiring_1_of_nat(int),Ma))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),Z))),Ma)))) # label(fact_968_dvd__int__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4452 (all M hAPP(nat,nat,suc,M) != zero_zero(nat)) # label(fact_1312_Zero__not__Suc) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4453 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),min)),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),arcsin),Xa))))) # label(fact_4943_isCont__arcsin) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4454 (all X_a (archim1727834104eiling(X_a) -> (all X all Z_1 hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),Z_1))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),Z_1)))) # label(fact_3619_floor__add__of__int) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4455 (all R_3 all X hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),R_3),hAPP(complex,complex,cnj,X)) = hAPP(complex,complex,cnj,hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),R_3),X))) # label(fact_4708_cnj_OscaleR) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4456 (all X all Y X = hAPP(complex,real,re,hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,X),Y))) # label(fact_3019_Re) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4457 (all R_3 all S_3 hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),R_3),hAPP(real,real,uminus_uminus(real),S_3)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),R_3),S_3)) # label(fact_1992_real__diff__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4458 (all A_1 all B hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),B))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),A_1)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))) # label(fact_788_zdiff__power3) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4459 (all Xa all Ya (one_one(real) = Ya & Xa = zero_zero(real) <-> hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Xa),Ya) = ii)) # label(fact_2783_Complex__eq__i) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4460 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 all B hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B))))))) # label(fact_993_abs__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4461 (all X_a (number_ring(X_a) -> (all A_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls))),A_1) = ti(X_a,A_1)))) # label(fact_495_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4462 (all X_a all One all Times all A_3 ti(X_a,One) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),hAPP(X_a,fun(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a))),power_1(X_a),One),Times),A_3),zero_zero(nat))) # label(fact_2866_power_Opower_Opower__0) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4463 (all M all N (hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N) = zero_zero(nat) -> (zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),M) -> N = M))) # label(fact_804_diffs0__imp__equal) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4464 (all X_a all Xa hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),Xa)) = ti(X_a,Xa)) # label(fact_4811_some__eq__trivial) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4465 (all X_a all X_b (real_normed_vector(X_b) -> (all Ra all G all A_3 all A_2 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),A_3),A_2)) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),Ra)),G)),hAPP(X_b,X_b,hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),Ra),A_3)),A_2)))))) # label(fact_4751_scaleR__right_Otendsto) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4466 (all X_b all X_a (real_normed_algebra(X_a) -> (all G all A_2 all Ya hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),G),A_2)),Ya) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(X_a,fun(X_b,X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(X_a,fun(X_b,X_a)),combc(X_b,X_a,X_a),hAPP(fun(X_b,X_a),fun(X_b,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_b),times_times(X_a)),G)),Ya)),A_2)))) # label(fact_4212_mult__left_Osetsum) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4467 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,K1)),hAPP(int,int,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]. 7.88/7.86 4468 (all X all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> hAPP(real,real,hAPP(nat,fun(real,real),root,N),hAPP(real,real,uminus_uminus(real),X)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)))) # label(fact_2692_real__root__minus) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4469 (all A_1 all B all C_1 all D_3 hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,C_1),D_3)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_1),C_1)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),D_3))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_1),D_3)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),C_1)))) # label(fact_2766_complex__mult) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4470 (all Xa (zero_zero(int) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Xa),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) <-> hBOOL(hAPP(int,bool,even_odd_even(int),Xa)))) # label(fact_1629_even__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.86 4471 (all P_5 all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_1),B)) -> -(hBOOL(hAPP(nat,bool,prime,P_5)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),A_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),B))))) # label(fact_3086_coprime__prime) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4472 (all X_a (linord581940658strict(X_a) -> (all A_3 all B_2 all 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(X_a),A_3),B_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B_2)))))))) # label(fact_443_mult__less__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4473 (all X_a all X_b (comm_semiring_1(X_b) -> (all F all I all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I),A_2)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),dvd_dvd(X_b),hAPP(X_a,X_b,F,I)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2)))))))) # label(fact_3980_dvd__setprod) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4474 (all X_b all X_a (linordered_field(X_a) -> (all F all A_2 hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),abs_abs(X_a)),F)),A_2) = hAPP(X_a,X_a,abs_abs(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),F),A_2))))) # label(fact_3975_abs__setprod) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4475 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) | Na = Ma <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),Na)))) # label(fact_401_le__eq__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4476 (all Na (zero_zero(nat) = Na <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),zero_zero(nat))))) # label(fact_396_le__0__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4477 (all N hAPP(real,nat,natfloor,hAPP(nat,real,real_1(nat),N)) = N) # label(fact_2374_natfloor__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4478 (all X_a (field_inverse_zero(X_a) & real_normed_field(X_a) -> (all A_1 all B hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(X_a,real,norm_norm(X_a),A_1)),hAPP(X_a,real,norm_norm(X_a),B)) = hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B))))) # label(fact_1984_norm__divide) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4479 (all K_4 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),pls) = ti(int,K_4)) # label(fact_78_add__Pls__right) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4480 (all X_a all X_b (real_normed_vector(X_b) -> (all F all A_2 (hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),zero_zero(real)),A_2)) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),A_2)))))) # label(fact_4475_tendsto__norm__zero__cancel) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4481 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))))) # label(fact_467_pos__zmult__pos) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4482 (all Ya hAPP(fun(real,bool),real,the(real),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),ord_less(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(real,real),fun(real,fun(real,bool))),combb(real,fun(real,bool),real),fequal(real)),tan)),Ya)))) = hAPP(real,real,arctan,Ya)) # label(fact_4159_arctan__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4483 (all X_a (real_algebra_1(X_a) -> (all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(real,X_a,of_real(X_a),X)),hAPP(real,X_a,of_real(X_a),Y)) = hAPP(real,X_a,of_real(X_a),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y))))) # label(fact_2797_of__real__add) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4484 (all M all N hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)),M))) # label(fact_1527_mod__less__eq__dividend) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4485 (all X_a (real_normed_algebra(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))))) # label(fact_1911_mult_Ominus__left) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4486 (all F all A_2 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),resSet,Ma),A_2)) -> ((all X_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),A_2)) -> (all Xa_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),A_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,F,X_1)),hAPP(int,int,F,Xa_1)),Ma)) -> ti(int,X_1) = ti(int,Xa_1)))))) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),resSet,Ma),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,int),fun(fun(int,bool),fun(int,bool)),image(int,int),F),A_2))))))) # label(fact_3668_ResSet__image) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4487 (all B all A_1 all N (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N)) -> (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),N) -> (exists X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),N)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),X_1)),B),N)) & (all Y_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),Y_1)),B),N)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y_1),N)) & 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))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))))))) # label(fact_3503_zcong__lineq__unique) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4488 (all Xa all P_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Xa),zero_zero(int)),P_4)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_4),Xa)))) # label(fact_909_zcong__eq__zdvd__prop) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4489 (all Xa hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),cos),Xa),hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,Xa))))) # label(fact_2651_DERIV__cos) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4490 (all A_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1)) -> (one_one(real) != A_1 -> hAPP(real,real,hAPP(real,fun(real,real),log,A_1),A_1) = one_one(real)))) # label(fact_2030_log__eq__one) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4491 (all X all Y (hBOOL(hAPP(int,bool,even_odd_even(int),Y)) -> hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y))))) # label(fact_1591_anything__times__even) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4492 (all X_a (number_ring(X_a) -> (all W hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),hAPP(int,X_a,number_number_of(X_a),W)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,pred,W))))) # label(fact_2935_number__of__pred) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4493 (all K all I all J hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,nat),fun(fun(nat,bool),fun(nat,bool)),image(nat,nat),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),I),J)) = hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),K))) # label(fact_4250_image__add__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4494 (all N all M all A_1 all B (hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B) = one_one(nat) -> one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A_1),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),M)))) # label(fact_3241_coprime__exp2__nat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4495 (all X all M hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),hAPP(int,int,hAPP(int,fun(int,int),standardRes,M),X)),M))) # label(fact_1350_StandardRes__prop1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4496 (all Nat all Nat_5 (hAPP(nat,nat,suc,Nat_5) = hAPP(nat,nat,suc,Nat) <-> Nat = Nat_5)) # label(fact_1276_nat_Oinject) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4497 (all X_a (preorder(X_a) -> (all X -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),X))))) # label(fact_536_order__less__irrefl) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4498 (all X_2 ((all R_1 (hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R_1)) -> (exists K_1 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N_1)) -> hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,abs_abs(rat),hAPP(nat,rat,X_2,N_1))),R_1)))))) -> hBOOL(hAPP(fun(nat,rat),bool,vanishes,X_2)))) # label(fact_4890_vanishesI) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4499 (all A_1 all P_5 (hBOOL(hAPP(int,bool,zprime,P_5)) -> (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),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),A_1)),one_one(int)),P_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))),P_5)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),one_one(int)),P_5)))))) # label(fact_797_zcong__square) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4500 (all X_a (linord20386208strict(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),A_1))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B))))))) # label(fact_448_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4501 (all X_a (linorder(X_a) -> (all Z all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),Ya)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),Xa)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Xa),Ya))))))) # label(fact_4824_min__less__iff__conj) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4502 (all X all A_1 hAPP(real,int,archim2085082626_floor(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(int,real,real_1(int),A_1))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(real,int,archim2085082626_floor(real),X)),A_1)) # label(fact_3390_floor__add) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4503 (all X all K_4 hAPP(real,real,sqrt,hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),K_4)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,sqrt,X)),K_4)) # label(fact_2130_real__sqrt__power) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4504 (all X_a all F all A_2 all Na (hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),A_2) = hAPP(nat,nat,suc,Na) -> (exists X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(X_a,nat,F,X_1))))))) # label(fact_4328_setsum__SucD) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4505 (all X_a (linordered_idom(X_a) -> (all Wa all Z (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Wa)),hAPP(int,X_a,ring_1_of_int(X_a),Z))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Wa),Z)))))) # label(fact_3605_of__int__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4506 (all X_b (number(X_b) & monoid_mult(X_b) -> (all V all W hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,W))) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(int,nat,number_number_of(nat),W))),hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(int,nat,number_number_of(nat),W)))))) # label(fact_3882_power__number__of__even__number__of) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4507 (all B all A_1 all A_6 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),A_6)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_6),B)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)))))) # label(fact_1101_zdiv__mono1__neg) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4508 (all X_a (metric_space(X_a) -> cauchy(X_a) = ti(fun(fun(nat,X_a),bool),cauchy(X_a)))) # label(tsy_c_SEQ_OCauchy_res) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4509 (all Xa all Ya ((hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Ya),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Xa))) & (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Ya)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),zero_zero(real)))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),Xa),Ya))))) # label(fact_1770_real__0__le__divide__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4510 (all A_3 all P_4 (hBOOL(hAPP(int,bool,zprime,P_4)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_3),zero_zero(int)),P_4)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_4),A_3)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),hAPP(fun(fun(int,bool),bool),fun(int,bool),complete_Sup_Sup(fun(int,bool)),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,A_3),P_4)))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_3),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),P_4))))))) # label(fact_3954_Union__SetS__setprod__prop1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4511 (all K_4 all I_1 all J_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_1),J_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),J_2),K_4)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_1),K_4))))) # label(fact_204_zle__trans) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4512 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),min)),Y)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),one_one(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,arccos,Y)),pi)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,arccos,Y)))))) # label(fact_2359_arccos__lt__bounded) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4513 (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N))))))) # label(fact_652_zero__le__even__power_H) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4514 (all X_a (field_char_0(X_a) -> (all Na all A_3 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),field_char_0_Rats(X_a))) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_3),Na)),field_char_0_Rats(X_a))))))) # label(fact_3713_Rats__power) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4515 (all Na all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Ma),Na)) <-> ti(int,Ma) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,Na),Ma)))) # label(fact_3482_zdvd__iff__zgcd) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4516 (all X_a (archim1727834104eiling(X_a) -> (all Xa all Va (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),Va)),one_one(X_a)))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archim2085082626_floor(X_a),Xa)),hAPP(int,int,number_number_of(int),Va))))))) # label(fact_2920_floor__le__number__of) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4517 (all X_a (number_ring(X_a) -> (all Ya (hAPP(int,X_a,number_number_of(X_a),Ya) = zero_zero(X_a) <-> hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),pls),hAPP(int,int,uminus_uminus(int),Ya))))))))) # label(fact_2182_eq__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4518 (all X_a (ordered_ring_abs(X_a) -> (all B all A_1 ((hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a)))) & (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)) = hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)))))) # label(fact_1011_abs__eq__mult) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4519 (all X_a (linordered_field(X_a) -> (all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),B)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)))))))) # label(fact_2607_le__imp__inverse__le) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4520 (all N (N != zero_zero(nat) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N) = hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) # label(fact_1508_Suc__Suc__mult__two__diff__two) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4521 (all Ta all A_2 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D)) -> (all X_1 ((all Xa_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D))) -> (all Xb (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2)) -> ti(int,X_1) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa_1))))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Ta),X_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Ta),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),D)))))))) # label(fact_4662_aset_I7_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4522 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),N)) -> N = zero_zero(nat) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)))) # label(fact_1231_divides__le) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4523 (all X_a (real_normed_vector(X_a) -> (all X_2 ((exists K_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_1)) & (exists N_2 all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,X_2,N_1)),hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(nat,X_a,X_2,N_2))))),K_1))))) <-> hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2)))))) # label(fact_3516_Bseq__iff3) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4524 (all X all Y hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,uminus_uminus(real),Y))) # label(fact_1993_minus__real__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4525 (all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),zero_zero(nat)),N) = zero_zero(nat)) # label(fact_602_mult__0) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4526 (all X_a (real_normed_field(X_a) -> (all C all F (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) -> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),F)),C))))))) # label(fact_4035_summable__divide) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4527 (all X_a (metric_space(X_a) -> (all X all Z_1 all Y all E_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Z_1)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Y),Z_1))),E_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y)),E_1)))))) # label(fact_3533_dist__triangle__lt) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4528 (all X_a (comm_monoid_add(X_a) -> (all F all Na hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),ord_atMost(nat),Na))),hAPP(nat,X_a,F,hAPP(nat,nat,suc,Na))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),ord_atMost(nat),hAPP(nat,nat,suc,Na)))))) # label(fact_4968_setsum__atMost__Suc) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4529 (all X_a (one(X_a) -> (all Xa (ti(X_a,Xa) = one_one(X_a) <-> ti(X_a,Xa) = one_one(X_a))))) # label(fact_302_one__reorient) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4530 (all A_1 all N (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N)) -> (hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),N) = one_one(int) -> (exists X_1 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),X_1)),one_one(int)),N)))))) # label(fact_3504_zcong__lineq__ex) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4531 (all X_a (field(X_a) & number_ring(X_a) -> (all X hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls))) = ti(X_a,X)))) # label(fact_1783_divide__Numeral1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4532 (all Ma all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Ma),Na))) <-> Na != zero_zero(nat) | zero_zero(nat) != Ma)) # label(fact_3284_gcd__pos__nat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4533 (all M all N hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),hAPP(nat,nat,suc,M)))) # label(fact_1394_diff__less__Suc) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4534 (all X_a (order(X_a) -> (all F (hBOOL(hAPP(fun(nat,X_a),bool,hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,X_a),bool),strict_mono(nat,X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(fun(nat,bool),fun(nat,bool))),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool))),combs(nat,fun(nat,bool),fun(nat,bool)),hAPP(fun(nat,fun(nat,fun(bool,bool))),fun(nat,fun(fun(nat,bool),fun(nat,bool))),hAPP(fun(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool))),fun(fun(nat,fun(nat,fun(bool,bool))),fun(nat,fun(fun(nat,bool),fun(nat,bool)))),combb(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),nat),combs(nat,bool,bool)),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,fun(bool,bool))),hAPP(fun(fun(nat,bool),fun(nat,fun(bool,bool))),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,fun(bool,bool)))),combb(fun(nat,bool),fun(nat,fun(bool,bool)),nat),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj)),dvd_dvd(nat)))),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),hAPP(fun(fun(nat,bool),fun(nat,bool)),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool))),combb(fun(nat,bool),fun(nat,bool),nat),hAPP(fun(bool,bool),fun(fun(nat,bool),fun(nat,bool)),combb(bool,bool,nat),fNot)),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),dvd_dvd(nat))))),F)) -> hBOOL(hAPP(fun(nat,X_a),bool,hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,X_a),bool),mono(nat,X_a),dvd_dvd(nat)),F)))))) # label(fact_4817_dvd_Ostrict__mono__mono) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4535 (all N all M zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),M))) # label(fact_833_diff__add__0) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4536 (all X hAPP(real,real,abs_abs(real),X) = hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),X))) # label(fact_2153_real__sqrt__abs2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4537 (all X_a (semiring_div(X_a) -> (all B all B_5 all A_1 all C_1 all A_6 (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_6),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C_1) -> (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B_5),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C_1) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_6),B_5)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C_1)))))) # label(fact_1290_mod__add__cong) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4538 (all F all A_3 all B_2 hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,F,A_3)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,F,B_2)),hAPP(real,real,F,A_3))),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B_2),A_3))),A_3)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,F,B_2)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,F,B_2)),hAPP(real,real,F,A_3))),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B_2),A_3))),B_2))) # label(fact_1772_lemma__MVT) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4539 (all Xa all Na (-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),Na))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Xa)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),Na)),Xa)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(int,nat,number_number_of(nat),Na)),hAPP(real,nat,natfloor,Xa))))))) # label(fact_1679_le__natfloor__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4540 (all X_a (metric_space(X_a) -> ti(fun(X_a,fun(X_a,real)),dist_dist(X_a)) = dist_dist(X_a))) # label(tsy_c_RealVector_Odist__class_Odist_1_res) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4541 (all N (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),ord_less_eq(int),one_one(int)),hAPP(int,int,fact_fact(int),N))))) # label(fact_3219_fact__ge__one__int) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4542 (all X_a all X_b (real_normed_vector(X_b) -> (all F all A_3 all A_2 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),A_3),A_2)) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),uminus_uminus(X_b)),F)),hAPP(X_b,X_b,uminus_uminus(X_b),A_3)),A_2)))))) # label(fact_4477_tendsto__minus) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4543 (all A_1 all N all P_5 (hBOOL(hAPP(nat,bool,prime,P_5)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A_1),N))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),A_1))))) # label(fact_3081_prime__dvd__power) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4544 (all Sb all Ta all A_3 all B_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A_3),B_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Sb),Ta)) -> hBOOL(hAPP(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),bool,hAPP(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),hAPP(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),hAPP(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A_3),Sb)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B_2),Ta))),pair_less))))) # label(fact_3510_pair__lessI2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4545 (all M all N -(all Q_4 all R_1 -hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M),N),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Q_4),R_1))))) # label(fact_3129_divmod__nat__rel__ex) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4546 (all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A_3),zero_zero(real)),sequentially)) -> (hBOOL(hAPP(fun(nat,real),bool,monoseq(real),A_3)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,A_3,zero_zero(nat)))) -> (all N_1 hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastAtMost(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_1)))),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_1)),one_one(nat)))))))))))) # label(fact_4547_summable__Leibniz_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4547 (all A_1 all B hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),B),A_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)) # label(fact_3250_gcd__commute__nat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4548 (all X_a (boolean_algebra(X_a) -> (all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),hAPP(X_a,X_a,uminus_uminus(X_a),X))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(X_a,X_a,uminus_uminus(X_a),Y))))))) # label(fact_2986_compl__less__swap1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4549 (all X_a (number_ring(X_a) & field_char_0(X_a) -> (all Wa hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(int,X_a,number_number_of(X_a),Wa)),field_char_0_Rats(X_a)))))) # label(fact_3710_Rats__number__of) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4550 (all Xa hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),sin_coeff)),hAPP(real,fun(nat,real),power_power(real),Xa))))) # label(fact_4048_summable__sin) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4551 (all Va all Na ((-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))),Na))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,number_number_of(nat),Va)),hAPP(nat,nat,suc,Na))))) # label(fact_3923_less__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4552 (all X_a (comm_semiring_1(X_a) -> (all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),zero_zero(X_a)),A_1)) -> zero_zero(X_a) = ti(X_a,A_1))))) # label(fact_721_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4553 (all N hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),N) = hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),pi),hAPP(nat,real,real_1(nat),N)))) # label(fact_2434_cos__npi2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4554 (all X_a (real_n1866405975lgebra(X_a) -> (all X all Y hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X)),hAPP(X_a,real,norm_norm(X_a),Y)) = hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y))))) # label(fact_1971_norm__mult) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4555 (all X_a (ring_1(X_a) -> (all A_1 hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) # label(fact_2035_power2__minus) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4556 (all X_a (archim1727834104eiling(X_a) -> (all Xa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archim2085082626_floor(X_a),Xa)),one_one(int))))))) # label(fact_2872_floor__le__one) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4557 (all Na (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Na)) -> hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(int,bool),nat,finite_card(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),Na))))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Na),one_one(int)))) # label(fact_3910_int__card__bdd__int__set__l__l) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4558 (all X_b all X_a all P_1 all P_4 (hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,hAPP(product_prod(X_b,X_a),X_a,product_snd(X_b,X_a),P_4)),hAPP(product_prod(X_b,X_a),X_b,product_fst(X_b,X_a),P_4))) -> -(all X_1 all Y_1 -hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,Y_1),X_1))))) # label(fact_3171_exE__realizer_H) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4559 (all X_a (archim1727834104eiling(X_a) -> (all Xa all Z (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),hAPP(int,X_a,ring_1_of_int(X_a),Z))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archim2085082626_floor(X_a),Xa)),Z)))))) # label(fact_3617_floor__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4560 (all X_a (linordered_field(X_a) -> (all X all Z_1 all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),Y))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),Z_1))))))) # label(fact_1871_mult__imp__div__pos__less) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4561 (all A_1 all B all P_5 all Q_3 (hAPP(rat,product_prod(int,int),quotient_of,hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_5),Q_3) -> hAPP(int,rat,hAPP(int,fun(int,rat),fract,P_5),Q_3) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B))) # label(fact_3757_quotient__of__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4562 (all X_a (real_vector(X_a) -> (all A_1 all B all X hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_1),B)),X) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),B),X))))) # label(fact_4716_real__vector_Oscale__scale) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4563 (all X_a (semiri456707255roduct(X_a) -> (all B_2 all A_3 (ti(X_a,B_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B_2),A_3) <-> ti(X_a,A_3) = zero_zero(X_a))))) # label(fact_317_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4564 (all X_a all X_b (comm_monoid_mult(X_b) -> (all Ya all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(X_b,fun(X_a,X_b),combk(X_b,X_a),Ya)),A_2) = hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),Ya),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)))))) # label(fact_3986_setprod__constant) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4565 (all Na all P_1 (-hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) -> (hBOOL(hAPP(nat,bool,P_1,Na)) -> (exists K_1 (hBOOL(hAPP(nat,bool,P_1,K_1)) & (all I_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),K_1)) -> -hBOOL(hAPP(nat,bool,P_1,I_2)))) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),Na))))))) # label(fact_659_ex__least__nat__le) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4566 (all X_a (metric_space(X_a) -> (all X all Z_1 all Y hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Z_1)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Y),Z_1))))))) # label(fact_3530_dist__triangle) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4567 (all P_1 hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(fun(nat,int),fun(nat,bool),hAPP(fun(int,bool),fun(fun(nat,int),fun(nat,bool)),combb(int,bool,nat),P_1),semiring_1_of_nat(int)))) = hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),P_1))) # label(fact_3907_Nat__Transfer_Otransfer__int__nat__set__functions_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4568 (all Ya all Xa (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Ya)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(int,nat,nat_1,Xa)),hAPP(int,nat,nat_1,Ya))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Xa),Ya)))))) # label(fact_1021_transfer__nat__int__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4569 (all X_a (real_normed_field(X_a) & banach(X_a) -> exp(X_a) = ti(fun(X_a,X_a),exp(X_a)))) # label(tsy_c_Transcendental_Oexp_res) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4570 (all Ma all No_1 all F ((all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,F,N_1)),hAPP(nat,real,F,hAPP(nat,nat,suc,N_1))))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,F,Ma)),hAPP(nat,real,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ma),No_1)))))) # label(fact_1651_lemma__f__mono__add) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4571 (all X_a (real_vector(X_a) -> ti(fun(real,fun(X_a,X_a)),scaleR_scaleR(X_a)) = scaleR_scaleR(X_a))) # label(tsy_c_RealVector_OscaleR__class_OscaleR_1_res) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4572 (all X_a (linorder(X_a) -> (all T_5 exists Z_2 all X_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Z_2)) -> -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),T_5),X_1)))))) # label(fact_2851_minf_I8_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4573 (all X_a (linord219039673up_add(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)),A_3)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3)))))) # label(fact_1948_minus__le__self__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4574 (all N all A_1 all B all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),N)) -> (hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N) = one_one(int) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),N))))))) # label(fact_3484_zcong__zgcd__zmult__zmod) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4575 (all X_a (order(X_a) -> (all X all Least ((all X_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Least),X_1))) -> ti(X_a,Least) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Least))))) # label(fact_4860_min__leastR) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4576 (all N (zero_zero(nat) != N -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),N) = one_one(nat))) # label(fact_3312_coprime__minus__one__nat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4577 (all X_a (archim1727834104eiling(X_a) -> (all X hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(X_a,int,archimedean_ceiling(X_a),X)),one_one(int)) = hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),one_one(X_a)))))) # label(fact_3332_ceiling__diff__one) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4578 (all X_a (linordered_idom(X_a) -> (all A_1 hAPP(X_a,X_a,sgn_sgn(X_a),hAPP(X_a,X_a,sgn_sgn(X_a),A_1)) = hAPP(X_a,X_a,sgn_sgn(X_a),A_1)))) # label(fact_2257_sgn__sgn) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4579 (exists K_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2)) & (all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(complex,real,im,X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_1)),K_2)))))) # label(fact_3013_Im_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4580 (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),hAPP(X_a,X_a,abs_abs(X_a),A_1)),zero_zero(X_a)))))) # label(fact_991_abs__not__less__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4581 (all Y hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),Y)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(real,real,tan,Y))) # label(fact_2449_tan__inverse) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4582 (all F all A_2 (hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2)) -> ((all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,F,X_1))))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co1399186613setsum(int,int),F),A_2)))))) # label(fact_4332_transfer__nat__int__sum__prod__closure_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4583 (all X_a all G all A_3 all A_2 (hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),A_3),A_2)) -> hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),hAPP(fun(X_a,complex),fun(X_a,complex),hAPP(fun(complex,complex),fun(fun(X_a,complex),fun(X_a,complex)),combb(complex,complex,X_a),cnj),G)),hAPP(complex,complex,cnj,A_3)),A_2)))) # label(fact_4531_cnj_Otendsto) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4584 (all X_a (semiring_div(X_a) -> ti(fun(X_a,fun(X_a,X_a)),div_mod(X_a)) = div_mod(X_a))) # label(tsy_c_Divides_Odiv__class_Omod_res) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4585 (all X_a all A_2 all B_1 (hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),B_1))) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)))) # label(fact_2843_PowD) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4586 (all B all D_3 all A_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_3),A_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_3),B)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)))))) # label(fact_2880_coprime__mul) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4587 (all X_a (metric_space(X_a) -> (all Y all Z_1 all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Y),Z_1)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Z_1))))))) # label(fact_3529_dist__triangle__alt) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4588 (all A_1 (zero_zero(int) != ti(int,A_1) -> one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),A_1))) # label(fact_1077_zdiv__self) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4589 (all X_a (dense_linorder(X_a) -> (all A_3 all B_2 all C all D_1 ((hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),D_1)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),A_3))) <-> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),A_3),B_2)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),C),D_1))))))) # label(fact_4553_atLeastLessThan__subseteq__atLeastAtMost__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4590 (all M all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,M)),N)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)))) # label(fact_1325_Suc__lessD) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4591 (all X_a (topological_space(X_a) -> (all A_3 hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),combi(X_a)),A_3),hAPP(X_a,filter(X_a),at(X_a),A_3)))))) # label(fact_4581_LIM__ident) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4592 (all Z_4 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_4)) -> hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z_1),Z_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,nat_1,Z_1)),hAPP(int,nat,nat_1,Z_4))))) # label(fact_1042_nat__add__distrib) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4593 (all X_a (real_normed_vector(X_a) -> (all X_2 (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),X_2)),zero_zero(real)),sequentially)) <-> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),zero_zero(X_a)),sequentially)))))) # label(fact_4472_LIMSEQ__norm__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4594 (all Q_3 ((Q_3 = zero_zero(rat) -> hAPP(rat,rat,sgn_sgn(rat),Q_3) = zero_zero(rat)) & (Q_3 != zero_zero(rat) -> (hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),Q_3)) -> hAPP(rat,rat,sgn_sgn(rat),Q_3) = one_one(rat)) & (-hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),Q_3)) -> hAPP(rat,rat,uminus_uminus(rat),one_one(rat)) = hAPP(rat,rat,sgn_sgn(rat),Q_3))))) # label(fact_3724_sgn__rat__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4595 (all K all Xa hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),sin),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),plus_plus(real)),K))),Xa),hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa),K))))) # label(fact_3867_DERIV__sin__add) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4596 (all X all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_5)) -> (hBOOL(hAPP(int,bool,zprime,P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_5)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_5),X)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_5),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(int,int,number_number_of(int),min)),P_5))))))) # label(fact_1071_Euler__part1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4597 (all M all N hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),M),N)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(nat,real,real_1(nat),M)),N)) # label(fact_2388_power__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4598 (all X_a all C_2 all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),C_2)) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),C_2))))) # label(fact_2203_subset__trans) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4599 (all N all A_1 all B (one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B) -> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),N)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),N)) = one_one(int))) # label(fact_3474_zgcd__1__power__distrib) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4600 (all Na hAPP(nat,fun(nat,bool),ord_atMost(nat),Na) = hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)),Na)) # label(fact_4967_atLeast0AtMost) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4601 (all X_2 (hBOOL(hAPP(fun(nat,complex),bool,summable(complex),X_2)) -> hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),re),X_2)) = hAPP(complex,real,re,hAPP(fun(nat,complex),complex,suminf(complex),X_2)))) # label(fact_4075_Re_Osuminf) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4602 (all P_1 all Na all K (hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Na),K))) <-> (zero_zero(nat) = K -> hBOOL(hAPP(nat,bool,P_1,Na))) & (K != zero_zero(nat) -> (all I_2 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J_1),K)) -> (hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),I_2)),J_1) = Na -> hBOOL(hAPP(nat,bool,P_1,J_1)))))))) # label(fact_1552_split__mod) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4603 (all X_b all X_a (real_normed_vector(X_b) & topological_space(X_a) -> (all Xa all G all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),G),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,real),fun(X_a,fun(X_b,X_b)),hAPP(fun(real,fun(X_b,X_b)),fun(fun(X_a,real),fun(X_a,fun(X_b,X_b))),combb(real,fun(X_b,X_b),X_a),scaleR_scaleR(X_b)),G)),Xa)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4771_scaleR__left_OLIM__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4604 (all W hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),W))) # label(fact_206_zle__refl) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4605 (all X_a (ordered_ab_group_add(X_a) -> (all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),B_2)),A_3)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)),B_2)))))) # label(fact_1901_minus__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4606 (all Xa (hBOOL(hAPP(int,bool,even_odd_even(int),Xa)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Xa)))) # label(fact_1626_int__even__iff__2__dvd) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4607 (all Y all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) # label(fact_2173_real__sqrt__sum__squares__ge2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4608 (all Z_1 hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,re,Z_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,im,Z_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(complex,real,norm_norm(complex),Z_1)) # label(fact_3043_cmod__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4609 (all Xa all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) -> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),hAPP(nat,fun(real,real),root,Na)),Xa))))) # label(fact_4939_isCont__root__pos) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4610 (all X (hBOOL(hAPP(nat,bool,even_odd_even(nat),X)) -> zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))))) # label(fact_2503_even__nat__mod__two__eq__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4611 (all N (zero_zero(nat) != N -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)))) # label(fact_142_gr0I) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4612 (all X_a (dvd(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B),A_1)) -> -(all K_1 ti(X_a,A_1) != hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),K_1)))))) # label(fact_2838_dvdE) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4613 (all Ma all Na (ti(int,Ma) != zero_zero(int) | zero_zero(int) != ti(int,Na) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),Ma),Na))))) # label(fact_4411_gcd__pos__int) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4614 (all X_a (monoid_mult(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),one_one(nat))))) # label(fact_160_power__one__right) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4615 (all M all N all K_4 hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),M),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),N),K_4))),N) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N)) # label(fact_3470_zgcd__zadd__zmult) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4616 (all X_a all X_b all Xa all Ya hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),Xa),Ya) = hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),hilbert_Eps(product_prod(X_a,X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),hAPP(fun(X_b,bool),fun(X_a,fun(X_b,bool)),hAPP(fun(X_a,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(X_a,fun(X_b,bool))),combc(X_a,fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(bool,bool)),fun(X_a,fun(fun(X_b,bool),fun(X_b,bool))),hAPP(fun(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_a,fun(bool,bool)),fun(X_a,fun(fun(X_b,bool),fun(X_b,bool)))),combb(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),X_a),combb(bool,bool,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(X_a,fun(X_a,bool),fequal(X_a),Xa)))),hAPP(X_b,fun(X_b,bool),fequal(X_b),Ya))))) # label(fact_4805_Eps__split__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4617 (all X_a (linord20386208strict(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B)))))))) # label(fact_453_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4618 (all Q_3 ((all A_4 all B_4 (hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_4),B_4) = Q_3 -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_4)) -> (zero_zero(int) != ti(int,A_4) -> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_4),B_4) != one_one(int))))) -> zero_zero(rat) = Q_3)) # label(fact_4438_Rat__cases__nonzero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4619 (all Xa all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (hBOOL(hAPP(nat,bool,even_odd_even(nat),Na)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Xa),zero_zero(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),root,Na)),Xa),hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,uminus_uminus(real),hAPP(nat,real,real_1(nat),Na))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,zero_zero(nat)))))))))))) # label(fact_2671_DERIV__even__real__root) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4620 (all X_a (divisi14063676e_zero(X_a) -> (all A_3 (hAPP(X_a,X_a,inverse_inverse(X_a),A_3) = zero_zero(X_a) <-> ti(X_a,A_3) = zero_zero(X_a))))) # label(fact_2574_inverse__nonzero__iff__nonzero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4621 (all Q all P (-hBOOL(P) | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q)) | -hBOOL(Q))) # label(help_fconj_1_1_U) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4622 (all X_a (semiring(X_a) -> (all A_1 all E_1 all B all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),E_1)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),E_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),E_1)),C_1))))) # label(fact_381_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4623 (all X_a all X_b all P_5 all Q_3 (hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),Q_3) = hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),P_5) -> (hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),Q_3) = hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),P_5) -> ti(product_prod(X_a,X_b),P_5) = ti(product_prod(X_a,X_b),Q_3)))) # label(fact_3160_prod__eqI) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4624 (all X_a all X_b (real_normed_vector(X_b) -> (all F all A_3 all A_2 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),uminus_uminus(X_b)),F)),hAPP(X_b,X_b,uminus_uminus(X_b),A_3)),A_2)) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),A_3),A_2)))))) # label(fact_4478_tendsto__minus__cancel) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4625 (all X_a (real_normed_vector(X_a) -> (all Xa all A_3 hBOOL(hAPP(real,bool,hAPP(fun(real,X_a),fun(real,bool),isCont(real,X_a),hAPP(X_a,fun(real,X_a),hAPP(fun(real,fun(X_a,X_a)),fun(X_a,fun(real,X_a)),combc(real,X_a,X_a),scaleR_scaleR(X_a)),Xa)),A_3))))) # label(fact_4936_scaleR__left_OisCont) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4626 (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),hAPP(int,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)),hAPP(int,X_a,number_number_of(X_a),Ya))))))) # label(fact_241_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4627 (all A_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B)) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),B) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B)))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B)) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A_1)))) # label(fact_1650_posDivAlg__eqn) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4628 (all X_a (ordere236663937imp_le(X_a) -> (all X all Y all Z_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),Z_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Z_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Y),Z_1))))) # label(fact_4841_min__add__distrib__left) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4629 (all Ma all A_3 all B_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),B_2)) -> -hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),B_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,A_3),Ma))))) # label(fact_3232_Bnor__mem__zle__swap) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4630 (all K (min = ti(int,K) <-> min = hAPP(int,int,bit1,K))) # label(fact_667_rel__simps_I47_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4631 (all M all K_4 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),J_2)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),K_4))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K_4)),hAPP(nat,nat,suc,J_2)))) # label(fact_1453_diff__Suc__diff__eq1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4632 -(hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int)))) -> -hBOOL(hAPP(int,bool,twoSqu1091635170sum2sq,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))))) # label(fact_580_smaller_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4633 (all V ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls)) -> hAPP(int,nat,number_number_of(nat),hAPP(int,int,succ,V)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),one_one(nat))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),one_one(nat)) = one_one(nat)))) # label(fact_338_nat__number__of__add__1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.87 4634 (all Ma all X_2 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),resSet,Ma),X_2)) <-> (all Y1 all Y2 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Y1),X_2)) & hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Y2),X_2)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Y1),Y2),Ma)) -> ti(int,Y2) = ti(int,Y1))))) # label(fact_2941_ResSet__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4635 (all X_a (divisi14063676e_zero(X_a) -> hAPP(X_a,X_a,inverse_inverse(X_a),zero_zero(X_a)) = zero_zero(X_a))) # label(fact_2573_inverse__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4636 (all X_a (real_normed_vector(X_a) & ring_1(X_a) -> (all C all Xa (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),C))),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa)))) -> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),semiring_1_of_nat(X_a))),C))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),minus_minus(nat)),hAPP(nat,nat,suc,zero_zero(nat)))))),hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),C))),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))))))))) # label(fact_4134_diffs__equiv) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4637 (all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) -> (A_1 != B -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))))) # label(fact_953_dvd_Ole__neq__trans) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4638 (all X_b all X_a all P_4 hAPP(product_prod(X_b,X_a),X_a,product_snd(X_b,X_a),P_4) = hAPP(product_prod(X_b,X_a),X_a,hAPP(fun(X_b,fun(X_a,X_a)),fun(product_prod(X_b,X_a),X_a),product_prod_case(X_b,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_b,fun(X_a,X_a)),combk(fun(X_a,X_a),X_b),combi(X_a))),P_4)) # label(fact_4034_snd__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4639 (all X_a all A_2 hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2) = hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),hAPP(nat,fun(X_a,nat),combk(nat,X_a),one_one(nat))),A_2)) # label(fact_4282_card__eq__setsum) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4640 (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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ma),Na)),Ma)) <-> one_one(nat) = Na))) # label(fact_879_dvd__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4641 (all X_a (order(X_a) -> (all Xa all Ya (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ya),Xa)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),Ya)) <-> ti(X_a,Xa) = ti(X_a,Ya))))) # label(fact_512_order__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4642 (all M all N ((hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)) != N -> hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,M)),N)) & (N = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,M)),N) = zero_zero(nat)))) # label(fact_1532_mod__Suc) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4643 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),zero_zero(int)))))) # label(fact_1105_div__nonpos__pos__le0) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4644 (all X_a (real_normed_algebra(X_a) -> (exists K_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2)) & (all A_4 all B_4 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_4),B_4))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),A_4)),hAPP(X_a,real,norm_norm(X_a),B_4))),K_2)))))))) # label(fact_2108_mult_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4645 (all X_a (real_normed_vector(X_a) -> (all X_2 all K_3 all N_3 ((all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_3),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,X_2,N_1))),K_3)))) -> hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_2)))))) # label(fact_3524_BseqI2_H) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4646 (all X_a (real_n1866405975lgebra(X_a) -> (all A_1 (ti(X_a,A_1) != zero_zero(X_a) -> hAPP(real,real,inverse_inverse(real),hAPP(X_a,real,norm_norm(X_a),A_1)) = hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)))))) # label(fact_2580_nonzero__norm__inverse) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4647 (all X_a (ord(X_a) -> min_1(X_a) = ti(fun(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,X_a))),min_1(X_a)))) # label(tsy_c_Orderings_Oord_Omin_res) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4648 (all M all N all R_4 all R_3 all S_4 all S_3 all T_6 all T_5 all K_4 ((exists Sn exists Tn hAPP(product_prod(int,int),product_prod(int,product_prod(int,int)),hAPP(int,fun(product_prod(int,int),product_prod(int,product_prod(int,int))),product_Pair(int,product_prod(int,int)),K_4),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Sn),Tn)) = hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),hAPP(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))),hAPP(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))))),xzgcda,M),N),R_4),R_3),S_4),S_3),T_6),T_5)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),R_3)) -> ti(int,K_4) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,R_4),R_3)))) # label(fact_3502_xzgcd__correct__aux2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4649 (all P_5 all Q_3 (hBOOL(hAPP(nat,bool,prime,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),P_5),Q_3))) -> one_one(nat) = P_5 | Q_3 = one_one(nat))) # label(fact_3080_prime__product) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4650 (all X_a (real_normed_vector(X_a) -> (all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(X_a,real,norm_norm(X_a),X)))))) # label(fact_1974_norm__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4651 (all X_a (real_normed_vector(X_a) -> (all X_2 ((all E (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E)) -> (exists M_3 all M_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),M_2)) -> (all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,X_2,M_2)),hAPP(nat,X_a,X_2,N_1)))),E)))))))) -> hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_2)))))) # label(fact_3342_CauchyI) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4652 (all K_4 hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),one_one(rat)),hAPP(int,rat,number_number_of(rat),K_4)) = hAPP(product_prod(int,int),rat,frct,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),hAPP(int,int,number_number_of(int),K_4)))) # label(fact_3723_Frct__code__post_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4653 (all A_1 all B ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))) -> (zero_zero(int) = ti(int,B) | zero_zero(int) = ti(int,A_1) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,A_1),B)) & (-(zero_zero(int) = ti(int,B) | zero_zero(int) = ti(int,A_1)) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),int_gcd,A_1),B))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),hAPP(int,int,hAPP(int,fun(int,int),int_gcd,A_1),B))) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,A_1),B))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,A_1),B)))) # label(fact_3952_norm__frac_Osimps) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4654 (all X_a (real_normed_vector(X_a) -> (all X hAPP(X_a,real,norm_norm(X_a),X) = hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X))))) # label(fact_1925_norm__minus__cancel) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4655 (all A_1 all N all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),N)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),N))) -> (zero_zero(nat) != N -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_1),B))))) # label(fact_1235_pow__divides__pow__int) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4656 (all X_a all X_b (real_normed_vector(X_b) -> (all G all B_2 all F all A_3 all A_2 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),A_3),A_2)) -> (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),B_2),A_2)) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),plus_plus(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_3),B_2)),A_2))))))) # label(fact_4500_tendsto__add) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4657 (all X_a all Q_1 all P_1 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),P_1))) | hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),Q_1))) -> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),P_1)),Q_1)))))) # label(fact_3842_finite__Collect__conjI) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4658 (all Na all Xa hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),power_power(real)),Na)),Xa),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),Na)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),hAPP(nat,nat,suc,zero_zero(nat)))))))) # label(fact_3920_DERIV__pow) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4659 (all X_b all X_a all X_c (comm_monoid_add(X_c) -> (all F all R_2 all T_3 all S (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),T_3)) -> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),hAPP(fun(X_b,fun(X_a,bool)),fun(X_b,X_c),hAPP(fun(X_b,fun(fun(X_a,bool),X_c)),fun(fun(X_b,fun(X_a,bool)),fun(X_b,X_c)),combs(X_b,fun(X_a,bool),X_c),hAPP(fun(X_b,fun(X_a,X_c)),fun(X_b,fun(fun(X_a,bool),X_c)),hAPP(fun(fun(X_a,X_c),fun(fun(X_a,bool),X_c)),fun(fun(X_b,fun(X_a,X_c)),fun(X_b,fun(fun(X_a,bool),X_c))),combb(fun(X_a,X_c),fun(fun(X_a,bool),X_c),X_b),big_co1399186613setsum(X_a,X_c)),hAPP(fun(X_a,fun(X_b,X_c)),fun(X_b,fun(X_a,X_c)),combc(X_a,X_b,X_c),F))),hAPP(fun(X_b,fun(X_a,bool)),fun(X_b,fun(X_a,bool)),hAPP(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_b,fun(X_a,bool)),fun(X_b,fun(X_a,bool))),combb(fun(X_a,bool),fun(X_a,bool),X_b),collect(X_a)),hAPP(fun(X_b,fun(X_a,bool)),fun(X_b,fun(X_a,bool)),hAPP(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_b,fun(X_a,bool)),fun(X_b,fun(X_a,bool))),combb(fun(X_a,bool),fun(X_a,bool),X_b),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),S)))),hAPP(fun(X_a,fun(X_b,bool)),fun(X_b,fun(X_a,bool)),combc(X_a,X_b,bool),R_2))))),T_3) = hAPP(fun(X_a,bool),X_c,hAPP(fun(X_a,X_c),fun(fun(X_a,bool),X_c),big_co1399186613setsum(X_a,X_c),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,X_c),hAPP(fun(X_a,fun(fun(X_b,bool),X_c)),fun(fun(X_a,fun(X_b,bool)),fun(X_a,X_c)),combs(X_a,fun(X_b,bool),X_c),hAPP(fun(X_a,fun(X_b,X_c)),fun(X_a,fun(fun(X_b,bool),X_c)),hAPP(fun(fun(X_b,X_c),fun(fun(X_b,bool),X_c)),fun(fun(X_a,fun(X_b,X_c)),fun(X_a,fun(fun(X_b,bool),X_c))),combb(fun(X_b,X_c),fun(fun(X_b,bool),X_c),X_a),big_co1399186613setsum(X_b,X_c)),F)),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_a),collect(X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_a),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),T_3)))),R_2)))),S)))))) # label(fact_4210_setsum__setsum__restrict) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4660 (all M all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N)),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N)) # label(fact_1550_Divides_Omod__div__equality_H) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4661 (all B all D_3 all A_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_3),A_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A_1),B))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_3),B))))) # label(fact_825_divides__add__revr) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4662 (all Y all X (hBOOL(hAPP(int,bool,nat_is_nat,X)) -> (hBOOL(hAPP(int,bool,nat_is_nat,Y)) -> hBOOL(hAPP(int,bool,nat_is_nat,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),Y)))))) # label(fact_4871_Divides_Otransfer__int__nat__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4663 (all X hAPP(int,real,real_1(int),hAPP(int,int,abs_abs(int),X)) = hAPP(real,real,abs_abs(real),hAPP(int,real,real_1(int),X))) # label(fact_3368_real__of__int__abs) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4664 (all A_1 all B ((hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) != zero_zero(int) -> hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),A_1)),B)) & (zero_zero(int) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) -> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),A_1)),B) = zero_zero(int)))) # label(fact_2089_zmod__zminus1__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4665 (all X_a (number_ring(X_a) -> (all Xa (hAPP(int,X_a,number_number_of(X_a),Xa) = zero_zero(X_a) <-> hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xa),hAPP(int,int,uminus_uminus(int),pls))))))))) # label(fact_2183_eq__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4666 (all L all U hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_gr676655987ssThan(int),L),U)))) # label(fact_3695_finite__greaterThanLessThan__int) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4667 (all A_1 all B all M hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),B) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),M)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),M)))),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),M)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),M)))) # label(fact_1459_zcong__zmod__aux) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4668 (all Na (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Na)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,nat_1,Na)),one_one(nat)) = hAPP(fun(int,bool),nat,finite_card(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less_eq(int)),Na)))))) # label(fact_3911_card__bdd__int__set__le) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4669 (all Va all V_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,number_number_of(nat),Va)),hAPP(int,nat,number_number_of(nat),V_1))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Va),V_1)) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Va),V_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),V_1))))) # label(fact_154_less__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4670 (all Q_3 all R_3 all B all C_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),R_3),B)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Q_3),C_1))),R_3)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C_1)))))) # label(fact_1553_mod__lemma) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4671 (all Z_1 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z_1),zero_zero(int)) = ti(int,Z_1)) # label(fact_82_zadd__0__right) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4672 -(all R_1 all V_3 -(hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,abs_abs(int),V_3))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))) & ti(int,V_3) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),x),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),R_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n)))))) # label(fact_925__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]. 7.88/7.88 4673 (all X_a (real_normed_algebra(X_a) -> (exists K_2 all A_4 all B_4 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_4),B_4))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),A_4)),hAPP(X_a,real,norm_norm(X_a),B_4))),K_2)))))) # label(fact_2112_mult_Obounded) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4674 (all X_a all X_b product_Pair(X_a,X_b) = ti(fun(X_a,fun(X_b,product_prod(X_a,X_b))),product_Pair(X_a,X_b))) # label(tsy_c_Product__Type_OPair_res) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4675 (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),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Xa),Ya)),zero_zero(real))))) # label(fact_830_real__le__eq__diff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4676 (all K_4 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),K_4)),K_4) = hAPP(int,int,bit1,K_4)) # label(fact_95_Bit1__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4677 (all X hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),pi)) = hAPP(real,real,tan,X)) # label(fact_2392_tan__periodic__pi) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4678 (all N all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> (zero_zero(real) = X -> zero_zero(real) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N)) & (zero_zero(real) != X -> hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N) = hAPP(real,real,hAPP(real,fun(real,real),powr,X),hAPP(nat,real,real_1(nat),N)))))) # label(fact_3445_powr__realpow2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4679 (all X_a (dist_norm(X_a) -> (all X all Y hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y) = hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y))))) # label(fact_3540_dist__norm) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4680 (all X_a (banach(X_a) & real_normed_field(X_a) -> (all Z all F all Xa (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),F)),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa)))) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),Z)),hAPP(X_a,real,norm_norm(X_a),Xa))) -> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),F)),hAPP(X_a,fun(nat,X_a),power_power(X_a),Z))))))))) # label(fact_4051_powser__inside) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4681 (all X_a (linordered_field(X_a) -> (all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1))))))) # label(fact_2586_inverse__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4682 (all X_a (archim1727834104eiling(X_a) -> (all Va all Xa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),Va)),Xa)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),Va)),hAPP(X_a,int,archimedean_ceiling(X_a),Xa))))))) # label(fact_3328_number__of__less__ceiling) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4683 (all X_a member(X_a) = ti(fun(X_a,fun(fun(X_a,bool),bool)),member(X_a))) # label(tsy_c_member_res) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4684 (all X_a (semiring_div(X_a) -> (all A_1 all B all C_1 ((ti(X_a,C_1) != zero_zero(X_a) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B))) & (zero_zero(X_a) = ti(X_a,C_1) -> zero_zero(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),B))))))) # label(fact_1089_div__mult__mult1__if) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4685 (all Ma all Na hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),Ma),hAPP(nat,nat,suc,Na)) = hAPP(nat,nat,hAPP(fun(nat,nat),fun(nat,nat),hAPP(nat,fun(fun(nat,nat),fun(nat,nat)),nat_case(nat),hAPP(nat,nat,suc,Na)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),suc),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),ord_max(nat)),Na))),Ma)) # label(fact_4987_max__Suc2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4686 (all X_a (linordered_semidom(X_a) -> (all A_1 all N all N_4 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),N_4)) -> (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_4)))))))) # label(fact_216_power__increasing) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4687 (all Na all G all Xa all Ma (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),G),Xa),Ma)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),G)),Na)),Xa),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),Na)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,G,Xa)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))))),Ma))))) # label(fact_3909_DERIV__fun__pow) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4688 (all X_a (linordered_field(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(X_a),zero_zero(X_a)),Y)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)))))))) # label(fact_1860_divide__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4689 (all X 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),zero_zero(nat)),N)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) -> hAPP(real,real,hAPP(nat,fun(real,real),root,M),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X)) = hAPP(real,real,hAPP(nat,fun(real,real),root,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)),X))))) # label(fact_2711_real__root__pos__mult__exp) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4690 (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),A_1),zero_zero(X_a))) -> hAPP(X_a,X_a,abs_abs(X_a),A_1) = hAPP(X_a,X_a,uminus_uminus(X_a),A_1))))) # label(fact_2002_abs__of__neg) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4691 (all I all P_1 all K (hBOOL(hAPP(nat,bool,P_1,K)) -> ((all N_1 (hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,suc,N_1))) -> hBOOL(hAPP(nat,bool,P_1,N_1)))) -> hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),K),I)))))) # label(fact_1693_zero__induct__lemma) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4692 (all X_a all F all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2)) -> hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),A_2)),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),B_1))))) # label(fact_4287_setsum__diff__nat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4693 (all A_1 all B all C_1 hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A_1),B)),C_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C_1))) # label(fact_1158_div__mult2__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4694 (all X_a (number_ring(X_a) -> (all Xa (hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xa),hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit1,pls)))))) <-> one_one(X_a) = hAPP(int,X_a,number_number_of(X_a),Xa))))) # label(fact_2118_eq__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4695 (all Xa all K (hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),K) = ti(int,Xa) -> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)))) # label(fact_1733_zEvenI) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4696 (all A_3 all B_2 all Na (Na != zero_zero(nat) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_3),Na)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B_2),Na))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_3),B_2))))) # label(fact_1234_pow__divides__eq__int) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4697 (all N all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),N)),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),N)),one_one(real)))) -> hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N),one_one(int)) = hAPP(real,int,archimedean_ceiling(real),X)))) # label(fact_3343_ceiling__eq2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4698 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B)))))) # label(fact_1979_abs__le__D2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4699 (all V_2 all K_4 all V ((-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V))) -> (-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V_2))) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V_2)),K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),V_2))),K_4)) & (hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V_2))) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V_2)),K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),K_4))) & (hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V))) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V_2)),K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V_2)),K_4)))) # label(fact_1621_nat__number__of__add__left) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4700 (all Na all Ma (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),Na)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),Ma)),one_one(real)))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Na),Ma)))) # label(fact_3417_int__le__real__less) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4701 (all X_a (real_algebra_1(X_a) -> (all Xa all Ya (Ya = Xa <-> hAPP(real,X_a,of_real(X_a),Ya) = hAPP(real,X_a,of_real(X_a),Xa))))) # label(fact_2791_of__real__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4702 (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Xa)) <-> hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,sqrt,Xa)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = Xa)) # label(fact_2167_real__sqrt__pow2__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4703 (all V all W ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W))) -> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,int,number_number_of(int),W)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,V))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,W)))) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W))) -> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),V)),one_one(int))),hAPP(int,int,number_number_of(int),W)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,V))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,W)))))) # label(fact_1128_zdiv__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4704 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) -> hAPP(real,real,hAPP(real,fun(real,real),log,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),log,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,exp(real),one_one(real)))),hAPP(real,real,ln,X)))) # label(fact_2047_log__base__10__eq2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4705 (all R_3 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),R_3),hAPP(int,real,real_1(int),hAPP(real,int,archimedean_ceiling(real),R_3))))) # label(fact_3370_real__of__int__ceiling__ge) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4706 (all Diff all F all Na all H (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),H),zero_zero(real))) & hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F & (all M_2 all T_4 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),H),T_4)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_4),zero_zero(real))) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),Na)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_2)),T_4),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_2)),T_4))))) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (exists T_4 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),H),T_4)) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),T_4),zero_zero(real))) & hAPP(real,real,F,H) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),H))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,Na),T_4)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),H),Na))))))) # label(fact_4310_Maclaurin__minus__objl) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4707 (all A_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_1),one_one(nat)))) # label(fact_2883_coprime__1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4708 (all X_a (comm_monoid_add(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),zero_zero(X_a)),A_1)))) # label(fact_322_add__0) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4709 (all Code_numeral_2 hAPP(code_code_numeral,nat,size_size(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,Code_numeral_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(code_code_numeral,nat,size_size(code_code_numeral),Code_numeral_2)),hAPP(nat,nat,suc,zero_zero(nat)))) # label(fact_2994_code__numeral_Osize_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4710 (all Y all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),Y)),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) # label(fact_2180_real__sqrt__ge__abs2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4711 (all Xa all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa),Ya)),zero_zero(real))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Ya),hAPP(real,real,uminus_uminus(real),Xa))))) # label(fact_2021_real__add__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4712 (all N all M hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),M),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M),N)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),N),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),M),N))) # label(fact_1435_zmult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4713 (all Z_4 all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1)) -> hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z_1),Z_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,nat_1,Z_1)),hAPP(int,nat,nat_1,Z_4)))) # label(fact_1041_nat__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4714 (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),hAPP(int,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),hAPP(int,int,bit1,pls))))))) # label(fact_240_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4715 (all X_b all X_a (ordere142940540dd_abs(X_a) -> (all F all A_2 hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),abs_abs(X_a)),F)),A_2) = hAPP(X_a,X_a,abs_abs(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),abs_abs(X_a)),F)),A_2))))) # label(fact_4246_abs__setsum__abs) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4716 (all X_a (banach(X_a) -> (all F (hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),F))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),F))),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),F)))))))) # label(fact_4050_summable__norm) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4717 (all X_a (metric_space(X_a) -> (all Xa all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Xa),Ya))) <-> ti(X_a,Xa) != ti(X_a,Ya))))) # label(fact_3537_zero__less__dist__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4718 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),zero_zero(int)))))) # label(fact_1080_div__neg__pos__less0) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4719 (all P all Q (-hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fdisj,P),Q)) | hBOOL(Q) | hBOOL(P))) # label(help_fdisj_3_1_U) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4720 (all T_1 ord(filter(T_1))) # label(arity_Limits_Ofilter___Orderings_Oord) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4721 (all X_a (real_normed_field(X_a) -> (all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),Xa)),one_one(real))) -> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa))))))) # label(fact_4052_summable__geometric) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4722 (all X_a all X_b all A_2 all B_1 (hBOOL(hAPP(fun(sum_sum(X_a,X_b),bool),bool,finite_finite_1(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_2),B_1))) <-> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_1)) & hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)))) # label(fact_3207_finite__Plus__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4723 (all X_a (real_vector(X_a) -> (all X all Y all Xa_2 hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y)),Xa_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),X),Xa_2)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Y),Xa_2))))) # label(fact_4696_scaleR__left_Odiff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4724 (all X_a (monoid_mult(X_a) -> (all Y all P_5 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),P_5),N)) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),P_5))),Y) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,N)),P_5)))))) # label(fact_1463_lemma__realpow__diff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4725 (all K all F all Na all Ma ((all I_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),Na)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),I_2)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_2),one_one(nat)))),hAPP(nat,int,F,I_2)))),one_one(int))))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),Na)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,F,Ma)),K)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),hAPP(nat,int,F,Na))) -> (exists I_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),Na)) & hAPP(nat,int,F,I_2) = ti(int,K) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),I_2))))))))) # label(fact_1068_nat__intermed__int__val) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4726 (all L_4 min != hAPP(int,int,bit0,L_4)) # label(fact_670_rel__simps_I42_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4727 (all X_a (real_normed_algebra(X_a) -> (all Xa_2 all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_2),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_2),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y))))) # label(fact_1249_mult__right_Odiff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4728 (all P_1 ((exists X1 hBOOL(hAPP(nat,bool,P_1,X1))) <-> (exists X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) & hBOOL(hAPP(nat,bool,P_1,hAPP(int,nat,nat_1,X_1))))))) # label(fact_1003_ex__nat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4729 (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_834_nat__dvd__not__less) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4730 (all X_a (real_normed_vector(X_a) -> (all Ra all A_3 hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_a),fun(X_a,bool),isCont(X_a,X_a),hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra)),A_3))))) # label(fact_4937_scaleR__right_OisCont) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4731 (all X all Y (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Y)) -> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),Y),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),Y)) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y))) # label(fact_4419_gcd__non__0__int) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4732 (all A_1 ((zero_zero(real) != A_1 -> (-hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1)) -> hAPP(real,real,uminus_uminus(real),one_one(real)) = hAPP(real,real,sgn_sgn(real),A_1)) & (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1)) -> one_one(real) = hAPP(real,real,sgn_sgn(real),A_1))) & (zero_zero(real) = A_1 -> zero_zero(real) = hAPP(real,real,sgn_sgn(real),A_1)))) # label(fact_2313_sgn__real__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4733 (all X_a (linordered_semidom(X_a) -> (all A_1 all N all B (hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),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)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) -> ti(X_a,B) = ti(X_a,A_1)))))))) # label(fact_304_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4734 (all N all X (-hBOOL(hAPP(int,bool,even_odd_even(int),X)) -> -hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),N))))) # label(fact_1602_odd__pow) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4735 (all X_a (real_normed_field(X_a) -> (all F (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),zero_zero(X_a)),sequentially)))))) # label(fact_4514_summable__LIMSEQ__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4736 (all Xa all Ya (hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,number_number_of(int),Ya)),hAPP(int,int,number_number_of(int),Xa)) = zero_zero(int) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,number_number_of(int),Xa)),hAPP(int,int,number_number_of(int),Ya))))) # label(fact_1429_zdvd__iff__zmod__eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4737 (all X all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),P_5)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),standardRes,P_5),X))))) # label(fact_1467_StandardRes__lbound) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4738 (all N all I_1 ((I_1 != zero_zero(code_code_numeral) -> hAPP(nat,nat,hAPP(code_code_numeral,fun(nat,nat),code_nat_of_aux,hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),minus_minus(code_code_numeral),I_1),one_one(code_code_numeral))),hAPP(nat,nat,suc,N)) = hAPP(nat,nat,hAPP(code_code_numeral,fun(nat,nat),code_nat_of_aux,I_1),N)) & (zero_zero(code_code_numeral) = I_1 -> N = hAPP(nat,nat,hAPP(code_code_numeral,fun(nat,nat),code_nat_of_aux,I_1),N)))) # label(fact_1743_nat__of__aux__code) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4739 (all X_a (linorder(X_a) -> (all Xa all Ya all Z (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Xa),Ya)),Z)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Xa),Z)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ya),Z)))))) # label(fact_4823_min__less__iff__disj) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4740 (all X ti(int,X) = hAPP(int,int,succ,hAPP(int,int,pred,X))) # label(fact_2924_succ__pred) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4741 (all K_4 all L_4 ((ti(int,L_4) = zero_zero(int) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),pdivmod,K_4),L_4) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),hAPP(int,int,abs_abs(int),K_4))) & (zero_zero(int) != ti(int,L_4) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,hAPP(int,int,abs_abs(int),K_4)),hAPP(int,int,abs_abs(int),L_4)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),pdivmod,K_4),L_4)))) # label(fact_1654_pdivmod__posDivAlg) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4742 (all X_a (real_normed_vector(X_a) -> (exists K_2 ((all A_4 all B_4 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_4),B_4))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),A_4)),hAPP(X_a,real,norm_norm(X_a),B_4))),K_2)))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2)))))) # label(fact_4778_scaleR_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4743 (all Q_3 ((hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),Q_3),zero_zero(rat))) -> hAPP(rat,rat,abs_abs(rat),Q_3) = hAPP(rat,rat,uminus_uminus(rat),Q_3)) & (-hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),Q_3),zero_zero(rat))) -> hAPP(rat,rat,abs_abs(rat),Q_3) = Q_3))) # label(fact_3734_abs__rat__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4744 (all X all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),pi)),Y)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),X)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real))) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,cos,Y)),hAPP(real,real,cos,X))))))) # label(fact_2318_cos__monotone__minus__pi__0) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4745 (all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(real,real,abs_abs(real),X))))) # label(fact_1123_abs__add__one__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4746 (all I 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),I),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),K))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K)),J))))) # label(fact_849_le__diff__conv2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4747 (all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y)) -> zero_zero(real) = hAPP(real,real,cos,hAPP(complex,real,arg,hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,zero_zero(real)),Y))))) # label(fact_2770_cos__arg__i__mult__zero__pos) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4748 (all Xa all Ya all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Ya))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya))))) # label(fact_2681_real__root__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4749 (all F all G all Na hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),F),hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),G),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))) = hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(fun(nat,real),fun(nat,fun(real,real))),combs(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),F)),G)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Na)))) # label(fact_4262_sum__split__even__odd) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4750 (all Xa all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)),one_one(real))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),one_one(real)))))) # label(fact_2702_real__root__le__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4751 (all X_a (real_normed_vector(X_a) -> (all Y_2 all B_2 all X_2 all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_2),A_3),sequentially)) -> (hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),Y_2),B_2),sequentially)) -> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),X_2)),Y_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_2)),sequentially))))))) # label(fact_4502_LIMSEQ__diff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4752 (all X_a (number_ring(X_a) -> (all Z_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z_1),Z_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) # label(fact_543_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4753 (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_283_int__le__0__conv) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4754 (all X_a (comm_semiring_1(X_a) -> (all Lx all Rx all Ry hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),Ry)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ry))))) # label(fact_359_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4755 (all M all N (ti(int,N) != zero_zero(int) -> hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),M),N)),N) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,M),N)),one_one(rat)))) # label(fact_3811_Fract__add__one) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4756 (all X_a all X_b (comm_monoid_add(X_b) -> (all G all H all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),plus_plus(X_b)),G)),H)),A_2) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),H),A_2)))))) # label(fact_4245_setsum_Odistrib) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4757 (all A_1 all B (zero_zero(nat) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B) -> (exists A_5 exists B_3 (one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_5),B_3) & hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B_3),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)) = B & hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)) = A_1)))) # label(fact_3337_gcd__coprime__exists__nat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4758 (all V ((hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V))) -> hAPP(nat,real,real_1(nat),hAPP(int,nat,number_number_of(nat),V)) = zero_zero(real)) & (-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V))) -> hAPP(int,real,number_number_of(real),V) = hAPP(nat,real,real_1(nat),hAPP(int,nat,number_number_of(nat),V))))) # label(fact_2423_real__of__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4759 (all X_a (number_ring(X_a) -> (all W hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(int,X_a,number_number_of(X_a),W)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,uminus_uminus(int),W))))) # label(fact_1928_number__of__minus) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4760 (all Qr_2 all M all N all Qr_1 (hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M),N),Qr_1)) -> (hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M),N),Qr_2)) -> Qr_1 = Qr_2))) # label(fact_3123_divmod__nat__rel__unique) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4761 (all M all K_4 all N (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N)) -> (ti(int,K_4) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N) -> (exists S_1 exists T_4 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),S_1),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),T_4),N)) = ti(int,K_4))))) # label(fact_3507_zgcd__ex__linear) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4762 (all X_b (uminus(X_b) -> uminus_uminus(X_b) = ti(fun(X_b,X_b),uminus_uminus(X_b)))) # label(tsy_c_Groups_Ouminus__class_Ouminus_res) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4763 (all X_a (number_ring(X_a) & field_inverse_zero(X_a) -> (all X hAPP(X_a,X_a,uminus_uminus(X_a),X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),hAPP(int,X_a,number_number_of(X_a),min))))) # label(fact_2018_divide__minus1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4764 (all X_b all X_a (order(X_b) & order(X_a) -> (all C all F all B_2 all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,B_2)),A_3)) -> (hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),C),B_2)) -> ((all X_1 all Y_1 (hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_1),X_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,Y_1)),hAPP(X_b,X_a,F,X_1))))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,C)),A_3)))))))) # label(fact_2841_xt4) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4765 (all B hAPP(int,int,hAPP(int,fun(int,int),div_div(int),zero_zero(int)),B) = zero_zero(int)) # label(fact_1072_zdiv__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4766 (all K_4 all M all N hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),N))) # label(fact_1528_mod__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4767 (all X_a (division_ring(X_a) -> (all A_1 (zero_zero(X_a) != ti(X_a,A_1) -> ti(X_a,A_1) = hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)))))) # label(fact_2576_nonzero__inverse__inverse__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4768 (all X_a (semiring_1(X_a) -> (all Z (ti(X_a,Z) = zero_zero(X_a) <-> hBOOL(hAPP(X_a,bool,iszero(X_a),Z)))))) # label(fact_2146_iszero__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4769 (all X_a (abs_if(X_a) -> (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))) -> hAPP(X_a,X_a,abs_abs(X_a),A_1) = ti(X_a,A_1)) & (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a))) -> hAPP(X_a,X_a,abs_abs(X_a),A_1) = hAPP(X_a,X_a,uminus_uminus(X_a),A_1)))))) # label(fact_2003_abs__if) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4770 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),A_1)) -> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = ti(int,A_1)))) # label(fact_1423_mod__neg__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4771 (all X_a (semiring_div(X_a) -> (all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_3),B_2)) <-> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B_2),A_3) = zero_zero(X_a))))) # label(fact_1357_dvd__eq__mod__eq__0) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4772 (all Ta all A_2 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ta),A_2)) -> (all X_1 ((all Xa_1 (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D))) -> (all Xb (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2)) -> hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa_1) != ti(int,X_1))))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),Ta)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),D)),Ta)))))))) # label(fact_4664_aset_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4773 (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,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N))))))) # label(fact_182_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4774 (all X_a (linorder(X_a) -> (all A_3 all B_2 all C all D_1 (hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),A_3),B_2) = hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),C),D_1) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_1)) -> ti(X_a,A_3) = ti(X_a,C))))))) # label(fact_4223_atLeastLessThan__inj_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4775 (all X_a (comm_semiring_1(X_a) -> (all X all Q_3 hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,suc,Q_3)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),Q_3))))) # label(fact_1368_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4776 (all X_a all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1)) -> (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)),hAPP(fun(X_a,bool),nat,finite_card(X_a),B_1)) = hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1))))) # label(fact_2736_card__Diff__subset) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4777 (all X_a (ordere236663937imp_le(X_a) -> (all A_1 all C_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C_1))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)))))) # label(fact_326_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4778 (all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),one_one(real))) -> hAPP(real,real,uminus_uminus(real),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),X))))) = hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),X)))) # label(fact_2041_aux5) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4779 (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)) -> hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Y),hAPP(real,real,ln,X)) = hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),powr,X),Y))))) # label(fact_3430_ln__powr) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4780 (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_1 (hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),R_1),N) = A_1 & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1))))))) # label(fact_341_realpow__pos__nth) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4781 (all X_a all A_2 all B_1 (ti(fun(X_a,bool),B_1) = ti(fun(X_a,bool),A_2) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2)))) # label(fact_2207_equalityD2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4782 (all X_a (semiring_div(X_a) -> (all A_1 all C_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C_1)),B)),C_1)))) # label(fact_1291_zmod__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4783 (all A_3 all D_1 all Na hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),Na)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_3),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)))),D_1)))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(fun(nat,bool),int,hAPP(fun(nat,int),fun(fun(nat,bool),int),big_co1399186613setsum(nat,int),hAPP(fun(nat,int),fun(nat,int),hAPP(fun(int,int),fun(fun(nat,int),fun(nat,int)),combb(int,int,nat),hAPP(int,fun(int,int),plus_plus(int),A_3)),hAPP(int,fun(nat,int),hAPP(fun(nat,fun(int,int)),fun(int,fun(nat,int)),combc(nat,int,int),hAPP(fun(nat,int),fun(nat,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(nat,int),fun(nat,fun(int,int))),combb(int,fun(int,int),nat),times_times(int)),semiring_1_of_nat(int))),D_1))),hAPP(nat,fun(nat,bool),ord_lessThan(nat),Na)))) # label(fact_4338_arith__series__int) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4784 (all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,sin,X))),one_one(real)))) # label(fact_2288_abs__sin__le__one) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4785 (all X_a all X_b (real_n2089651433ebra_1(X_b) & metric_space(X_a) -> (all A_3 all K all L_2 (ti(X_a,L_2) != ti(X_a,K) -> -hBOOL(hAPP(filter(X_b),bool,hAPP(X_a,fun(filter(X_b),bool),hAPP(fun(X_b,X_a),fun(X_a,fun(filter(X_b),bool)),tendsto(X_b,X_a),hAPP(X_a,fun(X_b,X_a),combk(X_a,X_b),K)),L_2),hAPP(X_b,filter(X_b),at(X_b),A_3))))))) # label(fact_4582_LIM__const__not__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4786 (all I_1 all K_4 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),K_4)) -> (hAPP(nat,nat,suc,I_1) != K_4 -> -(all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_1)) -> K_4 != hAPP(nat,nat,suc,J_1)))))) # label(fact_2753_lessE) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4787 (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)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) # label(fact_1181_Suc__n__div__2__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4788 (all X_a (dvd(X_a) -> (all A_1 all B all K_4 (ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),K_4) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B),A_1)))))) # label(fact_724_dvdI) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4789 (all I all J hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),hAPP(nat,nat,suc,I)),hAPP(nat,nat,suc,J)) = hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,nat),fun(fun(nat,bool),fun(nat,bool)),image(nat,nat),suc),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),I),J))) # label(fact_4252_image__Suc__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4790 (all P (-hBOOL(hAPP(bool,bool,fNot,P)) | -hBOOL(P))) # label(help_fNot_1_1_U) # label(axiom) # label(non_clause). [assumption]. 7.88/7.88 4791 (all X_a (real_normed_vector(X_a) -> (all R_3 exists K_2 ((all X_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_3),X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_1)),K_2)))) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2)))))) # label(fact_4785_scaleR__right_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4792 (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_174_of__nat__1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4793 (all I_1 all M hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),I_1))))) # label(fact_1384_less__add__Suc2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4794 (all X hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),zero_zero(int)) = hAPP(int,int,abs_abs(int),X)) # label(fact_4381_gcd__0__int) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4795 (all X_a (division_ring(X_a) -> (all A_1 all B (one_one(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B) -> ti(X_a,B) = hAPP(X_a,X_a,inverse_inverse(X_a),A_1))))) # label(fact_2596_inverse__unique) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4796 (all X_a all X_b (topological_space(X_b) -> (all K all A_2 hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),combk(X_b,X_a),K)),K),A_2))))) # label(fact_4486_tendsto__const) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4797 (all X_a (semiring_div(X_a) -> (all A_1 all B ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B)),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B))))) # label(fact_1520_mod__div__decomp) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4798 (all X_a (comm_monoid_add(X_a) & topological_space(X_a) -> (all Sb all F (hAPP(nat,X_a,F,zero_zero(nat)) = zero_zero(X_a) -> (hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),suc)),Sb)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),Sb))))))) # label(fact_4121_sums__Suc__imp) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4799 (all D_1 hAPP(int,fun(product_prod(int,int),bool),int_ge_less_than,D_1) = hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool),collect(product_prod(int,int)),hAPP(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_prod_case(int,int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),hAPP(fun(int,fun(fun(int,bool),fun(int,bool))),fun(fun(int,fun(int,bool)),fun(int,fun(int,bool))),combs(int,fun(int,bool),fun(int,bool)),hAPP(fun(int,fun(bool,bool)),fun(int,fun(fun(int,bool),fun(int,bool))),hAPP(fun(fun(bool,bool),fun(fun(int,bool),fun(int,bool))),fun(fun(int,fun(bool,bool)),fun(int,fun(fun(int,bool),fun(int,bool)))),combb(fun(bool,bool),fun(fun(int,bool),fun(int,bool)),int),combb(bool,bool,int)),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),D_1)))),ord_less(int))))) # label(fact_4102_int__ge__less__than__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4800 (all X 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_eq(real),zero_zero(real)),X)) -> hAPP(real,real,hAPP(nat,fun(real,real),root,N),hAPP(real,real,inverse_inverse(real),X)) = hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X))))) # label(fact_2719_real__root__inverse__lemma) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4801 (all X_a (linorder(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)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)))))) # label(fact_570_linorder__le__less__linear) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4802 (all Na all Xa exists T_4 (hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(fun(nat,real),fun(nat,fun(real,real))),combs(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),div_div(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),zero_zero(real)))),hAPP(real,fun(nat,real),power_power(real),Xa))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),Na))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),T_4),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,real_1(nat),Na))),pi)))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),Na)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa),Na))) = hAPP(real,real,cos,Xa) & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),T_4)),hAPP(real,real,abs_abs(real),Xa))))) # label(fact_4269_Maclaurin__cos__expansion) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4803 (all X_a (ordere142940540dd_abs(X_a) -> (all A_1 hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)) = hAPP(X_a,X_a,abs_abs(X_a),A_1)))) # label(fact_965_abs__idempotent) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4804 (all N all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,nat_1,Z_1)),N) = hAPP(int,nat,nat_1,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Z_1),N)))) # label(fact_1047_nat__power__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4805 (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_907_zdvd__bounds) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4806 (all A_1 all B hAPP(rat,rat,uminus_uminus(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,uminus_uminus(int),A_1)),B)) # label(fact_3758_minus__rat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4807 (all X_a (number_ring(X_a) -> (all V hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,number_number_of(int),V)) = hAPP(int,X_a,number_number_of(X_a),V)))) # label(fact_3606_of__int__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4808 (all Va (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Va),pls)) <-> zero_zero(nat) = hAPP(int,nat,number_number_of(nat),Va))) # label(fact_274_eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4809 (all A_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)),B)))) # label(fact_1360_pos__mod__bound) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4810 (all Ya all Xa (hBOOL(hAPP(int,bool,nat_is_nat,Xa)) -> hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),Xa),Ya))))) # label(fact_4880_SetInterval_Otransfer__int__nat__set__function__closures) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4811 (all A_3 (hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A_3),zero_zero(real)),sequentially)) -> (hBOOL(hAPP(fun(nat,real),bool,monoseq(real),A_3)) -> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,fun(nat,bool)),fun(nat,real),hAPP(fun(fun(nat,bool),real),fun(fun(nat,fun(nat,bool)),fun(nat,real)),combb(fun(nat,bool),real,nat),hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat))),hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),A_3))),sequentially))))) # label(fact_4527_summable__Leibniz_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4812 (all X_a (ring_div(X_a) -> (all A_1 all B hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B))),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B)))) # label(fact_1944_mod__minus__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4813 (all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) # label(fact_1174_div__2__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4814 (all X all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,X),Y)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),X),Y)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) # label(fact_2871_coprime__sos) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4815 (all X ((zero_zero(real) = X -> zero_zero(real) = hAPP(real,real,sgn_sgn(real),X)) & (zero_zero(real) != X -> (-hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) -> hAPP(real,real,sgn_sgn(real),X) = hAPP(real,real,uminus_uminus(real),one_one(real))) & (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) -> one_one(real) = hAPP(real,real,sgn_sgn(real),X))))) # label(fact_2312_real__sgn__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4816 (all X_a (semiring_div(X_a) -> (all A_1 all B all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C_1))),C_1)))) # label(fact_1293_mod__add__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4817 (all A_3 all Xa (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zEven)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Xa)) -> hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_3),hAPP(int,nat,nat_1,Xa)) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,uminus_uminus(int),A_3)),hAPP(int,nat,nat_1,Xa))))) # label(fact_2100_neg__even__power) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4818 (all X_a (real_normed_vector(X_a) -> (all A_1 zero_zero(X_a) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),zero_zero(X_a))))) # label(fact_4731_scaleR_Ozero__right) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4819 (all X_a all X all Y (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),fequal(X_a),X),Y)) | ti(X_a,X) = ti(X_a,Y))) # label(help_fequal_1_1_T) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4820 (all X_a (linordered_idom(X_a) -> (all Xa all Wa (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa),hAPP(int,nat,number_number_of(nat),Wa))),zero_zero(X_a))) <-> zero_zero(nat) != hAPP(int,nat,number_number_of(nat),Wa) & (hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,number_number_of(nat),Wa))) & zero_zero(X_a) = ti(X_a,Xa) | -hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,number_number_of(nat),Wa))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Xa),zero_zero(X_a)))))))) # label(fact_2518_power__le__zero__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4821 (all X_a (comple44059174attice(X_a) -> (all A_2 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(X_a,bool),X_a,complete_Sup_Sup(X_a),A_2)),B_2)) <-> (all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),B_2)))))))) # label(fact_1814_Sup__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4822 (all X_b all X_a all F all B_1 all A_2 ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2)) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(X_a,X_b,F,X_1)),B_1)))) -> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),F),A_2)),B_1)))) # label(fact_3679_image__subsetI) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4823 (all X hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) # label(fact_10_quartic__square__square) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4824 (all L all K ((ti(int,K) = zero_zero(int) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,K),L) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),zero_zero(int))) & (zero_zero(int) != ti(int,K) -> (zero_zero(int) = ti(int,L) -> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,K),L) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K)) & (zero_zero(int) != ti(int,L) -> hAPP(product_prod(int,int),product_prod(int,int),hAPP(fun(int,int),fun(product_prod(int,int),product_prod(int,int)),product_apsnd(int,int,int),hAPP(int,fun(int,int),times_times(int),hAPP(int,int,sgn_sgn(int),L))),hAPP(product_prod(int,int),product_prod(int,int),hAPP(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),hAPP(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),if(product_prod(int,int)),hAPP(bool,bool,hAPP(bool,fun(bool,bool),fdisj,hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),L)),hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))),hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),L),zero_zero(int))),hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),zero_zero(int))))),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),pdivmod,K),L)),hAPP(product_prod(int,int),product_prod(int,int),hAPP(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_prod_case(int,int,product_prod(int,int)),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int)))),combs(int,fun(int,product_prod(int,int)),fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),hAPP(fun(fun(int,fun(product_prod(int,int),product_prod(int,int))),fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int))))),combb(fun(int,fun(product_prod(int,int),product_prod(int,int))),fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int))),int),combs(int,product_prod(int,int),product_prod(int,int))),hAPP(fun(int,product_prod(int,int)),fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),hAPP(fun(product_prod(int,int),fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(fun(int,product_prod(int,int)),fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int))))),combb(product_prod(int,int),fun(int,fun(product_prod(int,int),product_prod(int,int))),int),hAPP(fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),fun(product_prod(int,int),fun(int,fun(product_prod(int,int),product_prod(int,int)))),combc(int,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),hAPP(fun(int,bool),fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),hAPP(fun(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),fun(fun(int,bool),fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))),combb(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),int),if(product_prod(int,int))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),fequal(int)),zero_zero(int))))),hAPP(int,fun(int,product_prod(int,int)),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int))),combc(int,int,product_prod(int,int)),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int)),uminus_uminus(int))),zero_zero(int))))),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(fun(int,int),fun(int,product_prod(int,int)))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combc(int,fun(int,int),fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(fun(int,int),fun(int,product_prod(int,int)))),hAPP(fun(fun(int,product_prod(int,int)),fun(fun(int,int),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(fun(int,int),fun(int,product_prod(int,int))))),combb(fun(int,product_prod(int,int)),fun(fun(int,int),fun(int,product_prod(int,int))),int),combb(int,product_prod(int,int),int)),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int)),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),hAPP(fun(int,int),fun(int,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(int,int),fun(int,fun(int,int))),combb(int,fun(int,int),int),minus_minus(int)),uminus_uminus(int))),one_one(int))))),hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,abs_abs(int),L))))),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),pdivmod,K),L)))) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,K),L))))) # label(fact_4090_divmod__int__pdivmod) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4825 (all X_a (semiring_div(X_a) -> (all A_1 all B all C_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C_1))),C_1)))) # label(fact_1285_mod__mult__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4826 (all X_a (real_normed_algebra(X_a) -> (all Ya all A_3 hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_a),fun(X_a,bool),isCont(X_a,X_a),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),times_times(X_a)),Ya)),A_3))))) # label(fact_4913_mult__left_OisCont) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4827 (all X_a all X_b all X_c ti(fun(fun(X_a,fun(X_b,X_c)),fun(X_b,fun(X_a,X_c))),combc(X_a,X_b,X_c)) = combc(X_a,X_b,X_c)) # label(tsy_c_COMBC_res) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4828 (all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),zero_zero(int))) -> zero_zero(nat) = hAPP(int,nat,nat_1,Z_1))) # label(fact_1017_nat__le__0) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4829 (all X_a (semiring_div(X_a) -> (all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B)) -> ti(X_a,B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),B),A_1)),A_1))))) # label(fact_1093_dvd__div__mult__self) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4830 (all V_2 all V ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_2)) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),V),V_2))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls)) -> zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_2))))) # label(fact_649_mult__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4831 (all X_a (archim1727834104eiling(X_a) -> (all X all Z_1 hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(X_a,int,archimedean_ceiling(X_a),X)),Z_1) = hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)))))) # label(fact_3624_ceiling__diff__of__int) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4832 (all X_a (ring_1(X_a) -> (all K_4 ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K_4),zero_zero(int))) -> hAPP(int,X_a,ring_1_of_int(X_a),K_4) = hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,nat_1,K_4))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K_4),zero_zero(int))) -> hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,nat_1,hAPP(int,int,uminus_uminus(int),K_4)))) = hAPP(int,X_a,ring_1_of_int(X_a),K_4)))))) # label(fact_3584_of__int__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4833 (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),Xa),Ya)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)))) # label(fact_959_dvd_Oless__le) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4834 (all X_a (preorder(X_a) -> (all X all 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_529_order__less__not__sym) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4835 (all X_a (linordered_ring(X_a) -> (all A_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),A_1)))))) # label(fact_423_zero__le__square) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4836 (all Na all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Na),Ma)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,real_1(nat),Na)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,real_1(nat),Ma)),one_one(real)))))) # label(fact_2407_nat__le__real__less) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4837 (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_858_Nat__Transfer_Otransfer__int__nat__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4838 (all X_a all X_b all X_c all X_d all X_e all Y -(all A_4 all B_4 all C_4 all D_2 all E ti(product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),Y) != hAPP(product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e))),product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),hAPP(X_a,fun(product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e))),product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e))))),product_Pair(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),A_4),hAPP(product_prod(X_c,product_prod(X_d,X_e)),product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e))),hAPP(X_b,fun(product_prod(X_c,product_prod(X_d,X_e)),product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),product_Pair(X_b,product_prod(X_c,product_prod(X_d,X_e))),B_4),hAPP(product_prod(X_d,X_e),product_prod(X_c,product_prod(X_d,X_e)),hAPP(X_c,fun(product_prod(X_d,X_e),product_prod(X_c,product_prod(X_d,X_e))),product_Pair(X_c,product_prod(X_d,X_e)),C_4),hAPP(X_e,product_prod(X_d,X_e),hAPP(X_d,fun(X_e,product_prod(X_d,X_e)),product_Pair(X_d,X_e),D_2),E)))))) # label(fact_3175_prod__cases5) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4839 (all Va all Na (hAPP(int,nat,number_number_of(nat),Va) = hAPP(nat,nat,suc,Na) <-> -hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))) & (-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))) -> Na = hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,Va)))))) # label(fact_3914_eq__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4840 (all X_a (linord1117847801e_zero(X_a) -> (all C_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),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)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),C_1),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),C_1),B))))))))) # label(fact_1888_divide__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4841 (all X_a (linorder(X_a) -> (all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)))) # label(fact_4832_min__max_Oinf__commute) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4842 (all B_2 all Xa (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Xa)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(real,fun(real,real),log,B_2)),Xa),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,ln,B_2)),Xa)))))) # label(fact_3933_DERIV__log) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4843 (all A_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_3),zero_zero(nat))) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A_3)) <-> zero_zero(nat) != A_3)) # label(fact_1253_gcd__lcm__complete__lattice__nat_Oless__top) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4844 (all X_a (field_char_0(X_a) -> (all A_3 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_3),field_char_0_Rats(X_a))) -> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)),field_char_0_Rats(X_a))))))) # label(fact_3714_Rats__minus) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4845 (all Z_1 hAPP(real,complex,of_real(complex),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,re,Z_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,im,Z_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),Z_1),hAPP(complex,complex,cnj,Z_1))) # label(fact_3016_complex__mult__cnj) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4846 (all Ma all Na (one_one(nat) = Ma & Na = one_one(nat) <-> one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ma),Na))) # label(fact_621_nat__1__eq__mult__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4847 (all X all D_3 (zero_zero(int) != ti(int,D_3) -> hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),D_3))),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),D_3))),hAPP(int,real,real_1(int),D_3))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,real_1(int),X)),hAPP(int,real,real_1(int),D_3)))) # label(fact_3436_real__of__int__div__aux) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4848 (all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(complex,real,re,X))),hAPP(complex,real,norm_norm(complex),X)))) # label(fact_3033_abs__Re__le__cmod) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4849 (all X_a (number(X_a) -> ti(fun(int,X_a),number_number_of(X_a)) = number_number_of(X_a))) # label(tsy_c_Int_Onumber__class_Onumber__of_res) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4850 (all A_1 all P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),P_5)) -> hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,nat_1,P_5)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,nat_1,P_5)),one_one(nat)))))) # label(fact_1060_Euler_Oaux__1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4851 (all Ya all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Ya))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),Ya))))) # label(fact_2704_real__root__gt__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4852 (all X_a (ring_1(X_a) -> (all N hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)) = one_one(X_a)))) # label(fact_2044_power__minus1__even) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4853 (all R_3 all A_1 hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A_1),R_3) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),A_1)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_3),R_3))) # label(fact_1757_real__divide__square__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4854 (all X all A_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1)) -> (A_1 != one_one(real) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) -> hAPP(real,real,hAPP(real,fun(real,real),powr,A_1),hAPP(real,real,hAPP(real,fun(real,real),log,A_1),X)) = X)))) # label(fact_3431_powr__log__cancel) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4855 (all X all Y all L_4 all M hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,uminus_uminus(real),L_4)),hAPP(real,real,uminus_uminus(real),M))))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,uminus_uminus(real),L_4)))),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Y),hAPP(real,real,uminus_uminus(real),M))))))) # label(fact_2026_abs__sum__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4856 (all X_a all Q_1 all P_1 ((exists X1 hBOOL(hAPP(X_a,bool,P_1,X1))) -> ((all X_1 (hBOOL(hAPP(X_a,bool,P_1,X_1)) -> hBOOL(hAPP(X_a,bool,Q_1,X_1)))) -> hBOOL(hAPP(X_a,bool,Q_1,hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1)))))) # label(fact_4892_someI2__ex) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4857 (all Xa all Ya all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (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(real),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Xa)),hAPP(real,real,hAPP(nat,fun(real,real),root,Na),Ya)))))) # label(fact_2684_real__root__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4858 (all Xa all Ya (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Xa),Ya)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ya),Xa)) <-> Ya = Xa)) # label(fact_961_dvd_Oeq__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4859 -(all T_4 hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),s),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),one_one(int)) != hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),m)),one_one(int))),T_4)) # label(fact_656__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]. 7.88/7.89 4860 (all X hAPP(complex,real,im,hAPP(complex,complex,uminus_uminus(complex),X)) = hAPP(real,real,uminus_uminus(real),hAPP(complex,real,im,X))) # label(fact_3005_Im_Ominus) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4861 (all X_b all X_a (real_normed_vector(X_b) & topological_space(X_a) -> (all G all F all A_3 (hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,real),fun(X_a,bool),isCont(X_a,real),F),A_3)) -> (hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),G),A_3)) -> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,real),fun(X_a,fun(X_b,X_b)),hAPP(fun(real,fun(X_b,X_b)),fun(fun(X_a,real),fun(X_a,fun(X_b,X_b))),combb(real,fun(X_b,X_b),X_a),scaleR_scaleR(X_b)),F)),G)),A_3))))))) # label(fact_4935_isCont__scaleR) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4862 (all Xa all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) -> (hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Xa),Na)),zOdd)) <-> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),zOdd))))) # label(fact_1224_power__preserves__odd) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4863 (all A_1 all X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),A_1)),X)) -> hAPP(real,nat,natceiling,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),hAPP(nat,real,real_1(nat),A_1))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(real,nat,natceiling,X)),A_1))) # label(fact_2415_natceiling__subtract) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4864 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),B)) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)))) # label(fact_3142_mod__neg__pos) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4865 (all X_a (real_n2089651433ebra_1(X_a) & banach(X_a) -> (all Xa hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(X_a,fun(nat,X_a),power_power(X_a),Xa)))))))) # label(fact_4774_summable__norm__exp) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4866 (all R_3 all S_3 all T_5 all R_4 all S_4 all M all T_6 all N (hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),S_4),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),T_6),N)) = ti(int,R_4) -> (hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),S_3),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),T_5),N)) = ti(int,R_3) -> hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),S_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),R_4),R_3)),S_3))),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),T_6),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),R_4),R_3)),T_5))),N)) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),R_4),R_3)))) # label(fact_1458_xzgcda__linear__aux2) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4867 (all X_a (linorder(X_a) -> (all C all D_1 all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_1)) -> (hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),C),D_1) = hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),A_3),B_2) <-> ti(X_a,D_1) = ti(X_a,B_2) & ti(X_a,A_3) = ti(X_a,C))))))) # label(fact_4224_atLeastLessThan__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4868 (all X_a (semiring_div(X_a) -> (all Z_1 all W all Y all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Y),X)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Z_1),W)) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),W),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),W)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Z_1))))))) # label(fact_1095_div__mult__div__if__dvd) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4869 (all Ma all X_1 all Xa_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X_1),Xa_1),Ma)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcongm,Ma),X_1),Xa_1)))) # label(fact_4897_zcongm__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4870 (all X hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,cos,X))),one_one(real)))) # label(fact_2290_abs__cos__le__one) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4871 (all A_3 all 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),A_3),zero_zero(int))) <-> hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_2)),zero_zero(rat)))))) # label(fact_3813_Fract__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4872 (all X_a (order(X_a) -> (all A_3 all B_2 all C all D_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),A_3),B_2)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),C),D_1))) <-> -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),A_3)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),D_1)))))) # label(fact_4550_atLeastatMost__subset__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4873 (all X all Y (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) -> -(-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X))))) # label(fact_934_dvd_Oless__asym) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4874 (all N all M hAPP(quickcheck_code_int,product_prod(quickcheck_code_int,quickcheck_code_int),hAPP(quickcheck_code_int,fun(quickcheck_code_int,product_prod(quickcheck_code_int,quickcheck_code_int)),quickc495462417de_int,N),M) = hAPP(quickcheck_code_int,product_prod(quickcheck_code_int,quickcheck_code_int),hAPP(quickcheck_code_int,fun(quickcheck_code_int,product_prod(quickcheck_code_int,quickcheck_code_int)),product_Pair(quickcheck_code_int,quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),N),M)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),N),M))) # label(fact_3200_div__mod__code__int__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4875 (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),hAPP(nat,nat,fact_fact(nat),M)),hAPP(nat,nat,fact_fact(nat),N))))) # label(fact_2539_fact__mono__nat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4876 (all X_a (topological_space(X_a) -> ti(fun(X_a,filter(X_a)),at(X_a)) = at(X_a))) # label(tsy_c_Limits_Oat_res) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4877 (all Na (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) <-> zero_zero(nat) != Na)) # label(fact_148_neq0__conv) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4878 (all X_b all X_a all B_1 all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> (hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_1)) -> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)),hAPP(fun(X_b,bool),nat,finite_card(X_b),B_1)) = hAPP(fun(sum_sum(X_a,X_b),bool),nat,finite_card(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_2),B_1))))) # label(fact_3223_card__Plus) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4879 (all X_a (linordered_idom(X_a) -> (all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_3)),B_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_3)),B_2)))))) # label(fact_1981_abs__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4880 (all X_a (real_field(X_a) & field_inverse_zero(X_a) -> (all X all Y hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(real,X_a,of_real(X_a),X)),hAPP(real,X_a,of_real(X_a),Y)) = hAPP(real,X_a,of_real(X_a),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),Y))))) # label(fact_2805_of__real__divide) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4881 (all K_4 hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,number_number_of(int),K_4)),one_one(int)) = hAPP(rat,product_prod(int,int),quotient_of,hAPP(int,rat,number_number_of(rat),K_4))) # label(fact_3743_quotient__of__number_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4882 (all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,number_number_of(int),min)),B) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),one_one(int)))) # label(fact_1501_zmod__minus1) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4883 (all X all Y all Xa_2 all Ya_1 hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa_2),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Ya_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa_2),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Ya_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) # label(fact_2177_real__sqrt__sum__squares__mult__squared__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4884 (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),hAPP(X_a,X_a,abs_abs(X_a),K))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Ma),K)))))) # label(fact_981_dvd__abs__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4885 (all B all A_1 all C_1 hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),B),C_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),C_1))) # label(fact_3246_gcd__left__commute__nat) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4886 (all X_a (real_normed_algebra(X_a) -> (all Ya all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_2)) -> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),X_2)),Ya) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),X_2)),Ya)))))) # label(fact_4013_mult__left_Osuminf) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4887 (all Xa all Ya (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,exp(real),Xa)),hAPP(real,real,exp(real),Ya))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Xa),Ya)))) # label(fact_1754_exp__le__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4888 (all X_b all X_a (topological_space(X_a) & real_normed_vector(X_b) & real_algebra_1(X_b) -> (all G all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),G),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,real),fun(X_a,X_b),hAPP(fun(real,X_b),fun(fun(X_a,real),fun(X_a,X_b)),combb(real,X_b,X_a),of_real(X_b)),G)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),A_3))))))) # label(fact_4638_of__real_OLIM__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4889 (all X_a (number_ring(X_a) -> (all V ((hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V))) -> zero_zero(X_a) = hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,number_number_of(nat),V))) & (-hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V))) -> hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,number_number_of(nat),V)) = hAPP(int,X_a,number_number_of(X_a),V)))))) # label(fact_1624_of__nat__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4890 (all X_b all X_c all X_a (real_normed_vector(X_b) & real_normed_vector(X_c) & topological_space(X_a) -> (all G all Ma all F all L all A_3 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),A_3))) -> ((all X_1 (ti(X_a,A_3) != ti(X_a,X_1) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_c,real,norm_norm(X_c),hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),minus_minus(X_c),hAPP(X_a,X_c,G,X_1)),Ma))),hAPP(X_b,real,norm_norm(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_a,X_b,F,X_1)),L)))))) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_c,fun(filter(X_a),bool),hAPP(fun(X_a,X_c),fun(X_c,fun(filter(X_a),bool)),tendsto(X_a,X_c),G),Ma),hAPP(X_a,filter(X_a),at(X_a),A_3)))))))) # label(fact_4679_LIM__imp__LIM) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4891 (all X_a (real_normed_algebra(X_a) -> (all Xa all X_2 (hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_2)) -> hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa)),X_2))))))) # label(fact_4127_mult__right_OCauchy) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4892 (all Y all X ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Y),X)) -> zero_zero(int) = hAPP(int,int,hAPP(int,fun(int,int),nat_tsub,X),Y)) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Y),X)) -> hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),Y) = hAPP(int,int,hAPP(int,fun(int,int),nat_tsub,X),Y)))) # label(fact_1506_tsub__def) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4893 (all X_a (cancel146912293up_add(X_a) -> (all A_1 all B all C_1 (hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B) -> ti(X_a,C_1) = ti(X_a,B))))) # label(fact_291_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4894 (all X_a all X_b lex_prod(X_a,X_b) = ti(fun(fun(product_prod(X_a,X_a),bool),fun(fun(product_prod(X_b,X_b),bool),fun(product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b)),bool))),lex_prod(X_a,X_b))) # label(tsy_c_Wellfounded_Olex__prod_res) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4895 (all X_2 ((all R_1 exists N_2 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),N_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_1),hAPP(nat,real,X_2,N_1))))) -> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),X_2)),zero_zero(real)),sequentially)))) # label(fact_4685_LIMSEQ__inverse__zero) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4896 (all X all M all Y (hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Y),M) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),M) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),Y),M)))) # label(fact_1319_Residues_Oaux) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4897 (all X_a all X_b (real_normed_algebra(X_b) -> (all G all B_2 all F all A_3 all A_2 (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),A_3),A_2)) -> (hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),B_2),A_2)) -> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_3),B_2)),A_2))))))) # label(fact_4481_mult_Otendsto) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4898 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K),L)),zero_zero(int))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)))) # label(fact_768_less__bin__lemma) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4899 (all Z_1 all W hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W),Z_1) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z_1),W)) # label(fact_47_zadd__commute) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4900 (all X_a (ordere453448008miring(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),zero_zero(X_a)))))))) # label(fact_427_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4901 (all X hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),X) = hAPP(int,int,abs_abs(int),X)) # label(fact_4390_gcd__idem__int) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4902 (all X_a all X_b (dvd(X_b) & comm_monoid_mult(X_b) -> (all F all G all A_2 ((all X_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),dvd_dvd(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,G,X_1))))) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),dvd_dvd(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),A_2))))))) # label(fact_4452_setprod__dvd__setprod) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4903 (all X_a (division_ring(X_a) -> (all A_1 hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),zero_zero(X_a)),A_1) = zero_zero(X_a)))) # label(fact_1826_divide__zero__left) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4904 (all X_a all A_2 all B_1 (hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1)) -> (exists B_4 hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_4),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),B_1),A_2)))))) # label(fact_2721_psubset__imp__ex__mem) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4905 (all X_a all X_b all X_c all X_d all X_e all X_f all Y -(all A_4 all B_4 all C_4 all D_2 all E all F_3 ti(product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),Y) != hAPP(product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),hAPP(X_a,fun(product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))))),product_Pair(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),A_4),hAPP(product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))),product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),hAPP(X_b,fun(product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))),product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),product_Pair(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),B_4),hAPP(product_prod(X_d,product_prod(X_e,X_f)),product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))),hAPP(X_c,fun(product_prod(X_d,product_prod(X_e,X_f)),product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),product_Pair(X_c,product_prod(X_d,product_prod(X_e,X_f))),C_4),hAPP(product_prod(X_e,X_f),product_prod(X_d,product_prod(X_e,X_f)),hAPP(X_d,fun(product_prod(X_e,X_f),product_prod(X_d,product_prod(X_e,X_f))),product_Pair(X_d,product_prod(X_e,X_f)),D_2),hAPP(X_f,product_prod(X_e,X_f),hAPP(X_e,fun(X_f,product_prod(X_e,X_f)),product_Pair(X_e,X_f),E),F_3))))))) # label(fact_3173_prod__cases6) # label(axiom) # label(non_clause). [assumption]. 7.88/7.89 4906 (all X_a all A_2 (hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) -> one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2))))) # label(fact_2732_neg__one__special) # label(axiom) # label(non_clause). [assumption]. 8.02/8.02 4907 (all X_a all X_b (order(X_a) & order(X_b) -> (all F all C all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2)) -> (hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,B_2)),C)) -> ((all X_1 all Y_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1)) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,F,Y_1))))) -> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,A_3)),C)))))))) # label(fact_2856_order__le__less__subst2) # label(axiom) # label(non_clause). [assumption]. 8.02/8.02 4908 (all X_a (comm_monoid_mult(X_a) & comm_monoid_add(X_a) -> (all F all G all A_2 all B_1 (B_1 = A_2 -> (hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,B_1)) -> ((all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) -> hAPP(int,X_a,F,X_1) = hAPP(int,X_a,G,X_1))) -> hAPP(fun(int,bool),X_a,hAPP(fun(int,X_a),fun(fun(int,bool),X_a),big_co1399186613setsum(int,X_a),F),A_2) = hAPP(fun(int,bool),X_a,hAPP(fun(int,X_a),fun(fun(int,bool),X_a),big_co1399186613setsum(int,X_a),G),B_1))))))) # label(fact_4324_transfer__nat__int__sum__prod__cong_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.02/8.02 4909 -(zero_zero(int) != hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),n))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) # label(conj_0) # label(negated_conjecture) # label(non_clause). [assumption]. 8.02/8.02 8.02/8.02 ============================== end of process non-clausal formulas === 8.02/8.02 8.02/8.02 ============================== PROCESS INITIAL CLAUSES =============== 8.02/8.02 8.02/8.02 ============================== PREDICATE ELIMINATION ================= 8.02/8.02 4910 -linordered_semidom(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(nat,nat,suc,C)) != hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),D),hAPP(nat,nat,suc,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)) | ti(A,B) = ti(A,D) # label(fact_1424_power__inject__base) # label(axiom). [clausify(199)]. 8.02/8.02 4911 linordered_semidom(rat) # label(arity_Rat_Orat___Rings_Olinordered__semidom) # label(axiom). [assumption]. 8.02/8.02 Derived: hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(nat,nat,suc,B)) != hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),C),hAPP(nat,nat,suc,B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),C)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | ti(rat,A) = ti(rat,C). [resolve(4910,a,4911,a)]. 8.02/8.02 4912 -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_171_less__imp__of__nat__less) # label(axiom). [clausify(235)]. 8.02/8.02 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(nat,rat,semiring_1_of_nat(rat),A)),hAPP(nat,rat,semiring_1_of_nat(rat),B))). [resolve(4912,a,4911,a)]. 8.02/8.02 4913 -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,hAPP(A,fun(nat,A),power_power(A),D),C)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),D),B))) # label(fact_113_power__strict__decreasing) # label(axiom). [clausify(291)]. 8.02/8.02 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),C)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),C),one_one(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),C),B)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),C),A))). [resolve(4913,a,4911,a)]. 8.18/8.15 4914 -linordered_semidom(A) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f94(B,A,C,D)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),C),D))) # label(fact_4176_setprod__pos) # label(axiom). [clausify(437)]. 8.18/8.15 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f94(A,rat,B,C)),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),B),C))). [resolve(4914,a,4911,a)]. 8.18/8.15 4915 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(B,A,C,f94(B,A,C,D)))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),C),D))) # label(fact_4176_setprod__pos) # label(axiom). [clausify(437)]. 8.18/8.15 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(A,rat,B,f94(A,rat,B,C)))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),B),C))). [resolve(4915,a,4911,a)]. 8.18/8.15 4916 -linordered_semidom(A) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f173(B,A,C,D)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),C),D))) # label(fact_4178_setprod__nonneg) # label(axiom). [clausify(1008)]. 8.18/8.15 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f173(A,rat,B,C)),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),B),C))). [resolve(4916,a,4911,a)]. 8.18/8.15 4917 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(B,A,C,f173(B,A,C,D)))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),C),D))) # label(fact_4178_setprod__nonneg) # label(axiom). [clausify(1008)]. 8.18/8.15 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(A,rat,B,f173(A,rat,B,C)))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),B),C))). [resolve(4917,a,4911,a)]. 8.18/8.15 4918 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(nat,nat,suc,C))),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),D),hAPP(nat,nat,suc,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),B),D)) # label(fact_1425_power__le__imp__le__base) # label(axiom). [clausify(1068)]. 8.18/8.15 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(nat,nat,suc,B))),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),C),hAPP(nat,nat,suc,B)))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),C)). [resolve(4918,a,4911,a)]. 8.18/8.15 4919 -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,hAPP(A,fun(nat,A),power_power(A),B),D)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),D))) # label(fact_251_power__mono) # label(axiom). [clausify(1085)]. 8.18/8.15 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),C)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),B),C))). [resolve(4919,a,4911,a)]. 8.26/8.27 4920 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),one_one(A))) # label(fact_181_zero__less__one) # label(axiom). [clausify(1120)]. 8.26/8.27 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),one_one(rat))). [resolve(4920,a,4911,a)]. 8.26/8.27 4921 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C))) # label(fact_459_less__1__mult) # label(axiom). [clausify(1129)]. 8.26/8.27 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B))). [resolve(4921,a,4911,a)]. 8.26/8.27 4922 linordered_semidom(int) # label(arity_Int_Oint___Rings_Olinordered__semidom) # label(axiom). [assumption]. 8.26/8.27 Derived: hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),hAPP(nat,nat,suc,B)) != hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),C),hAPP(nat,nat,suc,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)) | ti(int,A) = ti(int,C). [resolve(4922,a,4910,a)]. 8.26/8.27 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,hAPP(int,fun(nat,int),power_power(int),C),B)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),C),A))). [resolve(4922,a,4913,a)]. 8.26/8.27 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f94(A,int,B,C)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),B),C))). [resolve(4922,a,4914,a)]. 8.26/8.27 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(A,int,B,f94(A,int,B,C)))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),B),C))). [resolve(4922,a,4915,a)]. 8.26/8.27 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f173(A,int,B,C)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),B),C))). [resolve(4922,a,4916,a)]. 8.26/8.27 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(A,int,B,f173(A,int,B,C)))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),B),C))). [resolve(4922,a,4917,a)]. 8.26/8.27 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),hAPP(nat,nat,suc,B))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),C),hAPP(nat,nat,suc,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),A),C)). [resolve(4922,a,4918,a)]. 8.26/8.27 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),C)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),C))). [resolve(4922,a,4919,a)]. 8.42/8.37 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),one_one(int))). [resolve(4922,a,4920,a)]. 8.42/8.37 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B))). [resolve(4922,a,4921,a)]. 8.42/8.37 4923 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C)),hAPP(nat,A,hAPP(A,fun(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_272_power__less__imp__less__base) # label(axiom). [clausify(1571)]. 8.42/8.37 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),C),B))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),C)). [resolve(4923,a,4911,a)]. 8.42/8.37 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),B)),hAPP(nat,int,hAPP(int,fun(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(4923,a,4922,a)]. 8.42/8.37 4924 -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_189_of__nat__less__0__iff) # label(axiom). [clausify(1731)]. 8.42/8.37 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(nat,rat,semiring_1_of_nat(rat),A)),zero_zero(rat))). [resolve(4924,a,4911,a)]. 8.42/8.37 4925 -linordered_semidom(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) != hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,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_313_power2__eq__imp__eq) # label(axiom). [clausify(1761)]. 8.42/8.37 Derived: hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) != hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | ti(rat,A) = ti(rat,B). [resolve(4925,a,4911,a)]. 8.42/8.37 Derived: hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) != hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,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(4925,a,4922,a)]. 8.42/8.37 4926 -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,hAPP(A,fun(nat,A),power_power(A),B),D)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),D))) # label(fact_310_power__strict__mono) # label(axiom). [clausify(1820)]. 8.42/8.37 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),C)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),B),C))). [resolve(4926,a,4911,a)]. 8.55/8.55 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,hAPP(int,fun(nat,int),power_power(int),A),C)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),C))). [resolve(4926,a,4922,a)]. 8.55/8.55 4927 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(nat,nat,suc,C)))) # label(fact_1430_power__gt1) # label(axiom). [clausify(1839)]. 8.55/8.55 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(nat,nat,suc,B)))). [resolve(4927,a,4911,a)]. 8.55/8.55 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),hAPP(nat,nat,suc,B)))). [resolve(4927,a,4922,a)]. 8.55/8.55 4928 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,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_332_power2__less__imp__less) # label(axiom). [clausify(1899)]. 8.55/8.55 4929 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))) # label(fact_114_zero__less__two) # label(axiom). [clausify(2110)]. 8.55/8.55 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),one_one(rat)),one_one(rat)))). [resolve(4929,a,4911,a)]. 8.55/8.55 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),one_one(int)))). [resolve(4929,a,4922,a)]. 8.55/8.55 4930 -linordered_semidom(A) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),f373(A,B,C,D,E)),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(fun(nat,bool),A,hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),C),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),E),B)))),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(nat,A,semiring_1_of_nat(A),E)),D))) # label(fact_4300_real__setsum__nat__ivl__bounded2) # label(axiom). [clausify(2227)]. 8.55/8.55 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),f373(rat,A,B,C,D)),D)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(fun(nat,bool),rat,hAPP(fun(nat,rat),fun(fun(nat,bool),rat),big_co1399186613setsum(nat,rat),B),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),D),A)))),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(nat,rat,semiring_1_of_nat(rat),D)),C))). [resolve(4930,a,4911,a)]. 8.55/8.55 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),f373(int,A,B,C,D)),D)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(fun(nat,bool),int,hAPP(fun(nat,int),fun(fun(nat,bool),int),big_co1399186613setsum(nat,int),B),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),D),A)))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),D)),C))). [resolve(4930,a,4922,a)]. 8.61/8.64 4931 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(nat,A,B,f373(A,C,B,D,E))),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(fun(nat,bool),A,hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),B),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),E),C)))),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(nat,A,semiring_1_of_nat(A),E)),D))) # label(fact_4300_real__setsum__nat__ivl__bounded2) # label(axiom). [clausify(2227)]. 8.61/8.64 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(nat,rat,A,f373(rat,B,A,C,D))),C)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(fun(nat,bool),rat,hAPP(fun(nat,rat),fun(fun(nat,bool),rat),big_co1399186613setsum(nat,rat),A),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),D),B)))),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(nat,rat,semiring_1_of_nat(rat),D)),C))). [resolve(4931,a,4911,a)]. 8.61/8.64 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,A,f373(int,B,A,C,D))),C)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(fun(nat,bool),int,hAPP(fun(nat,int),fun(fun(nat,bool),int),big_co1399186613setsum(nat,int),A),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),D),B)))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),D)),C))). [resolve(4931,a,4922,a)]. 8.61/8.64 4932 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,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_314_power2__le__imp__le) # label(axiom). [clausify(2235)]. 8.61/8.64 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)). [resolve(4932,a,4911,a)]. 8.61/8.64 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,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(4932,a,4922,a)]. 8.61/8.64 4933 -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),hAPP(nat,nat,fact_fact(nat),B)))) # label(fact_2549_of__nat__fact__ge__zero) # label(axiom). [clausify(2389)]. 8.79/8.76 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(nat,rat,semiring_1_of_nat(rat),hAPP(nat,nat,fact_fact(nat),A)))). [resolve(4933,a,4911,a)]. 8.79/8.76 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,fact_fact(nat),A)))). [resolve(4933,a,4922,a)]. 8.79/8.76 4934 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C)))) # label(fact_492_power__gt1__lemma) # label(axiom). [clausify(2477)]. 8.79/8.76 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B)))). [resolve(4934,a,4911,a)]. 8.79/8.76 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),B)))). [resolve(4934,a,4922,a)]. 8.79/8.76 4935 -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_170_of__nat__less__iff) # label(axiom). [clausify(2496)]. 8.79/8.76 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(nat,rat,semiring_1_of_nat(rat),A)),hAPP(nat,rat,semiring_1_of_nat(rat),B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)). [resolve(4935,a,4911,a)]. 8.79/8.76 4936 -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_170_of__nat__less__iff) # label(axiom). [clausify(2496)]. 8.79/8.76 4937 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(nat,A,semiring_1_of_nat(A),B)),hAPP(nat,A,semiring_1_of_nat(A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) # label(fact_208_of__nat__le__iff) # label(axiom). [clausify(2555)]. 8.79/8.76 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(nat,rat,semiring_1_of_nat(rat),A)),hAPP(nat,rat,semiring_1_of_nat(rat),B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)). [resolve(4937,a,4911,a)]. 8.79/8.76 4938 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(nat,A,semiring_1_of_nat(A),B)),hAPP(nat,A,semiring_1_of_nat(A),C))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) # label(fact_208_of__nat__le__iff) # label(axiom). [clausify(2555)]. 8.79/8.76 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(nat,rat,semiring_1_of_nat(rat),A)),hAPP(nat,rat,semiring_1_of_nat(rat),B))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)). [resolve(4938,a,4911,a)]. 8.79/8.76 4939 -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,hAPP(A,fun(nat,A),power_power(A),B),C))) # label(fact_253_one__le__power) # label(axiom). [clausify(2606)]. 8.79/8.76 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B))). [resolve(4939,a,4911,a)]. 8.79/8.76 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,hAPP(int,fun(nat,int),power_power(int),A),B))). [resolve(4939,a,4922,a)]. 8.92/8.89 4940 -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,hAPP(A,fun(nat,A),power_power(A),B),C)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),D))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),C),D)) # label(fact_186_power__less__imp__less__exp) # label(axiom). [clausify(2616)]. 8.92/8.89 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)). [resolve(4940,a,4911,a)]. 8.92/8.89 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,hAPP(int,fun(nat,int),power_power(int),A),B)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)). [resolve(4940,a,4922,a)]. 8.92/8.89 4941 -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_110_of__nat__0__less__iff) # label(axiom). [clausify(2767)]. 8.92/8.89 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(nat,rat,semiring_1_of_nat(rat),A))). [resolve(4941,a,4911,a)]. 8.92/8.89 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(4941,a,4922,a)]. 8.92/8.89 4942 -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_110_of__nat__0__less__iff) # label(axiom). [clausify(2767)]. 8.92/8.89 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(nat,rat,semiring_1_of_nat(rat),A))). [resolve(4942,a,4911,a)]. 8.92/8.89 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(4942,a,4922,a)]. 8.92/8.89 4943 linordered_semidom(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semidom) # label(axiom). [assumption]. 8.92/8.89 Derived: hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(nat,nat,suc,B)) != hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),C),hAPP(nat,nat,suc,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)) | ti(real,A) = ti(real,C). [resolve(4943,a,4910,a)]. 8.92/8.89 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(4943,a,4912,a)]. 8.92/8.89 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,hAPP(real,fun(nat,real),power_power(real),C),B)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),C),A))). [resolve(4943,a,4913,a)]. 8.92/8.89 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f94(A,real,B,C)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),B),C))). [resolve(4943,a,4914,a)]. 8.92/8.90 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(A,real,B,f94(A,real,B,C)))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),B),C))). [resolve(4943,a,4915,a)]. 8.92/8.90 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f173(A,real,B,C)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),B),C))). [resolve(4943,a,4916,a)]. 8.92/8.90 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(A,real,B,f173(A,real,B,C)))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),B),C))). [resolve(4943,a,4917,a)]. 8.92/8.90 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(nat,nat,suc,B))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),C),hAPP(nat,nat,suc,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),A),C)). [resolve(4943,a,4918,a)]. 8.92/8.90 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),C)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),C))). [resolve(4943,a,4919,a)]. 8.92/8.90 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),one_one(real))). [resolve(4943,a,4920,a)]. 8.92/8.90 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B))). [resolve(4943,a,4921,a)]. 8.92/8.90 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B)),hAPP(nat,real,hAPP(real,fun(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(4943,a,4923,a)]. 8.92/8.90 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(4943,a,4924,a)]. 8.92/8.90 Derived: hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) != hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,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(4943,a,4925,a)]. 8.92/8.90 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),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,hAPP(real,fun(nat,real),power_power(real),A),C)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),C))). [resolve(4943,a,4926,a)]. 8.92/8.90 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(nat,nat,suc,B)))). [resolve(4943,a,4927,a)]. 8.92/8.91 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),one_one(real)))). [resolve(4943,a,4929,a)]. 8.92/8.91 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),f373(real,A,B,C,D)),D)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),B),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),D),A)))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,semiring_1_of_nat(real),D)),C))). [resolve(4943,a,4930,a)]. 8.92/8.91 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,A,f373(real,B,A,C,D))),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),A),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),D),B)))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,semiring_1_of_nat(real),D)),C))). [resolve(4943,a,4931,a)]. 8.92/8.91 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,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(4943,a,4932,a)]. 8.92/8.91 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),hAPP(nat,nat,fact_fact(nat),A)))). [resolve(4943,a,4933,a)]. 8.92/8.91 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B)))). [resolve(4943,a,4934,a)]. 8.92/8.91 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(4943,a,4935,a)]. 8.92/8.91 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,semiring_1_of_nat(real),A)),hAPP(nat,real,semiring_1_of_nat(real),B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)). [resolve(4943,a,4937,a)]. 8.92/8.91 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,semiring_1_of_nat(real),A)),hAPP(nat,real,semiring_1_of_nat(real),B))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)). [resolve(4943,a,4938,a)]. 8.92/8.91 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,hAPP(real,fun(nat,real),power_power(real),A),B))). [resolve(4943,a,4939,a)]. 8.92/8.91 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,hAPP(real,fun(nat,real),power_power(real),A),B)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)). [resolve(4943,a,4940,a)]. 8.92/8.91 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(4943,a,4941,a)]. 9.06/9.01 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(4943,a,4942,a)]. 9.06/9.01 4944 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),one_one(A))) # label(fact_248_zero__le__one) # label(axiom). [clausify(2842)]. 9.06/9.01 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),one_one(rat))). [resolve(4944,a,4911,a)]. 9.06/9.01 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),one_one(int))). [resolve(4944,a,4922,a)]. 9.06/9.01 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),one_one(real))). [resolve(4944,a,4943,a)]. 9.06/9.01 4945 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | C != D | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),D) # label(fact_184_power__inject__exp) # label(axiom). [clausify(2938)]. 9.06/9.01 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),A)) | B != C | hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B) = hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),C). [resolve(4945,a,4911,a)]. 9.06/9.01 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A)) | B != C | hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),B) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),C). [resolve(4945,a,4922,a)]. 9.06/9.01 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A)) | B != C | hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),C). [resolve(4945,a,4943,a)]. 9.06/9.01 4946 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | C = D | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C) != hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),D) # label(fact_184_power__inject__exp) # label(axiom). [clausify(2938)]. 9.06/9.01 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),A)) | B = C | hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B) != hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),C). [resolve(4946,a,4911,a)]. 9.06/9.01 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A)) | B = C | hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),B) != hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),C). [resolve(4946,a,4922,a)]. 9.06/9.01 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A)) | B = C | hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B) != hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),C). [resolve(4946,a,4943,a)]. 9.06/9.01 4947 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),D))) # label(fact_179_pos__add__strict) # label(axiom). [clausify(2954)]. 9.06/9.01 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),C))). [resolve(4947,a,4911,a)]. 9.06/9.01 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),C))). [resolve(4947,a,4922,a)]. 9.06/9.01 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),C))). [resolve(4947,a,4943,a)]. 9.08/9.05 4948 -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_172_of__nat__less__imp__less) # label(axiom). [clausify(2964)]. 9.08/9.05 4949 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(nat,A,semiring_1_of_nat(A),hAPP(nat,nat,fact_fact(nat),B)))) # label(fact_2550_of__nat__fact__gt__zero) # label(axiom). [clausify(3002)]. 9.08/9.05 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(nat,rat,semiring_1_of_nat(rat),hAPP(nat,nat,fact_fact(nat),A)))). [resolve(4949,a,4911,a)]. 9.08/9.05 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,fact_fact(nat),A)))). [resolve(4949,a,4922,a)]. 9.08/9.05 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,semiring_1_of_nat(real),hAPP(nat,nat,fact_fact(nat),A)))). [resolve(4949,a,4943,a)]. 9.08/9.05 4950 -linordered_semidom(A) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f504(B,A,C,D,E)),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),D),E)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),C),E))) # label(fact_4177_setprod__mono) # label(axiom). [clausify(3010)]. 9.08/9.05 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f504(A,rat,B,C,D)),D)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),C),D)),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),B),D))). [resolve(4950,a,4911,a)]. 9.08/9.05 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f504(A,int,B,C,D)),D)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),C),D)),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),B),D))). [resolve(4950,a,4922,a)]. 9.08/9.05 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f504(A,real,B,C,D)),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),C),D)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),B),D))). [resolve(4950,a,4943,a)]. 9.08/9.05 4951 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(B,A,C,f504(B,A,D,C,E)))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(B,A,C,f504(B,A,D,C,E))),hAPP(B,A,D,f504(B,A,D,C,E)))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),C),E)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),D),E))) # label(fact_4177_setprod__mono) # label(axiom). [clausify(3010)]. 9.08/9.05 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(A,rat,B,f504(A,rat,C,B,D)))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(A,rat,B,f504(A,rat,C,B,D))),hAPP(A,rat,C,f504(A,rat,C,B,D)))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),B),D)),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),C),D))). [resolve(4951,a,4911,a)]. 9.08/9.05 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(A,int,B,f504(A,int,C,B,D)))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(A,int,B,f504(A,int,C,B,D))),hAPP(A,int,C,f504(A,int,C,B,D)))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),B),D)),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),C),D))). [resolve(4951,a,4922,a)]. 9.48/9.45 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(A,real,B,f504(A,real,C,B,D)))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(A,real,B,f504(A,real,C,B,D))),hAPP(A,real,C,f504(A,real,C,B,D)))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),B),D)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),C),D))). [resolve(4951,a,4943,a)]. 9.48/9.45 4952 -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_254_of__nat__0__le__iff) # label(axiom). [clausify(3164)]. 9.48/9.45 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(nat,rat,semiring_1_of_nat(rat),A))). [resolve(4952,a,4911,a)]. 9.48/9.45 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),A))). [resolve(4952,a,4922,a)]. 9.48/9.45 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(4952,a,4943,a)]. 9.48/9.45 4953 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),one_one(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C))),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C))) # label(fact_538_power__Suc__less) # label(axiom). [clausify(3287)]. 9.48/9.45 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),one_one(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B))),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B))). [resolve(4953,a,4911,a)]. 9.48/9.45 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),one_one(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),B))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),B))). [resolve(4953,a,4922,a)]. 9.48/9.45 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),one_one(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B))). [resolve(4953,a,4943,a)]. 9.48/9.45 4954 -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,hAPP(A,fun(nat,A),power_power(A),D),C)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),D),B))) # label(fact_237_power__decreasing) # label(axiom). [clausify(3436)]. 9.48/9.45 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),C)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),C),one_one(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),C),B)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),C),A))). [resolve(4954,a,4911,a)]. 9.63/9.57 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,hAPP(int,fun(nat,int),power_power(int),C),B)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),C),A))). [resolve(4954,a,4922,a)]. 9.63/9.57 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),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,hAPP(real,fun(nat,real),power_power(real),C),B)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),C),A))). [resolve(4954,a,4943,a)]. 9.63/9.57 4955 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),one_one(A)))) # label(fact_183_less__add__one) # label(axiom). [clausify(3505)]. 9.63/9.57 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),one_one(rat)))). [resolve(4955,a,4911,a)]. 9.63/9.57 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),one_one(int)))). [resolve(4955,a,4922,a)]. 9.63/9.57 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),one_one(real)))). [resolve(4955,a,4943,a)]. 9.63/9.57 4956 -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,hAPP(A,fun(nat,A),power_power(A),D),B)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),D),C))) # label(fact_187_power__strict__increasing) # label(axiom). [clausify(3517)]. 9.63/9.57 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),C),A)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),C),B))). [resolve(4956,a,4911,a)]. 9.63/9.57 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,hAPP(int,fun(nat,int),power_power(int),C),A)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),C),B))). [resolve(4956,a,4922,a)]. 9.63/9.57 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,hAPP(real,fun(nat,real),power_power(real),C),A)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),C),B))). [resolve(4956,a,4943,a)]. 9.63/9.57 4957 linordered_semidom(code_code_numeral) # label(arity_Code__Numeral_Ocode__numeral___Rings_Olinordered__semidom) # label(axiom). [assumption]. 9.63/9.57 Derived: hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),hAPP(nat,nat,suc,B)) != hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),C),hAPP(nat,nat,suc,B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),C)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),A)) | ti(code_code_numeral,A) = ti(code_code_numeral,C). [resolve(4957,a,4910,a)]. 9.63/9.57 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),A)),hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),B))). [resolve(4957,a,4912,a)]. 9.63/9.58 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),zero_zero(code_code_numeral)),C)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),C),one_one(code_code_numeral))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),C),B)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),C),A))). [resolve(4957,a,4913,a)]. 9.63/9.58 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f94(A,code_code_numeral,B,C)),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),zero_zero(code_code_numeral)),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),B),C))). [resolve(4957,a,4914,a)]. 9.63/9.58 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),zero_zero(code_code_numeral)),hAPP(A,code_code_numeral,B,f94(A,code_code_numeral,B,C)))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),zero_zero(code_code_numeral)),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),B),C))). [resolve(4957,a,4915,a)]. 9.63/9.58 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f173(A,code_code_numeral,B,C)),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),B),C))). [resolve(4957,a,4916,a)]. 9.63/9.58 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),hAPP(A,code_code_numeral,B,f173(A,code_code_numeral,B,C)))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),B),C))). [resolve(4957,a,4917,a)]. 9.63/9.58 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),hAPP(nat,nat,suc,B))),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),C),hAPP(nat,nat,suc,B)))) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),C)). [resolve(4957,a,4918,a)]. 9.63/9.58 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),A)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),C)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),B),C))). [resolve(4957,a,4919,a)]. 9.63/9.58 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),zero_zero(code_code_numeral)),one_one(code_code_numeral))). [resolve(4957,a,4920,a)]. 9.63/9.58 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),one_one(code_code_numeral)),A)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),one_one(code_code_numeral)),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),one_one(code_code_numeral)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B))). [resolve(4957,a,4921,a)]. 9.63/9.58 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),C),B))) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),C)). [resolve(4957,a,4923,a)]. 9.63/9.58 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),A)),zero_zero(code_code_numeral))). [resolve(4957,a,4924,a)]. 9.63/9.58 Derived: hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) != hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),A)) | ti(code_code_numeral,A) = ti(code_code_numeral,B). [resolve(4957,a,4925,a)]. 9.63/9.58 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),C)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),B),C))). [resolve(4957,a,4926,a)]. 9.64/9.58 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),one_one(code_code_numeral)),A)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),one_one(code_code_numeral)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),hAPP(nat,nat,suc,B)))). [resolve(4957,a,4927,a)]. 9.64/9.58 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),zero_zero(code_code_numeral)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),one_one(code_code_numeral)),one_one(code_code_numeral)))). [resolve(4957,a,4929,a)]. 9.64/9.58 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),f373(code_code_numeral,A,B,C,D)),D)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(fun(nat,bool),code_code_numeral,hAPP(fun(nat,code_code_numeral),fun(fun(nat,bool),code_code_numeral),big_co1399186613setsum(nat,code_code_numeral),B),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),D),A)))),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),D)),C))). [resolve(4957,a,4930,a)]. 9.64/9.58 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(nat,code_code_numeral,A,f373(code_code_numeral,B,A,C,D))),C)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(fun(nat,bool),code_code_numeral,hAPP(fun(nat,code_code_numeral),fun(fun(nat,bool),code_code_numeral),big_co1399186613setsum(nat,code_code_numeral),A),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),D),B)))),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),D)),C))). [resolve(4957,a,4931,a)]. 9.64/9.58 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),B)). [resolve(4957,a,4932,a)]. 9.64/9.58 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),hAPP(nat,nat,fact_fact(nat),A)))). [resolve(4957,a,4933,a)]. 9.64/9.58 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),one_one(code_code_numeral)),A)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),one_one(code_code_numeral)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B)))). [resolve(4957,a,4934,a)]. 9.64/9.58 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),A)),hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)). [resolve(4957,a,4935,a)]. 9.64/9.59 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),A)),hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)). [resolve(4957,a,4937,a)]. 9.64/9.59 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),A)),hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),B))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)). [resolve(4957,a,4938,a)]. 9.64/9.59 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),one_one(code_code_numeral)),A)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),one_one(code_code_numeral)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B))). [resolve(4957,a,4939,a)]. 9.64/9.59 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),one_one(code_code_numeral)),A)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)). [resolve(4957,a,4940,a)]. 9.64/9.59 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),zero_zero(code_code_numeral)),hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),A))). [resolve(4957,a,4941,a)]. 9.64/9.59 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),zero_zero(code_code_numeral)),hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),A))). [resolve(4957,a,4942,a)]. 9.64/9.59 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),one_one(code_code_numeral))). [resolve(4957,a,4944,a)]. 9.64/9.59 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),one_one(code_code_numeral)),A)) | B != C | hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B) = hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),C). [resolve(4957,a,4945,a)]. 9.64/9.59 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),one_one(code_code_numeral)),A)) | B = C | hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B) != hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),C). [resolve(4957,a,4946,a)]. 9.64/9.59 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),zero_zero(code_code_numeral)),A)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),B),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),B),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),C))). [resolve(4957,a,4947,a)]. 9.64/9.59 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),zero_zero(code_code_numeral)),hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),hAPP(nat,nat,fact_fact(nat),A)))). [resolve(4957,a,4949,a)]. 9.64/9.59 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f504(A,code_code_numeral,B,C,D)),D)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),C),D)),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),B),D))). [resolve(4957,a,4950,a)]. 9.64/9.59 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),hAPP(A,code_code_numeral,B,f504(A,code_code_numeral,C,B,D)))) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(A,code_code_numeral,B,f504(A,code_code_numeral,C,B,D))),hAPP(A,code_code_numeral,C,f504(A,code_code_numeral,C,B,D)))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),B),D)),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),C),D))). [resolve(4957,a,4951,a)]. 9.64/9.59 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),A))). [resolve(4957,a,4952,a)]. 9.64/9.59 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),zero_zero(code_code_numeral)),A)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),one_one(code_code_numeral))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B))),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B))). [resolve(4957,a,4953,a)]. 9.64/9.59 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),C)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),C),one_one(code_code_numeral))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),C),B)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),C),A))). [resolve(4957,a,4954,a)]. 9.64/9.59 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),one_one(code_code_numeral)))). [resolve(4957,a,4955,a)]. 9.64/9.59 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),one_one(code_code_numeral)),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),C),A)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),C),B))). [resolve(4957,a,4956,a)]. 9.68/9.67 4958 -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,hAPP(A,fun(nat,A),power_power(A),B),C)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),D))) # label(fact_239_power__increasing__iff) # label(axiom). [clausify(3640)]. 9.68/9.67 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),C))). [resolve(4958,a,4911,a)]. 9.68/9.67 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,hAPP(int,fun(nat,int),power_power(int),A),B)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),C))). [resolve(4958,a,4922,a)]. 9.68/9.67 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,hAPP(real,fun(nat,real),power_power(real),A),B)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),C))). [resolve(4958,a,4943,a)]. 9.68/9.67 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),one_one(code_code_numeral)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),C))). [resolve(4958,a,4957,a)]. 9.68/9.67 4959 -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,hAPP(A,fun(nat,A),power_power(A),B),C)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),D))) # label(fact_239_power__increasing__iff) # label(axiom). [clausify(3640)]. 9.68/9.67 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),C))). [resolve(4959,a,4911,a)]. 9.68/9.67 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,hAPP(int,fun(nat,int),power_power(int),A),B)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),C))). [resolve(4959,a,4922,a)]. 9.68/9.67 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,hAPP(real,fun(nat,real),power_power(real),A),B)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),C))). [resolve(4959,a,4943,a)]. 9.68/9.67 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),one_one(code_code_numeral)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),C))). [resolve(4959,a,4957,a)]. 9.85/9.84 4960 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),zero_zero(A))) # label(fact_180_not__one__less__zero) # label(axiom). [clausify(3676)]. 9.85/9.84 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),zero_zero(rat))). [resolve(4960,a,4911,a)]. 9.85/9.84 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),zero_zero(int))). [resolve(4960,a,4922,a)]. 9.85/9.84 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),zero_zero(real))). [resolve(4960,a,4943,a)]. 9.85/9.84 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),one_one(code_code_numeral)),zero_zero(code_code_numeral))). [resolve(4960,a,4957,a)]. 9.85/9.84 4961 -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,hAPP(A,fun(nat,A),power_power(A),B),C)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),D))) # label(fact_185_power__strict__increasing__iff) # label(axiom). [clausify(3677)]. 9.85/9.84 4962 -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,hAPP(A,fun(nat,A),power_power(A),B),C)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),D))) # label(fact_185_power__strict__increasing__iff) # label(axiom). [clausify(3677)]. 9.85/9.84 4963 linordered_semidom(quickcheck_code_int) # label(arity_Quickcheck__Narrowing_Ocode__int___Rings_Olinordered__semidom) # label(axiom). [assumption]. 9.85/9.84 Derived: hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),hAPP(nat,nat,suc,B)) != hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),C),hAPP(nat,nat,suc,B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),C)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),A)) | ti(quickcheck_code_int,A) = ti(quickcheck_code_int,C). [resolve(4963,a,4910,a)]. 9.85/9.84 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),A)),hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),B))). [resolve(4963,a,4912,a)]. 9.85/9.84 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),zero_zero(quickcheck_code_int)),C)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),C),one_one(quickcheck_code_int))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),C),B)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),C),A))). [resolve(4963,a,4913,a)]. 9.85/9.84 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f94(A,quickcheck_code_int,B,C)),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),B),C))). [resolve(4963,a,4914,a)]. 9.85/9.84 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(A,quickcheck_code_int,B,f94(A,quickcheck_code_int,B,C)))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),B),C))). [resolve(4963,a,4915,a)]. 9.85/9.84 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f173(A,quickcheck_code_int,B,C)),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),B),C))). [resolve(4963,a,4916,a)]. 9.85/9.84 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(A,quickcheck_code_int,B,f173(A,quickcheck_code_int,B,C)))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),B),C))). [resolve(4963,a,4917,a)]. 9.85/9.84 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),hAPP(nat,nat,suc,B))),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),C),hAPP(nat,nat,suc,B)))) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),C)). [resolve(4963,a,4918,a)]. 9.85/9.84 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),A)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),C)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),B),C))). [resolve(4963,a,4919,a)]. 9.85/9.84 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),zero_zero(quickcheck_code_int)),one_one(quickcheck_code_int))). [resolve(4963,a,4920,a)]. 9.85/9.84 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),one_one(quickcheck_code_int)),A)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),one_one(quickcheck_code_int)),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),one_one(quickcheck_code_int)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B))). [resolve(4963,a,4921,a)]. 9.85/9.85 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),C),B))) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),C)). [resolve(4963,a,4923,a)]. 9.85/9.85 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),A)),zero_zero(quickcheck_code_int))). [resolve(4963,a,4924,a)]. 9.85/9.85 Derived: hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) != hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),A)) | ti(quickcheck_code_int,A) = ti(quickcheck_code_int,B). [resolve(4963,a,4925,a)]. 9.85/9.85 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),C)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),B),C))). [resolve(4963,a,4926,a)]. 9.85/9.85 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),one_one(quickcheck_code_int)),A)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),one_one(quickcheck_code_int)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),hAPP(nat,nat,suc,B)))). [resolve(4963,a,4927,a)]. 9.85/9.85 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),one_one(quickcheck_code_int)),one_one(quickcheck_code_int)))). [resolve(4963,a,4929,a)]. 9.85/9.85 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),f373(quickcheck_code_int,A,B,C,D)),D)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(fun(nat,bool),quickcheck_code_int,hAPP(fun(nat,quickcheck_code_int),fun(fun(nat,bool),quickcheck_code_int),big_co1399186613setsum(nat,quickcheck_code_int),B),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),D),A)))),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),D)),C))). [resolve(4963,a,4930,a)]. 9.85/9.85 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(nat,quickcheck_code_int,A,f373(quickcheck_code_int,B,A,C,D))),C)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(fun(nat,bool),quickcheck_code_int,hAPP(fun(nat,quickcheck_code_int),fun(fun(nat,bool),quickcheck_code_int),big_co1399186613setsum(nat,quickcheck_code_int),A),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),D),B)))),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),D)),C))). [resolve(4963,a,4931,a)]. 9.85/9.85 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),B)). [resolve(4963,a,4932,a)]. 9.85/9.85 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),hAPP(nat,nat,fact_fact(nat),A)))). [resolve(4963,a,4933,a)]. 9.85/9.85 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),one_one(quickcheck_code_int)),A)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),one_one(quickcheck_code_int)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B)))). [resolve(4963,a,4934,a)]. 9.85/9.85 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),A)),hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)). [resolve(4963,a,4935,a)]. 9.85/9.85 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),A)),hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)). [resolve(4963,a,4937,a)]. 9.85/9.85 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),A)),hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),B))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)). [resolve(4963,a,4938,a)]. 9.85/9.85 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),one_one(quickcheck_code_int)),A)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),one_one(quickcheck_code_int)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B))). [resolve(4963,a,4939,a)]. 9.85/9.85 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),one_one(quickcheck_code_int)),A)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)). [resolve(4963,a,4940,a)]. 9.85/9.85 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),A))). [resolve(4963,a,4941,a)]. 9.85/9.85 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),A))). [resolve(4963,a,4942,a)]. 9.85/9.85 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),one_one(quickcheck_code_int))). [resolve(4963,a,4944,a)]. 9.85/9.85 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),one_one(quickcheck_code_int)),A)) | B != C | hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B) = hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),C). [resolve(4963,a,4945,a)]. 9.85/9.85 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),one_one(quickcheck_code_int)),A)) | B = C | hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B) != hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),C). [resolve(4963,a,4946,a)]. 9.85/9.85 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),zero_zero(quickcheck_code_int)),A)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),B),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),B),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),C))). [resolve(4963,a,4947,a)]. 9.85/9.85 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),hAPP(nat,nat,fact_fact(nat),A)))). [resolve(4963,a,4949,a)]. 9.85/9.86 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f504(A,quickcheck_code_int,B,C,D)),D)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),C),D)),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),B),D))). [resolve(4963,a,4950,a)]. 9.85/9.86 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(A,quickcheck_code_int,B,f504(A,quickcheck_code_int,C,B,D)))) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(A,quickcheck_code_int,B,f504(A,quickcheck_code_int,C,B,D))),hAPP(A,quickcheck_code_int,C,f504(A,quickcheck_code_int,C,B,D)))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),B),D)),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),C),D))). [resolve(4963,a,4951,a)]. 9.85/9.86 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),A))). [resolve(4963,a,4952,a)]. 9.85/9.86 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),zero_zero(quickcheck_code_int)),A)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),one_one(quickcheck_code_int))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B))),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B))). [resolve(4963,a,4953,a)]. 9.85/9.86 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),C)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),C),one_one(quickcheck_code_int))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),C),B)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),C),A))). [resolve(4963,a,4954,a)]. 9.85/9.86 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),one_one(quickcheck_code_int)))). [resolve(4963,a,4955,a)]. 9.85/9.86 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),one_one(quickcheck_code_int)),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),C),A)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),C),B))). [resolve(4963,a,4956,a)]. 9.93/9.88 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),one_one(quickcheck_code_int)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),C))). [resolve(4963,a,4958,a)]. 9.93/9.88 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),one_one(quickcheck_code_int)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),C))). [resolve(4963,a,4959,a)]. 9.93/9.88 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),one_one(quickcheck_code_int)),zero_zero(quickcheck_code_int))). [resolve(4963,a,4960,a)]. 9.93/9.88 4964 -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),B),one_one(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(nat,nat,suc,C))),B)) # label(fact_1456_realpow__Suc__le__self) # label(axiom). [clausify(3864)]. 9.93/9.88 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),one_one(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(nat,nat,suc,B))),A)). [resolve(4964,a,4911,a)]. 9.93/9.88 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),one_one(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),hAPP(nat,nat,suc,B))),A)). [resolve(4964,a,4922,a)]. 9.93/9.88 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),one_one(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(nat,nat,suc,B))),A)). [resolve(4964,a,4943,a)]. 9.93/9.88 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),A)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),one_one(code_code_numeral))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),hAPP(nat,nat,suc,B))),A)). [resolve(4964,a,4957,a)]. 10.16/10.13 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),A)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),one_one(quickcheck_code_int))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),hAPP(nat,nat,suc,B))),A)). [resolve(4964,a,4963,a)]. 10.16/10.13 4965 -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_255_zero__le__imp__of__nat) # label(axiom). [clausify(3908)]. 10.16/10.13 4966 -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,hAPP(A,fun(nat,A),power_power(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C)))) # label(fact_493_power__less__power__Suc) # label(axiom). [clausify(3918)]. 10.16/10.13 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B)))). [resolve(4966,a,4911,a)]. 10.16/10.13 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,hAPP(int,fun(nat,int),power_power(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),B)))). [resolve(4966,a,4922,a)]. 10.16/10.13 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B)))). [resolve(4966,a,4943,a)]. 10.16/10.13 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),one_one(code_code_numeral)),A)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B)))). [resolve(4966,a,4957,a)]. 10.16/10.13 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),one_one(quickcheck_code_int)),A)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B)))). [resolve(4966,a,4963,a)]. 10.16/10.13 4967 linordered_semidom(nat) # label(arity_Nat_Onat___Rings_Olinordered__semidom) # label(axiom). [assumption]. 10.16/10.13 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),hAPP(nat,nat,suc,B)) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),C),hAPP(nat,nat,suc,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)) | ti(nat,A) = ti(nat,C). [resolve(4967,a,4910,a)]. 10.16/10.14 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,semiring_1_of_nat(nat),A)),hAPP(nat,nat,semiring_1_of_nat(nat),B))). [resolve(4967,a,4912,a)]. 10.16/10.14 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),C),one_one(nat))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),C),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),C),A))). [resolve(4967,a,4913,a)]. 10.16/10.14 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f94(A,nat,B,C)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),B),C))). [resolve(4967,a,4914,a)]. 10.16/10.14 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(A,nat,B,f94(A,nat,B,C)))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),B),C))). [resolve(4967,a,4915,a)]. 10.16/10.14 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),hAPP(nat,nat,suc,B))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),C),hAPP(nat,nat,suc,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),A),C)). [resolve(4967,a,4918,a)]. 10.16/10.14 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,hAPP(nat,fun(nat,nat),power_power(nat),A),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),C))). [resolve(4967,a,4919,a)]. 10.16/10.14 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),one_one(nat))). [resolve(4967,a,4920,a)]. 10.16/10.14 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B))). [resolve(4967,a,4921,a)]. 10.16/10.14 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(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(4967,a,4923,a)]. 10.16/10.14 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,semiring_1_of_nat(nat),A)),zero_zero(nat))). [resolve(4967,a,4924,a)]. 10.16/10.14 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,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(4967,a,4925,a)]. 10.16/10.14 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_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,hAPP(nat,fun(nat,nat),power_power(nat),A),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),C))). [resolve(4967,a,4926,a)]. 10.16/10.14 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),hAPP(nat,nat,suc,B)))). [resolve(4967,a,4927,a)]. 10.16/10.14 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))). [resolve(4967,a,4929,a)]. 10.16/10.14 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),f373(nat,A,B,C,D)),D)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),B),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),D),A)))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,semiring_1_of_nat(nat),D)),C))). [resolve(4967,a,4930,a)]. 10.16/10.14 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,A,f373(nat,B,A,C,D))),C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),A),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),D),B)))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,semiring_1_of_nat(nat),D)),C))). [resolve(4967,a,4931,a)]. 10.16/10.14 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,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(4967,a,4932,a)]. 10.16/10.14 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),B)))). [resolve(4967,a,4934,a)]. 10.16/10.14 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(4967,a,4935,a)]. 10.16/10.14 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,semiring_1_of_nat(nat),A)),hAPP(nat,nat,semiring_1_of_nat(nat),B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)). [resolve(4967,a,4937,a)]. 10.16/10.14 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,semiring_1_of_nat(nat),A)),hAPP(nat,nat,semiring_1_of_nat(nat),B))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)). [resolve(4967,a,4938,a)]. 10.16/10.14 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,hAPP(nat,fun(nat,nat),power_power(nat),A),B))). [resolve(4967,a,4939,a)]. 10.16/10.14 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,hAPP(nat,fun(nat,nat),power_power(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)). [resolve(4967,a,4940,a)]. 10.16/10.14 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(4967,a,4941,a)]. 10.16/10.14 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(4967,a,4942,a)]. 10.16/10.15 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),A)) | B != C | hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),C). [resolve(4967,a,4945,a)]. 10.16/10.15 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),A)) | B = C | hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),B) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),C). [resolve(4967,a,4946,a)]. 10.16/10.15 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),C))). [resolve(4967,a,4947,a)]. 10.16/10.15 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,semiring_1_of_nat(nat),hAPP(nat,nat,fact_fact(nat),A)))). [resolve(4967,a,4949,a)]. 10.16/10.15 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f504(A,nat,B,C,D)),D)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),C),D)),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),B),D))). [resolve(4967,a,4950,a)]. 10.16/10.15 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),hAPP(A,nat,B,f504(A,nat,C,B,D)))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(A,nat,B,f504(A,nat,C,B,D))),hAPP(A,nat,C,f504(A,nat,C,B,D)))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),B),D)),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),C),D))). [resolve(4967,a,4951,a)]. 10.16/10.15 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),one_one(nat))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),B))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),B))). [resolve(4967,a,4953,a)]. 10.16/10.15 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),one_one(nat))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),C),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),C),A))). [resolve(4967,a,4954,a)]. 10.16/10.15 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),one_one(nat)))). [resolve(4967,a,4955,a)]. 10.16/10.15 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),C),A)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),C),B))). [resolve(4967,a,4956,a)]. 10.16/10.15 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,hAPP(nat,fun(nat,nat),power_power(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),C))). [resolve(4967,a,4958,a)]. 10.16/10.15 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,hAPP(nat,fun(nat,nat),power_power(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),C))). [resolve(4967,a,4959,a)]. 10.25/10.24 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),zero_zero(nat))). [resolve(4967,a,4960,a)]. 10.25/10.24 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),A),one_one(nat))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),hAPP(nat,nat,suc,B))),A)). [resolve(4967,a,4964,a)]. 10.25/10.24 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,hAPP(nat,fun(nat,nat),power_power(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),B)))). [resolve(4967,a,4966,a)]. 10.25/10.24 4968 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),one_one(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(nat,nat,suc,C))),one_one(A))) # label(fact_1457_power__Suc__less__one) # label(axiom). [clausify(4128)]. 10.25/10.24 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),one_one(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(nat,nat,suc,B))),one_one(rat))). [resolve(4968,a,4911,a)]. 10.25/10.24 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),one_one(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),hAPP(nat,nat,suc,B))),one_one(int))). [resolve(4968,a,4922,a)]. 10.25/10.24 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),one_one(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(nat,nat,suc,B))),one_one(real))). [resolve(4968,a,4943,a)]. 10.25/10.24 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),zero_zero(code_code_numeral)),A)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),one_one(code_code_numeral))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),hAPP(nat,nat,suc,B))),one_one(code_code_numeral))). [resolve(4968,a,4957,a)]. 10.25/10.24 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),zero_zero(quickcheck_code_int)),A)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),one_one(quickcheck_code_int))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),hAPP(nat,nat,suc,B))),one_one(quickcheck_code_int))). [resolve(4968,a,4963,a)]. 10.25/10.24 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),one_one(nat))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),hAPP(nat,nat,suc,B))),one_one(nat))). [resolve(4968,a,4967,a)]. 10.25/10.24 4969 -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,hAPP(A,fun(nat,A),power_power(A),B),C)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),D))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) # label(fact_238_power__le__imp__le__exp) # label(axiom). [clausify(4202)]. 10.46/10.42 4970 -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,hAPP(A,fun(nat,A),power_power(A),B),C))) # label(fact_252_zero__le__power) # label(axiom). [clausify(4246)]. 10.46/10.42 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B))). [resolve(4970,a,4911,a)]. 10.46/10.42 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,hAPP(int,fun(nat,int),power_power(int),A),B))). [resolve(4970,a,4922,a)]. 10.46/10.42 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,hAPP(real,fun(nat,real),power_power(real),A),B))). [resolve(4970,a,4943,a)]. 10.46/10.42 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),A)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B))). [resolve(4970,a,4957,a)]. 10.46/10.42 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),A)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B))). [resolve(4970,a,4963,a)]. 10.46/10.42 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,hAPP(nat,fun(nat,nat),power_power(nat),A),B))). [resolve(4970,a,4967,a)]. 10.46/10.42 4971 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),zero_zero(A))) # label(fact_249_not__one__le__zero) # label(axiom). [clausify(4296)]. 10.46/10.42 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),zero_zero(rat))). [resolve(4971,a,4911,a)]. 10.46/10.42 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),zero_zero(int))). [resolve(4971,a,4922,a)]. 10.46/10.42 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),zero_zero(real))). [resolve(4971,a,4943,a)]. 10.46/10.42 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),one_one(code_code_numeral)),zero_zero(code_code_numeral))). [resolve(4971,a,4957,a)]. 10.46/10.42 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),one_one(quickcheck_code_int)),zero_zero(quickcheck_code_int))). [resolve(4971,a,4963,a)]. 10.46/10.42 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),zero_zero(nat))). [resolve(4971,a,4967,a)]. 10.46/10.42 4972 -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,hAPP(A,fun(nat,A),power_power(A),B),C))) # label(fact_111_one__less__power) # label(axiom). [clausify(4386)]. 10.46/10.42 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B))). [resolve(4972,a,4911,a)]. 10.78/10.72 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,hAPP(int,fun(nat,int),power_power(int),A),B))). [resolve(4972,a,4922,a)]. 10.78/10.72 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,hAPP(real,fun(nat,real),power_power(real),A),B))). [resolve(4972,a,4943,a)]. 10.78/10.72 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),one_one(code_code_numeral)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),one_one(code_code_numeral)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B))). [resolve(4972,a,4957,a)]. 10.78/10.72 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),one_one(quickcheck_code_int)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),one_one(quickcheck_code_int)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B))). [resolve(4972,a,4963,a)]. 10.78/10.72 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,hAPP(nat,fun(nat,nat),power_power(nat),A),B))). [resolve(4972,a,4967,a)]. 10.78/10.72 4973 -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,hAPP(A,fun(nat,A),power_power(A),D),B)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),D),C))) # label(fact_216_power__increasing) # label(axiom). [clausify(4686)]. 10.78/10.72 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),C),A)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),C),B))). [resolve(4973,a,4911,a)]. 10.78/10.72 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,hAPP(int,fun(nat,int),power_power(int),C),A)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),C),B))). [resolve(4973,a,4922,a)]. 10.78/10.72 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,hAPP(real,fun(nat,real),power_power(real),C),A)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),C),B))). [resolve(4973,a,4943,a)]. 10.78/10.72 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),one_one(code_code_numeral)),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),C),A)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),C),B))). [resolve(4973,a,4957,a)]. 10.78/10.78 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),one_one(quickcheck_code_int)),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),C),A)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),C),B))). [resolve(4973,a,4963,a)]. 10.78/10.78 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),C),A)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),C),B))). [resolve(4973,a,4967,a)]. 10.78/10.78 4974 -linordered_semidom(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C) != hAPP(nat,A,hAPP(A,fun(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_304_power__eq__imp__eq__base) # label(axiom). [clausify(4733)]. 10.78/10.78 Derived: hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B) != hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),C),B) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),C)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | ti(rat,A) = ti(rat,C). [resolve(4974,a,4911,a)]. 10.78/10.78 Derived: hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),B) != hAPP(nat,int,hAPP(int,fun(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(4974,a,4922,a)]. 10.78/10.78 Derived: hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B) != hAPP(nat,real,hAPP(real,fun(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(4974,a,4943,a)]. 10.78/10.78 Derived: hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B) != hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),C),B) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),C)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | ti(code_code_numeral,A) = ti(code_code_numeral,C). [resolve(4974,a,4957,a)]. 10.78/10.78 Derived: hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B) != hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),C),B) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),C)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | ti(quickcheck_code_int,A) = ti(quickcheck_code_int,C). [resolve(4974,a,4963,a)]. 11.60/11.54 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),B) != hAPP(nat,nat,hAPP(nat,fun(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(4974,a,4967,a)]. 11.60/11.54 4975 -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,hAPP(A,fun(nat,A),power_power(A),B),C))) # label(fact_182_zero__less__power) # label(axiom). [clausify(4773)]. 11.60/11.54 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B))). [resolve(4975,a,4911,a)]. 11.60/11.54 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),B))). [resolve(4975,a,4922,a)]. 11.60/11.54 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B))). [resolve(4975,a,4943,a)]. 11.60/11.54 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),zero_zero(code_code_numeral)),A)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),zero_zero(code_code_numeral)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B))). [resolve(4975,a,4957,a)]. 11.60/11.54 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),zero_zero(quickcheck_code_int)),A)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B))). [resolve(4975,a,4963,a)]. 11.60/11.54 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),B))). [resolve(4975,a,4967,a)]. 11.60/11.54 4976 number_ring(real) # label(arity_RealDef_Oreal___Int_Onumber__ring) # label(axiom). [assumption]. 11.60/11.54 4977 -number_ring(A) | hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),hAPP(int,int,uminus_uminus(int),B))) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),one_one(A)),hAPP(int,A,number_number_of(A),B)) # label(fact_2104_diff__special_I1_J) # label(axiom). [clausify(1)]. 11.60/11.54 4978 -number_ring(A) | -ring_char_0(A) | -hBOOL(hAPP(A,bool,iszero(A),hAPP(int,A,number_number_of(A),hAPP(int,int,bit0,B)))) | hBOOL(hAPP(A,bool,iszero(A),hAPP(int,A,number_number_of(A),B))) # label(fact_2157_iszero__number__of__Bit0) # label(axiom). [clausify(84)]. 11.60/11.54 4979 -number_ring(A) | -ring_char_0(A) | hBOOL(hAPP(A,bool,iszero(A),hAPP(int,A,number_number_of(A),hAPP(int,int,bit0,B)))) | -hBOOL(hAPP(A,bool,iszero(A),hAPP(int,A,number_number_of(A),B))) # label(fact_2157_iszero__number__of__Bit0) # label(axiom). [clausify(84)]. 11.60/11.54 4980 -number_ring(A) | -field(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),hAPP(int,A,number_number_of(A),hAPP(int,int,bit1,pls))) # label(fact_1782_divide__numeral__1) # label(axiom). [clausify(144)]. 11.60/11.54 4981 -number_ring(A) | one_one(A) = hAPP(int,A,number_number_of(A),hAPP(int,int,bit1,pls)) # label(fact_13_numeral__1__eq__1) # label(axiom). [clausify(163)]. 11.60/11.54 4982 -number_ring(A) | hAPP(int,A,number_number_of(A),min) = hAPP(A,A,uminus_uminus(A),one_one(A)) # label(fact_2016_arith__simps_I31_J) # label(axiom). [clausify(243)]. 11.60/11.54 4983 -number_ring(A) | -linordered_idom(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,pls)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),hAPP(int,A,number_number_of(A),B))) # label(fact_63_less__special_I2_J) # label(axiom). [clausify(348)]. 11.60/11.54 4984 -number_ring(A) | -linordered_idom(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,pls)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),hAPP(int,A,number_number_of(A),B))) # label(fact_63_less__special_I2_J) # label(axiom). [clausify(348)]. 11.60/11.54 4985 -number_ring(A) | one_one(A) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(int,A,number_number_of(A),min)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B)) # label(fact_695_power__m1__even) # label(axiom). [clausify(360)]. 11.60/11.54 4986 -number_ring(A) | hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(int,A,number_number_of(A),B)),hAPP(int,A,number_number_of(A),C)) # label(fact_370_arith__simps_I32_J) # label(axiom). [clausify(365)]. 11.60/11.54 4987 -number_ring(A) | -hBOOL(hAPP(A,bool,iszero(A),hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),hAPP(int,int,uminus_uminus(int),C))))) | hAPP(int,A,number_number_of(A),C) = hAPP(int,A,number_number_of(A),B) # label(fact_2162_eq__number__of__eq) # label(axiom). [clausify(393)]. 11.60/11.54 4988 -number_ring(A) | hBOOL(hAPP(A,bool,iszero(A),hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),hAPP(int,int,uminus_uminus(int),C))))) | hAPP(int,A,number_number_of(A),C) != hAPP(int,A,number_number_of(A),B) # label(fact_2162_eq__number__of__eq) # label(axiom). [clausify(393)]. 11.60/11.54 4989 -number_ring(A) | hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),hAPP(int,int,uminus_uminus(int),C))) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(int,A,number_number_of(A),B)),hAPP(int,A,number_number_of(A),C)) # label(fact_2082_diff__number__of__eq) # label(axiom). [clausify(401)]. 11.60/11.54 4990 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(int,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_235_le__special_I3_J) # label(axiom). [clausify(417)]. 11.60/11.54 4991 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(int,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_235_le__special_I3_J) # label(axiom). [clausify(417)]. 11.60/11.54 4992 -number_ring(A) | hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(int,A,number_number_of(A),B)),hAPP(int,A,number_number_of(A),C)) # label(fact_716_number__of__diff) # label(axiom). [clausify(449)]. 11.60/11.54 4993 -number_ring(A) | hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(int,A,number_number_of(A),B)),hAPP(int,A,number_number_of(A),C)) # label(fact_91_add__number__of__eq) # label(axiom). [clausify(518)]. 11.60/11.54 4994 -number_ring(A) | zero_zero(A) = hAPP(int,A,number_number_of(A),pls) # label(fact_86_semiring__norm_I112_J) # label(axiom). [clausify(561)]. 11.60/11.54 4995 -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),hAPP(int,A,number_number_of(A),B)),hAPP(int,A,number_number_of(A),C))) # label(fact_18_less__number__of) # label(axiom). [clausify(581)]. 11.60/11.56 4996 -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),hAPP(int,A,number_number_of(A),B)),hAPP(int,A,number_number_of(A),C))) # label(fact_18_less__number__of) # label(axiom). [clausify(581)]. 11.60/11.56 Derived: hAPP(int,real,number_number_of(real),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),hAPP(int,int,uminus_uminus(int),A))) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(int,real,number_number_of(real),A)). [resolve(4976,a,4977,a)]. 11.60/11.56 Derived: -ring_char_0(real) | -hBOOL(hAPP(real,bool,iszero(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,A)))) | hBOOL(hAPP(real,bool,iszero(real),hAPP(int,real,number_number_of(real),A))). [resolve(4976,a,4978,a)]. 11.60/11.56 Derived: -ring_char_0(real) | hBOOL(hAPP(real,bool,iszero(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,A)))) | -hBOOL(hAPP(real,bool,iszero(real),hAPP(int,real,number_number_of(real),A))). [resolve(4976,a,4979,a)]. 11.60/11.56 Derived: -field(real) | ti(real,A) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,pls))). [resolve(4976,a,4980,a)]. 11.60/11.56 Derived: one_one(real) = hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,pls)). [resolve(4976,a,4981,a)]. 11.60/11.56 Derived: hAPP(int,real,number_number_of(real),min) = hAPP(real,real,uminus_uminus(real),one_one(real)). [resolve(4976,a,4982,a)]. 11.60/11.56 Derived: -linordered_idom(real) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,pls)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),hAPP(int,real,number_number_of(real),A))). [resolve(4976,a,4983,a)]. 11.60/11.56 Derived: -linordered_idom(real) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,pls)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),hAPP(int,real,number_number_of(real),A))). [resolve(4976,a,4984,a)]. 11.60/11.56 Derived: one_one(real) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A)). [resolve(4976,a,4985,a)]. 11.60/11.56 Derived: hAPP(int,real,number_number_of(real),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),A)),hAPP(int,real,number_number_of(real),B)). [resolve(4976,a,4986,a)]. 11.60/11.56 Derived: -hBOOL(hAPP(real,bool,iszero(real),hAPP(int,real,number_number_of(real),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),B))))) | hAPP(int,real,number_number_of(real),B) = hAPP(int,real,number_number_of(real),A). [resolve(4976,a,4987,a)]. 11.60/11.56 Derived: hBOOL(hAPP(real,bool,iszero(real),hAPP(int,real,number_number_of(real),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),B))))) | hAPP(int,real,number_number_of(real),B) != hAPP(int,real,number_number_of(real),A). [resolve(4976,a,4988,a)]. 11.60/11.56 Derived: hAPP(int,real,number_number_of(real),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),B))) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,number_number_of(real),A)),hAPP(int,real,number_number_of(real),B)). [resolve(4976,a,4989,a)]. 11.60/11.56 Derived: -linordered_idom(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,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(4976,a,4990,b)]. 11.60/11.56 Derived: -linordered_idom(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,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(4976,a,4991,b)]. 11.60/11.60 Derived: hAPP(int,real,number_number_of(real),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,number_number_of(real),A)),hAPP(int,real,number_number_of(real),B)). [resolve(4976,a,4992,a)]. 11.60/11.60 Derived: hAPP(int,real,number_number_of(real),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,number_number_of(real),A)),hAPP(int,real,number_number_of(real),B)). [resolve(4976,a,4993,a)]. 11.60/11.60 Derived: zero_zero(real) = hAPP(int,real,number_number_of(real),pls). [resolve(4976,a,4994,a)]. 11.60/11.60 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),hAPP(int,real,number_number_of(real),A)),hAPP(int,real,number_number_of(real),B))). [resolve(4976,a,4995,b)]. 11.60/11.60 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),hAPP(int,real,number_number_of(real),A)),hAPP(int,real,number_number_of(real),B))). [resolve(4976,a,4996,b)]. 11.60/11.60 4997 -number_ring(A) | hBOOL(hAPP(nat,bool,even_odd_even(nat),B)) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(int,A,number_number_of(A),min)),B) = hAPP(int,A,number_number_of(A),min) # label(fact_2512_neg__one__even__odd__power) # label(axiom). [clausify(600)]. 11.60/11.60 Derived: hBOOL(hAPP(nat,bool,even_odd_even(nat),A)) | hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),A) = hAPP(int,real,number_number_of(real),min). [resolve(4997,a,4976,a)]. 11.60/11.60 4998 -number_ring(A) | -hBOOL(hAPP(nat,bool,even_odd_even(nat),B)) | one_one(A) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(int,A,number_number_of(A),min)),B) # label(fact_2512_neg__one__even__odd__power) # label(axiom). [clausify(600)]. 11.60/11.60 Derived: -hBOOL(hAPP(nat,bool,even_odd_even(nat),A)) | one_one(real) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),A). [resolve(4998,a,4976,a)]. 11.60/11.60 4999 -number_ring(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(int,A,number_number_of(A),pls)),B) # label(fact_87_add__numeral__0) # label(axiom). [clausify(664)]. 11.60/11.60 Derived: ti(real,A) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,number_number_of(real),pls)),A). [resolve(4999,a,4976,a)]. 11.60/11.60 5000 number_ring(int) # label(arity_Int_Oint___Int_Onumber__ring) # label(axiom). [assumption]. 11.60/11.60 Derived: hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),hAPP(int,int,uminus_uminus(int),A))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),one_one(int)),hAPP(int,int,number_number_of(int),A)). [resolve(5000,a,4977,a)]. 11.60/11.60 Derived: -ring_char_0(int) | -hBOOL(hAPP(int,bool,iszero(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,A)))) | hBOOL(hAPP(int,bool,iszero(int),hAPP(int,int,number_number_of(int),A))). [resolve(5000,a,4978,a)]. 11.60/11.60 Derived: -ring_char_0(int) | hBOOL(hAPP(int,bool,iszero(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,A)))) | -hBOOL(hAPP(int,bool,iszero(int),hAPP(int,int,number_number_of(int),A))). [resolve(5000,a,4979,a)]. 11.60/11.60 Derived: -field(int) | ti(int,A) = hAPP(int,int,hAPP(int,fun(int,int),inverse_divide(int),A),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,pls))). [resolve(5000,a,4980,a)]. 11.60/11.60 Derived: one_one(int) = hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,pls)). [resolve(5000,a,4981,a)]. 11.60/11.60 Derived: hAPP(int,int,number_number_of(int),min) = hAPP(int,int,uminus_uminus(int),one_one(int)). [resolve(5000,a,4982,a)]. 11.60/11.60 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,pls)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(int,int,number_number_of(int),A))). [resolve(5000,a,4983,a)]. 11.60/11.60 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,pls)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(int,int,number_number_of(int),A))). [resolve(5000,a,4984,a)]. 11.68/11.64 Derived: one_one(int) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A)). [resolve(5000,a,4985,a)]. 11.68/11.64 Derived: -hBOOL(hAPP(int,bool,iszero(int),hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),B))))) | hAPP(int,int,number_number_of(int),B) = hAPP(int,int,number_number_of(int),A). [resolve(5000,a,4987,a)]. 11.68/11.64 Derived: hBOOL(hAPP(int,bool,iszero(int),hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),B))))) | hAPP(int,int,number_number_of(int),B) != hAPP(int,int,number_number_of(int),A). [resolve(5000,a,4988,a)]. 11.68/11.64 Derived: hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),B))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,number_number_of(int),A)),hAPP(int,int,number_number_of(int),B)). [resolve(5000,a,4989,a)]. 11.68/11.64 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,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(5000,a,4990,b)]. 11.68/11.64 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,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(5000,a,4991,b)]. 11.68/11.64 Derived: hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,number_number_of(int),A)),hAPP(int,int,number_number_of(int),B)). [resolve(5000,a,4992,a)]. 11.68/11.64 Derived: zero_zero(int) = hAPP(int,int,number_number_of(int),pls). [resolve(5000,a,4994,a)]. 11.68/11.64 Derived: hBOOL(hAPP(nat,bool,even_odd_even(nat),A)) | hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),A) = hAPP(int,int,number_number_of(int),min). [resolve(5000,a,4997,a)]. 11.68/11.64 Derived: -hBOOL(hAPP(nat,bool,even_odd_even(nat),A)) | one_one(int) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),A). [resolve(5000,a,4998,a)]. 11.68/11.64 Derived: ti(int,A) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),pls)),A). [resolve(5000,a,4999,a)]. 11.68/11.64 5001 -number_ring(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(int,A,number_number_of(A),hAPP(int,int,bit1,pls))) # label(fact_494_mult__numeral__1__right) # label(axiom). [clausify(837)]. 11.68/11.64 Derived: ti(real,A) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,pls))). [resolve(5001,a,4976,a)]. 11.68/11.64 Derived: ti(int,A) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,pls))). [resolve(5001,a,5000,a)]. 11.68/11.64 5002 -linord1117847801e_zero(A) | -number_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),hAPP(int,A,number_number_of(A),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) # label(fact_1807_half__gt__zero__iff) # label(axiom). [clausify(871)]. 11.68/11.64 Derived: -linord1117847801e_zero(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)). [resolve(5002,b,4976,a)]. 11.68/11.64 Derived: -linord1117847801e_zero(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),inverse_divide(int),A),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)). [resolve(5002,b,5000,a)]. 11.73/11.69 5003 -linord1117847801e_zero(A) | -number_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),hAPP(int,A,number_number_of(A),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) # label(fact_1807_half__gt__zero__iff) # label(axiom). [clausify(871)]. 11.73/11.69 Derived: -linord1117847801e_zero(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)). [resolve(5003,b,4976,a)]. 11.73/11.69 Derived: -linord1117847801e_zero(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),inverse_divide(int),A),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)). [resolve(5003,b,5000,a)]. 11.73/11.69 5004 -number_ring(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(int,A,number_number_of(A),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B)),C)) # label(fact_787_power2__diff) # label(axiom). [clausify(895)]. 11.73/11.69 Derived: hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A)),B)). [resolve(5004,a,4976,a)]. 11.73/11.69 Derived: hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A)),B)). [resolve(5004,a,5000,a)]. 11.73/11.69 5005 -number_ring(A) | -ring_char_0(A) | hAPP(int,A,number_number_of(A),B) != hAPP(int,A,number_number_of(A),C) | ti(int,B) = ti(int,C) # label(fact_40_eq__number__of) # label(axiom). [clausify(1071)]. 11.73/11.69 Derived: -ring_char_0(real) | hAPP(int,real,number_number_of(real),A) != hAPP(int,real,number_number_of(real),B) | ti(int,A) = ti(int,B). [resolve(5005,a,4976,a)]. 11.73/11.77 Derived: -ring_char_0(int) | hAPP(int,int,number_number_of(int),A) != hAPP(int,int,number_number_of(int),B) | ti(int,A) = ti(int,B). [resolve(5005,a,5000,a)]. 11.73/11.77 5006 -number_ring(A) | -ring_char_0(A) | hAPP(int,A,number_number_of(A),B) = hAPP(int,A,number_number_of(A),C) | ti(int,B) != ti(int,C) # label(fact_40_eq__number__of) # label(axiom). [clausify(1071)]. 11.73/11.77 Derived: -ring_char_0(real) | hAPP(int,real,number_number_of(real),A) = hAPP(int,real,number_number_of(real),B) | ti(int,A) != ti(int,B). [resolve(5006,a,4976,a)]. 11.73/11.77 Derived: -ring_char_0(int) | hAPP(int,int,number_number_of(int),A) = hAPP(int,int,number_number_of(int),B) | ti(int,A) != ti(int,B). [resolve(5006,a,5000,a)]. 11.73/11.77 5007 -field_inverse_zero(A) | -number_ring(A) | hAPP(A,A,uminus_uminus(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),one_one(A)),B)) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),hAPP(int,A,number_number_of(A),min)),B) # label(fact_2032_minus1__divide) # label(axiom). [clausify(1115)]. 11.73/11.77 Derived: -field_inverse_zero(real) | hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),A)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,number_number_of(real),min)),A). [resolve(5007,b,4976,a)]. 11.73/11.77 Derived: -field_inverse_zero(int) | hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),inverse_divide(int),one_one(int)),A)) = hAPP(int,int,hAPP(int,fun(int,int),inverse_divide(int),hAPP(int,int,number_number_of(int),min)),A). [resolve(5007,b,5000,a)]. 11.73/11.77 5008 -number_ring(A) | hAPP(A,A,uminus_uminus(A),B) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(int,A,number_number_of(A),min)),B) # label(fact_2013_mult__minus1) # label(axiom). [clausify(1157)]. 11.73/11.77 Derived: hAPP(real,real,uminus_uminus(real),A) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),min)),A). [resolve(5008,a,4976,a)]. 11.73/11.77 Derived: hAPP(int,int,uminus_uminus(int),A) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),min)),A). [resolve(5008,a,5000,a)]. 11.73/11.77 5009 -number_ring(A) | -real_n2089651433ebra_1(A) | hAPP(real,real,abs_abs(real),hAPP(int,real,number_number_of(real),B)) = hAPP(A,real,norm_norm(A),hAPP(int,A,number_number_of(A),B)) # label(fact_2019_norm__number__of) # label(axiom). [clausify(1166)]. 11.73/11.77 Derived: -real_n2089651433ebra_1(int) | hAPP(real,real,abs_abs(real),hAPP(int,real,number_number_of(real),A)) = hAPP(int,real,norm_norm(int),hAPP(int,int,number_number_of(int),A)). [resolve(5009,a,5000,a)]. 11.73/11.77 5010 -number_ring(A) | -hBOOL(hAPP(A,bool,iszero(A),hAPP(int,A,number_number_of(A),hAPP(int,int,bit1,pls)))) # label(fact_2160_not__iszero__Numeral1) # label(axiom). [clausify(1291)]. 11.73/11.77 Derived: -hBOOL(hAPP(real,bool,iszero(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,pls)))). [resolve(5010,a,4976,a)]. 11.73/11.77 Derived: -hBOOL(hAPP(int,bool,iszero(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,pls)))). [resolve(5010,a,5000,a)]. 11.73/11.77 5011 -number_ring(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(int,A,number_number_of(A),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),B) # label(fact_545_mult__2) # label(axiom). [clausify(1295)]. 11.73/11.77 Derived: hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),A). [resolve(5011,a,4976,a)]. 11.73/11.77 Derived: hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),A). [resolve(5011,a,5000,a)]. 11.73/11.77 5012 number_ring(complex) # label(arity_Complex_Ocomplex___Int_Onumber__ring) # label(axiom). [assumption]. 11.73/11.77 Derived: hAPP(int,complex,number_number_of(complex),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),hAPP(int,int,uminus_uminus(int),A))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),one_one(complex)),hAPP(int,complex,number_number_of(complex),A)). [resolve(5012,a,4977,a)]. 11.83/11.78 Derived: -ring_char_0(complex) | -hBOOL(hAPP(complex,bool,iszero(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,bit0,A)))) | hBOOL(hAPP(complex,bool,iszero(complex),hAPP(int,complex,number_number_of(complex),A))). [resolve(5012,a,4978,a)]. 11.83/11.78 Derived: -ring_char_0(complex) | hBOOL(hAPP(complex,bool,iszero(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,bit0,A)))) | -hBOOL(hAPP(complex,bool,iszero(complex),hAPP(int,complex,number_number_of(complex),A))). [resolve(5012,a,4979,a)]. 11.83/11.78 Derived: -field(complex) | ti(complex,A) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),A),hAPP(int,complex,number_number_of(complex),hAPP(int,int,bit1,pls))). [resolve(5012,a,4980,a)]. 11.83/11.78 Derived: one_one(complex) = hAPP(int,complex,number_number_of(complex),hAPP(int,int,bit1,pls)). [resolve(5012,a,4981,a)]. 11.83/11.78 Derived: hAPP(int,complex,number_number_of(complex),min) = hAPP(complex,complex,uminus_uminus(complex),one_one(complex)). [resolve(5012,a,4982,a)]. 11.83/11.78 Derived: -linordered_idom(complex) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,pls)),A)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less(complex),one_one(complex)),hAPP(int,complex,number_number_of(complex),A))). [resolve(5012,a,4983,a)]. 11.83/11.78 Derived: -linordered_idom(complex) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,pls)),A)) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less(complex),one_one(complex)),hAPP(int,complex,number_number_of(complex),A))). [resolve(5012,a,4984,a)]. 11.83/11.78 Derived: one_one(complex) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(int,complex,number_number_of(complex),min)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A)). [resolve(5012,a,4985,a)]. 11.83/11.78 Derived: hAPP(int,complex,number_number_of(complex),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(int,complex,number_number_of(complex),A)),hAPP(int,complex,number_number_of(complex),B)). [resolve(5012,a,4986,a)]. 11.83/11.78 Derived: -hBOOL(hAPP(complex,bool,iszero(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),B))))) | hAPP(int,complex,number_number_of(complex),B) = hAPP(int,complex,number_number_of(complex),A). [resolve(5012,a,4987,a)]. 11.83/11.78 Derived: hBOOL(hAPP(complex,bool,iszero(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),B))))) | hAPP(int,complex,number_number_of(complex),B) != hAPP(int,complex,number_number_of(complex),A). [resolve(5012,a,4988,a)]. 11.83/11.78 Derived: hAPP(int,complex,number_number_of(complex),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),B))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(int,complex,number_number_of(complex),A)),hAPP(int,complex,number_number_of(complex),B)). [resolve(5012,a,4989,a)]. 11.83/11.78 Derived: -linordered_idom(complex) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less_eq(complex),hAPP(int,complex,number_number_of(complex),A)),zero_zero(complex))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),pls)). [resolve(5012,a,4990,b)]. 11.83/11.78 Derived: -linordered_idom(complex) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less_eq(complex),hAPP(int,complex,number_number_of(complex),A)),zero_zero(complex))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),pls)). [resolve(5012,a,4991,b)]. 11.83/11.78 Derived: hAPP(int,complex,number_number_of(complex),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(int,complex,number_number_of(complex),A)),hAPP(int,complex,number_number_of(complex),B)). [resolve(5012,a,4992,a)]. 11.86/11.79 Derived: hAPP(int,complex,number_number_of(complex),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(int,complex,number_number_of(complex),A)),hAPP(int,complex,number_number_of(complex),B)). [resolve(5012,a,4993,a)]. 11.86/11.79 Derived: zero_zero(complex) = hAPP(int,complex,number_number_of(complex),pls). [resolve(5012,a,4994,a)]. 11.86/11.79 Derived: -linordered_idom(complex) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less(complex),hAPP(int,complex,number_number_of(complex),A)),hAPP(int,complex,number_number_of(complex),B))). [resolve(5012,a,4995,b)]. 11.86/11.79 Derived: -linordered_idom(complex) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less(complex),hAPP(int,complex,number_number_of(complex),A)),hAPP(int,complex,number_number_of(complex),B))). [resolve(5012,a,4996,b)]. 11.86/11.79 Derived: hBOOL(hAPP(nat,bool,even_odd_even(nat),A)) | hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(int,complex,number_number_of(complex),min)),A) = hAPP(int,complex,number_number_of(complex),min). [resolve(5012,a,4997,a)]. 11.86/11.79 Derived: -hBOOL(hAPP(nat,bool,even_odd_even(nat),A)) | one_one(complex) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(int,complex,number_number_of(complex),min)),A). [resolve(5012,a,4998,a)]. 11.86/11.79 Derived: ti(complex,A) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(int,complex,number_number_of(complex),pls)),A). [resolve(5012,a,4999,a)]. 11.86/11.79 Derived: ti(complex,A) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),hAPP(int,complex,number_number_of(complex),hAPP(int,int,bit1,pls))). [resolve(5012,a,5001,a)]. 11.86/11.79 Derived: -linord1117847801e_zero(complex) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less(complex),zero_zero(complex)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),A),hAPP(int,complex,number_number_of(complex),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less(complex),zero_zero(complex)),A)). [resolve(5012,a,5002,b)]. 11.86/11.79 Derived: -linord1117847801e_zero(complex) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less(complex),zero_zero(complex)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),A),hAPP(int,complex,number_number_of(complex),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less(complex),zero_zero(complex)),A)). [resolve(5012,a,5003,b)]. 11.86/11.79 Derived: hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A)),B)). [resolve(5012,a,5004,a)]. 11.86/11.79 Derived: -ring_char_0(complex) | hAPP(int,complex,number_number_of(complex),A) != hAPP(int,complex,number_number_of(complex),B) | ti(int,A) = ti(int,B). [resolve(5012,a,5005,a)]. 11.86/11.79 Derived: -ring_char_0(complex) | hAPP(int,complex,number_number_of(complex),A) = hAPP(int,complex,number_number_of(complex),B) | ti(int,A) != ti(int,B). [resolve(5012,a,5006,a)]. 11.86/11.83 Derived: -field_inverse_zero(complex) | hAPP(complex,complex,uminus_uminus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),one_one(complex)),A)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),hAPP(int,complex,number_number_of(complex),min)),A). [resolve(5012,a,5007,b)]. 11.86/11.83 Derived: hAPP(complex,complex,uminus_uminus(complex),A) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(int,complex,number_number_of(complex),min)),A). [resolve(5012,a,5008,a)]. 11.86/11.83 Derived: -real_n2089651433ebra_1(complex) | hAPP(real,real,abs_abs(real),hAPP(int,real,number_number_of(real),A)) = hAPP(complex,real,norm_norm(complex),hAPP(int,complex,number_number_of(complex),A)). [resolve(5012,a,5009,a)]. 11.86/11.83 Derived: -hBOOL(hAPP(complex,bool,iszero(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,bit1,pls)))). [resolve(5012,a,5010,a)]. 11.86/11.83 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),A). [resolve(5012,a,5011,a)]. 11.86/11.83 5013 -number_ring(A) | -linordered_idom(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(int,A,number_number_of(A),B))) # label(fact_236_le__special_I1_J) # label(axiom). [clausify(1412)]. 11.86/11.83 Derived: -linordered_idom(real) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(int,real,number_number_of(real),A))). [resolve(5013,a,4976,a)]. 11.86/11.83 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))). [resolve(5013,a,5000,a)]. 11.86/11.83 Derived: -linordered_idom(complex) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less_eq(complex),zero_zero(complex)),hAPP(int,complex,number_number_of(complex),A))). [resolve(5013,a,5012,a)]. 11.86/11.83 5014 -number_ring(A) | -linordered_idom(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(int,A,number_number_of(A),B))) # label(fact_236_le__special_I1_J) # label(axiom). [clausify(1412)]. 11.86/11.83 Derived: -linordered_idom(real) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(int,real,number_number_of(real),A))). [resolve(5014,a,4976,a)]. 11.86/11.83 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))). [resolve(5014,a,5000,a)]. 11.86/11.83 Derived: -linordered_idom(complex) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less_eq(complex),zero_zero(complex)),hAPP(int,complex,number_number_of(complex),A))). [resolve(5014,a,5012,a)]. 11.86/11.83 5015 -ring_char_0(A) | -number_ring(A) | -hBOOL(hAPP(A,bool,iszero(A),hAPP(int,A,number_number_of(A),hAPP(int,int,bit1,B)))) # label(fact_2155_iszero__number__of__Bit1) # label(axiom). [clausify(1418)]. 11.86/11.83 Derived: -ring_char_0(real) | -hBOOL(hAPP(real,bool,iszero(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,A)))). [resolve(5015,b,4976,a)]. 11.86/11.83 Derived: -ring_char_0(int) | -hBOOL(hAPP(int,bool,iszero(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,A)))). [resolve(5015,b,5000,a)]. 11.86/11.83 Derived: -ring_char_0(complex) | -hBOOL(hAPP(complex,bool,iszero(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,bit1,A)))). [resolve(5015,b,5012,a)]. 12.03/12.03 5016 -linordered_idom(A) | -number_ring(A) | one_one(A) = hAPP(A,A,abs_abs(A),hAPP(int,A,number_number_of(A),min)) # label(fact_1015_abs__minus__one) # label(axiom). [clausify(1630)]. 12.03/12.03 Derived: -linordered_idom(real) | one_one(real) = hAPP(real,real,abs_abs(real),hAPP(int,real,number_number_of(real),min)). [resolve(5016,b,4976,a)]. 12.03/12.03 Derived: -linordered_idom(int) | one_one(int) = hAPP(int,int,abs_abs(int),hAPP(int,int,number_number_of(int),min)). [resolve(5016,b,5000,a)]. 12.03/12.03 Derived: -linordered_idom(complex) | one_one(complex) = hAPP(complex,complex,abs_abs(complex),hAPP(int,complex,number_number_of(complex),min)). [resolve(5016,b,5012,a)]. 12.03/12.03 5017 -number_ring(A) | hBOOL(hAPP(A,bool,iszero(A),hAPP(int,A,number_number_of(A),pls))) # label(fact_2156_iszero__Numeral0) # label(axiom). [clausify(1890)]. 12.03/12.03 Derived: hBOOL(hAPP(real,bool,iszero(real),hAPP(int,real,number_number_of(real),pls))). [resolve(5017,a,4976,a)]. 12.03/12.03 Derived: hBOOL(hAPP(int,bool,iszero(int),hAPP(int,int,number_number_of(int),pls))). [resolve(5017,a,5000,a)]. 12.03/12.03 Derived: hBOOL(hAPP(complex,bool,iszero(complex),hAPP(int,complex,number_number_of(complex),pls))). [resolve(5017,a,5012,a)]. 12.03/12.03 5018 -number_ring(A) | hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(int,A,number_number_of(A),B)),hAPP(int,A,number_number_of(A),C)) # label(fact_371_number__of__mult) # label(axiom). [clausify(1900)]. 12.03/12.03 Derived: hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),A)),hAPP(int,int,number_number_of(int),B)). [resolve(5018,a,5000,a)]. 12.03/12.03 5019 -number_ring(A) | -real_algebra_1(A) | hAPP(real,A,of_real(A),hAPP(int,real,number_number_of(real),B)) = hAPP(int,A,number_number_of(A),B) # label(fact_2796_of__real__number__of__eq) # label(axiom). [clausify(1938)]. 12.03/12.03 Derived: -real_algebra_1(real) | hAPP(real,real,of_real(real),hAPP(int,real,number_number_of(real),A)) = hAPP(int,real,number_number_of(real),A). [resolve(5019,a,4976,a)]. 12.03/12.03 Derived: -real_algebra_1(int) | hAPP(real,int,of_real(int),hAPP(int,real,number_number_of(real),A)) = hAPP(int,int,number_number_of(int),A). [resolve(5019,a,5000,a)]. 12.03/12.03 Derived: -real_algebra_1(complex) | hAPP(real,complex,of_real(complex),hAPP(int,real,number_number_of(real),A)) = hAPP(int,complex,number_number_of(complex),A). [resolve(5019,a,5012,a)]. 12.03/12.03 5020 -number_ring(A) | hAPP(int,A,number_number_of(A),min) = hAPP(A,A,uminus_uminus(A),one_one(A)) # label(fact_2017_number__of__Min) # label(axiom). [clausify(1981)]. 12.03/12.03 5021 -number_ring(A) | hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),C))),D) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(int,A,number_number_of(A),B)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(int,A,number_number_of(A),C)),D)) # label(fact_90_add__number__of__left) # label(axiom). [clausify(1984)]. 12.03/12.03 Derived: hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,number_number_of(real),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B))),C) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,number_number_of(real),A)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,number_number_of(real),B)),C)). [resolve(5021,a,4976,a)]. 12.03/12.03 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B))),C) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),A)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),B)),C)). [resolve(5021,a,5000,a)]. 12.03/12.03 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B))),C) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(int,complex,number_number_of(complex),A)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(int,complex,number_number_of(complex),B)),C)). [resolve(5021,a,5012,a)]. 12.27/12.23 5022 -number_ring(A) | hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(int,A,number_number_of(A),B)),hAPP(int,A,number_number_of(A),C)) # label(fact_92_number__of__add) # label(axiom). [clausify(1991)]. 12.27/12.23 Derived: hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),A)),hAPP(int,int,number_number_of(int),B)). [resolve(5022,a,5000,a)]. 12.27/12.23 5023 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),hAPP(int,int,bit1,pls))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(int,A,number_number_of(A),B)),one_one(A))) # label(fact_62_less__special_I4_J) # label(axiom). [clausify(2005)]. 12.27/12.23 Derived: -linordered_idom(real) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(int,int,bit1,pls))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),A)),one_one(real))). [resolve(5023,b,4976,a)]. 12.27/12.23 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(int,int,bit1,pls))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),A)),one_one(int))). [resolve(5023,b,5000,a)]. 12.27/12.23 Derived: -linordered_idom(complex) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(int,int,bit1,pls))) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less(complex),hAPP(int,complex,number_number_of(complex),A)),one_one(complex))). [resolve(5023,b,5012,a)]. 12.27/12.23 5024 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),hAPP(int,int,bit1,pls))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(int,A,number_number_of(A),B)),one_one(A))) # label(fact_62_less__special_I4_J) # label(axiom). [clausify(2005)]. 12.27/12.23 Derived: -linordered_idom(real) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(int,int,bit1,pls))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),A)),one_one(real))). [resolve(5024,b,4976,a)]. 12.27/12.23 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(int,int,bit1,pls))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),A)),one_one(int))). [resolve(5024,b,5000,a)]. 12.27/12.23 Derived: -linordered_idom(complex) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(int,int,bit1,pls))) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less(complex),hAPP(int,complex,number_number_of(complex),A)),one_one(complex))). [resolve(5024,b,5012,a)]. 12.27/12.23 5025 -number_ring(A) | -field(A) | hAPP(int,A,number_number_of(A),hAPP(int,int,bit1,pls)) = hAPP(A,A,inverse_inverse(A),hAPP(int,A,number_number_of(A),hAPP(int,int,bit1,pls))) # label(fact_2621_inverse__numeral__1) # label(axiom). [clausify(2233)]. 12.27/12.23 Derived: -field(real) | hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,pls)) = hAPP(real,real,inverse_inverse(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,pls))). [resolve(5025,a,4976,a)]. 12.27/12.23 Derived: -field(int) | hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,pls)) = hAPP(int,int,inverse_inverse(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,pls))). [resolve(5025,a,5000,a)]. 12.27/12.23 Derived: -field(complex) | hAPP(int,complex,number_number_of(complex),hAPP(int,int,bit1,pls)) = hAPP(complex,complex,inverse_inverse(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,bit1,pls))). [resolve(5025,a,5012,a)]. 12.27/12.23 5026 -number_ring(A) | one_one(A) != hAPP(int,A,number_number_of(A),B) | hBOOL(hAPP(A,bool,iszero(A),hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),hAPP(int,int,uminus_uminus(int),B))))) # label(fact_2119_eq__special_I2_J) # label(axiom). [clausify(2431)]. 12.27/12.26 Derived: one_one(real) != hAPP(int,real,number_number_of(real),A) | hBOOL(hAPP(real,bool,iszero(real),hAPP(int,real,number_number_of(real),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),hAPP(int,int,uminus_uminus(int),A))))). [resolve(5026,a,4976,a)]. 12.27/12.26 Derived: one_one(int) != hAPP(int,int,number_number_of(int),A) | hBOOL(hAPP(int,bool,iszero(int),hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),hAPP(int,int,uminus_uminus(int),A))))). [resolve(5026,a,5000,a)]. 12.27/12.26 Derived: one_one(complex) != hAPP(int,complex,number_number_of(complex),A) | hBOOL(hAPP(complex,bool,iszero(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),hAPP(int,int,uminus_uminus(int),A))))). [resolve(5026,a,5012,a)]. 12.27/12.26 5027 -number_ring(A) | one_one(A) = hAPP(int,A,number_number_of(A),B) | -hBOOL(hAPP(A,bool,iszero(A),hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),hAPP(int,int,uminus_uminus(int),B))))) # label(fact_2119_eq__special_I2_J) # label(axiom). [clausify(2431)]. 12.27/12.26 Derived: one_one(real) = hAPP(int,real,number_number_of(real),A) | -hBOOL(hAPP(real,bool,iszero(real),hAPP(int,real,number_number_of(real),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),hAPP(int,int,uminus_uminus(int),A))))). [resolve(5027,a,4976,a)]. 12.27/12.26 Derived: one_one(int) = hAPP(int,int,number_number_of(int),A) | -hBOOL(hAPP(int,bool,iszero(int),hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),hAPP(int,int,uminus_uminus(int),A))))). [resolve(5027,a,5000,a)]. 12.27/12.26 Derived: one_one(complex) = hAPP(int,complex,number_number_of(complex),A) | -hBOOL(hAPP(complex,bool,iszero(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),hAPP(int,int,uminus_uminus(int),A))))). [resolve(5027,a,5012,a)]. 12.27/12.26 5028 -number_ring(A) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),hAPP(int,A,number_number_of(A),B)),ring_1_Ints(A))) # label(fact_3829_Ints__number__of) # label(axiom). [clausify(2435)]. 12.27/12.26 Derived: hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),hAPP(int,real,number_number_of(real),A)),ring_1_Ints(real))). [resolve(5028,a,4976,a)]. 12.27/12.26 Derived: hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,number_number_of(int),A)),ring_1_Ints(int))). [resolve(5028,a,5000,a)]. 12.27/12.26 Derived: hBOOL(hAPP(fun(complex,bool),bool,hAPP(complex,fun(fun(complex,bool),bool),member(complex),hAPP(int,complex,number_number_of(complex),A)),ring_1_Ints(complex))). [resolve(5028,a,5012,a)]. 12.27/12.26 5029 -number_ring(A) | hAPP(int,A,ring_1_of_int(A),B) = hAPP(int,A,number_number_of(A),B) # label(fact_3594_number__of__eq) # label(axiom). [clausify(2454)]. 12.27/12.26 Derived: hAPP(int,real,ring_1_of_int(real),A) = hAPP(int,real,number_number_of(real),A). [resolve(5029,a,4976,a)]. 12.27/12.26 Derived: hAPP(int,int,ring_1_of_int(int),A) = hAPP(int,int,number_number_of(int),A). [resolve(5029,a,5000,a)]. 12.27/12.26 Derived: hAPP(int,complex,ring_1_of_int(complex),A) = hAPP(int,complex,number_number_of(complex),A). [resolve(5029,a,5012,a)]. 12.27/12.26 5030 number_ring(rat) # label(arity_Rat_Orat___Int_Onumber__ring) # label(axiom). [assumption]. 12.27/12.26 Derived: hAPP(int,rat,number_number_of(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),hAPP(int,int,uminus_uminus(int),A))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),one_one(rat)),hAPP(int,rat,number_number_of(rat),A)). [resolve(5030,a,4977,a)]. 12.27/12.26 Derived: -ring_char_0(rat) | -hBOOL(hAPP(rat,bool,iszero(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit0,A)))) | hBOOL(hAPP(rat,bool,iszero(rat),hAPP(int,rat,number_number_of(rat),A))). [resolve(5030,a,4978,a)]. 12.27/12.26 Derived: -ring_char_0(rat) | hBOOL(hAPP(rat,bool,iszero(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit0,A)))) | -hBOOL(hAPP(rat,bool,iszero(rat),hAPP(int,rat,number_number_of(rat),A))). [resolve(5030,a,4979,a)]. 12.34/12.27 Derived: -field(rat) | ti(rat,A) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit1,pls))). [resolve(5030,a,4980,a)]. 12.34/12.27 Derived: one_one(rat) = hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit1,pls)). [resolve(5030,a,4981,a)]. 12.34/12.27 Derived: hAPP(int,rat,number_number_of(rat),min) = hAPP(rat,rat,uminus_uminus(rat),one_one(rat)). [resolve(5030,a,4982,a)]. 12.34/12.27 Derived: -linordered_idom(rat) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,pls)),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),hAPP(int,rat,number_number_of(rat),A))). [resolve(5030,a,4983,a)]. 12.34/12.27 Derived: -linordered_idom(rat) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,pls)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),hAPP(int,rat,number_number_of(rat),A))). [resolve(5030,a,4984,a)]. 12.34/12.27 Derived: one_one(rat) = hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),hAPP(int,rat,number_number_of(rat),min)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A)). [resolve(5030,a,4985,a)]. 12.34/12.27 Derived: hAPP(int,rat,number_number_of(rat),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(int,rat,number_number_of(rat),A)),hAPP(int,rat,number_number_of(rat),B)). [resolve(5030,a,4986,a)]. 12.34/12.27 Derived: -hBOOL(hAPP(rat,bool,iszero(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),B))))) | hAPP(int,rat,number_number_of(rat),B) = hAPP(int,rat,number_number_of(rat),A). [resolve(5030,a,4987,a)]. 12.34/12.27 Derived: hBOOL(hAPP(rat,bool,iszero(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),B))))) | hAPP(int,rat,number_number_of(rat),B) != hAPP(int,rat,number_number_of(rat),A). [resolve(5030,a,4988,a)]. 12.34/12.27 Derived: hAPP(int,rat,number_number_of(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),B))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(int,rat,number_number_of(rat),A)),hAPP(int,rat,number_number_of(rat),B)). [resolve(5030,a,4989,a)]. 12.34/12.27 Derived: -linordered_idom(rat) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,number_number_of(rat),A)),zero_zero(rat))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),pls)). [resolve(5030,a,4990,b)]. 12.34/12.27 Derived: -linordered_idom(rat) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,number_number_of(rat),A)),zero_zero(rat))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),pls)). [resolve(5030,a,4991,b)]. 12.34/12.27 Derived: hAPP(int,rat,number_number_of(rat),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(int,rat,number_number_of(rat),A)),hAPP(int,rat,number_number_of(rat),B)). [resolve(5030,a,4992,a)]. 12.34/12.27 Derived: hAPP(int,rat,number_number_of(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,number_number_of(rat),A)),hAPP(int,rat,number_number_of(rat),B)). [resolve(5030,a,4993,a)]. 12.34/12.27 Derived: zero_zero(rat) = hAPP(int,rat,number_number_of(rat),pls). [resolve(5030,a,4994,a)]. 12.34/12.27 Derived: -linordered_idom(rat) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,number_number_of(rat),A)),hAPP(int,rat,number_number_of(rat),B))). [resolve(5030,a,4995,b)]. 12.34/12.27 Derived: -linordered_idom(rat) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,number_number_of(rat),A)),hAPP(int,rat,number_number_of(rat),B))). [resolve(5030,a,4996,b)]. 12.34/12.28 Derived: hBOOL(hAPP(nat,bool,even_odd_even(nat),A)) | hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),hAPP(int,rat,number_number_of(rat),min)),A) = hAPP(int,rat,number_number_of(rat),min). [resolve(5030,a,4997,a)]. 12.34/12.28 Derived: -hBOOL(hAPP(nat,bool,even_odd_even(nat),A)) | one_one(rat) = hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),hAPP(int,rat,number_number_of(rat),min)),A). [resolve(5030,a,4998,a)]. 12.34/12.28 Derived: ti(rat,A) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,number_number_of(rat),pls)),A). [resolve(5030,a,4999,a)]. 12.34/12.28 Derived: ti(rat,A) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit1,pls))). [resolve(5030,a,5001,a)]. 12.34/12.28 Derived: -linord1117847801e_zero(rat) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)). [resolve(5030,a,5002,b)]. 12.34/12.28 Derived: -linord1117847801e_zero(rat) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)). [resolve(5030,a,5003,b)]. 12.34/12.28 Derived: hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A)),B)). [resolve(5030,a,5004,a)]. 12.34/12.28 Derived: -ring_char_0(rat) | hAPP(int,rat,number_number_of(rat),A) != hAPP(int,rat,number_number_of(rat),B) | ti(int,A) = ti(int,B). [resolve(5030,a,5005,a)]. 12.34/12.28 Derived: -ring_char_0(rat) | hAPP(int,rat,number_number_of(rat),A) = hAPP(int,rat,number_number_of(rat),B) | ti(int,A) != ti(int,B). [resolve(5030,a,5006,a)]. 12.34/12.28 Derived: -field_inverse_zero(rat) | hAPP(rat,rat,uminus_uminus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),one_one(rat)),A)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),hAPP(int,rat,number_number_of(rat),min)),A). [resolve(5030,a,5007,b)]. 12.34/12.28 Derived: hAPP(rat,rat,uminus_uminus(rat),A) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(int,rat,number_number_of(rat),min)),A). [resolve(5030,a,5008,a)]. 12.34/12.28 Derived: -real_n2089651433ebra_1(rat) | hAPP(real,real,abs_abs(real),hAPP(int,real,number_number_of(real),A)) = hAPP(rat,real,norm_norm(rat),hAPP(int,rat,number_number_of(rat),A)). [resolve(5030,a,5009,a)]. 12.34/12.28 Derived: -hBOOL(hAPP(rat,bool,iszero(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit1,pls)))). [resolve(5030,a,5010,a)]. 12.34/12.28 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),A). [resolve(5030,a,5011,a)]. 12.34/12.28 Derived: -linordered_idom(rat) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(int,rat,number_number_of(rat),A))). [resolve(5030,a,5013,a)]. 12.34/12.28 Derived: -linordered_idom(rat) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(int,rat,number_number_of(rat),A))). [resolve(5030,a,5014,a)]. 12.34/12.33 Derived: -ring_char_0(rat) | -hBOOL(hAPP(rat,bool,iszero(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit1,A)))). [resolve(5030,a,5015,b)]. 12.34/12.33 Derived: -linordered_idom(rat) | one_one(rat) = hAPP(rat,rat,abs_abs(rat),hAPP(int,rat,number_number_of(rat),min)). [resolve(5030,a,5016,b)]. 12.34/12.33 Derived: hBOOL(hAPP(rat,bool,iszero(rat),hAPP(int,rat,number_number_of(rat),pls))). [resolve(5030,a,5017,a)]. 12.34/12.33 Derived: -real_algebra_1(rat) | hAPP(real,rat,of_real(rat),hAPP(int,real,number_number_of(real),A)) = hAPP(int,rat,number_number_of(rat),A). [resolve(5030,a,5019,a)]. 12.34/12.33 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B))),C) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,number_number_of(rat),A)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,number_number_of(rat),B)),C)). [resolve(5030,a,5021,a)]. 12.34/12.33 Derived: -linordered_idom(rat) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(int,int,bit1,pls))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,number_number_of(rat),A)),one_one(rat))). [resolve(5030,a,5023,b)]. 12.34/12.33 Derived: -linordered_idom(rat) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(int,int,bit1,pls))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,number_number_of(rat),A)),one_one(rat))). [resolve(5030,a,5024,b)]. 12.34/12.33 Derived: -field(rat) | hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit1,pls)) = hAPP(rat,rat,inverse_inverse(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit1,pls))). [resolve(5030,a,5025,a)]. 12.34/12.33 Derived: one_one(rat) != hAPP(int,rat,number_number_of(rat),A) | hBOOL(hAPP(rat,bool,iszero(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),hAPP(int,int,uminus_uminus(int),A))))). [resolve(5030,a,5026,a)]. 12.34/12.33 Derived: one_one(rat) = hAPP(int,rat,number_number_of(rat),A) | -hBOOL(hAPP(rat,bool,iszero(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),hAPP(int,int,uminus_uminus(int),A))))). [resolve(5030,a,5027,a)]. 12.34/12.33 Derived: hBOOL(hAPP(fun(rat,bool),bool,hAPP(rat,fun(fun(rat,bool),bool),member(rat),hAPP(int,rat,number_number_of(rat),A)),ring_1_Ints(rat))). [resolve(5030,a,5028,a)]. 12.34/12.33 Derived: hAPP(int,rat,ring_1_of_int(rat),A) = hAPP(int,rat,number_number_of(rat),A). [resolve(5030,a,5029,a)]. 12.34/12.33 5031 -number_ring(A) | hAPP(int,A,number_number_of(A),hAPP(int,int,succ,B)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),one_one(A)),hAPP(int,A,number_number_of(A),B)) # label(fact_308_number__of__succ) # label(axiom). [clausify(2498)]. 12.34/12.33 Derived: hAPP(int,real,number_number_of(real),hAPP(int,int,succ,A)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(int,real,number_number_of(real),A)). [resolve(5031,a,4976,a)]. 12.34/12.33 Derived: hAPP(int,int,number_number_of(int),hAPP(int,int,succ,A)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(int,int,number_number_of(int),A)). [resolve(5031,a,5000,a)]. 12.34/12.33 Derived: hAPP(int,complex,number_number_of(complex),hAPP(int,int,succ,A)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),one_one(complex)),hAPP(int,complex,number_number_of(complex),A)). [resolve(5031,a,5012,a)]. 12.34/12.33 Derived: hAPP(int,rat,number_number_of(rat),hAPP(int,int,succ,A)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),one_one(rat)),hAPP(int,rat,number_number_of(rat),A)). [resolve(5031,a,5030,a)]. 12.34/12.33 5032 -number_ring(A) | hAPP(int,A,number_number_of(A),hAPP(int,int,bit0,B)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),zero_zero(A)),hAPP(int,A,number_number_of(A),B))),hAPP(int,A,number_number_of(A),B)) # label(fact_102_number__of__Bit0) # label(axiom). [clausify(2528)]. 12.34/12.33 Derived: hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,A)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),zero_zero(real)),hAPP(int,real,number_number_of(real),A))),hAPP(int,real,number_number_of(real),A)). [resolve(5032,a,4976,a)]. 12.48/12.41 Derived: hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,A)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))),hAPP(int,int,number_number_of(int),A)). [resolve(5032,a,5000,a)]. 12.48/12.41 Derived: hAPP(int,complex,number_number_of(complex),hAPP(int,int,bit0,A)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),zero_zero(complex)),hAPP(int,complex,number_number_of(complex),A))),hAPP(int,complex,number_number_of(complex),A)). [resolve(5032,a,5012,a)]. 12.48/12.41 Derived: hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit0,A)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),zero_zero(rat)),hAPP(int,rat,number_number_of(rat),A))),hAPP(int,rat,number_number_of(rat),A)). [resolve(5032,a,5030,a)]. 12.48/12.41 5033 -number_ring(A) | hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),B)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),one_one(A)),hAPP(int,A,number_number_of(A),B)) # label(fact_6_add__special_I2_J) # label(axiom). [clausify(2680)]. 12.48/12.41 Derived: hAPP(int,real,number_number_of(real),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),A)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(int,real,number_number_of(real),A)). [resolve(5033,a,4976,a)]. 12.48/12.41 Derived: hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),A)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(int,int,number_number_of(int),A)). [resolve(5033,a,5000,a)]. 12.48/12.41 Derived: hAPP(int,complex,number_number_of(complex),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),A)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),one_one(complex)),hAPP(int,complex,number_number_of(complex),A)). [resolve(5033,a,5012,a)]. 12.48/12.41 Derived: hAPP(int,rat,number_number_of(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),A)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),one_one(rat)),hAPP(int,rat,number_number_of(rat),A)). [resolve(5033,a,5030,a)]. 12.48/12.41 5034 -number_ring(A) | -linordered_idom(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),pls)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(int,A,number_number_of(A),B)),zero_zero(A))) # label(fact_49_less__special_I3_J) # label(axiom). [clausify(2687)]. 12.48/12.41 Derived: -linordered_idom(real) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),A)),zero_zero(real))). [resolve(5034,a,4976,a)]. 12.48/12.41 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),A)),zero_zero(int))). [resolve(5034,a,5000,a)]. 12.48/12.41 Derived: -linordered_idom(complex) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less(complex),hAPP(int,complex,number_number_of(complex),A)),zero_zero(complex))). [resolve(5034,a,5012,a)]. 12.48/12.41 Derived: -linordered_idom(rat) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,number_number_of(rat),A)),zero_zero(rat))). [resolve(5034,a,5030,a)]. 12.48/12.41 5035 -number_ring(A) | -linordered_idom(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),pls)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(int,A,number_number_of(A),B)),zero_zero(A))) # label(fact_49_less__special_I3_J) # label(axiom). [clausify(2687)]. 12.48/12.41 Derived: -linordered_idom(real) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),A)),zero_zero(real))). [resolve(5035,a,4976,a)]. 12.60/12.56 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),A)),zero_zero(int))). [resolve(5035,a,5000,a)]. 12.60/12.56 Derived: -linordered_idom(complex) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less(complex),hAPP(int,complex,number_number_of(complex),A)),zero_zero(complex))). [resolve(5035,a,5012,a)]. 12.60/12.56 Derived: -linordered_idom(rat) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,number_number_of(rat),A)),zero_zero(rat))). [resolve(5035,a,5030,a)]. 12.60/12.56 5036 -number_ring(A) | -linord1117847801e_zero(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),one_one(A)),hAPP(int,A,number_number_of(A),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) # label(fact_1813_half) # label(axiom). [clausify(2736)]. 12.60/12.56 Derived: -linord1117847801e_zero(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))). [resolve(5036,a,4976,a)]. 12.60/12.56 Derived: -linord1117847801e_zero(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),inverse_divide(int),one_one(int)),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))). [resolve(5036,a,5000,a)]. 12.60/12.56 Derived: -linord1117847801e_zero(complex) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less(complex),zero_zero(complex)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),one_one(complex)),hAPP(int,complex,number_number_of(complex),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))). [resolve(5036,a,5012,a)]. 12.60/12.56 Derived: -linord1117847801e_zero(rat) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),one_one(rat)),hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))). [resolve(5036,a,5030,a)]. 12.60/12.56 5037 -number_ring(A) | hAPP(A,A,uminus_uminus(A),B) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(int,A,number_number_of(A),min)) # label(fact_2014_mult__minus1__right) # label(axiom). [clausify(2772)]. 12.60/12.56 Derived: hAPP(real,real,uminus_uminus(real),A) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),hAPP(int,real,number_number_of(real),min)). [resolve(5037,a,4976,a)]. 12.60/12.56 Derived: hAPP(int,int,uminus_uminus(int),A) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),hAPP(int,int,number_number_of(int),min)). [resolve(5037,a,5000,a)]. 12.60/12.56 Derived: hAPP(complex,complex,uminus_uminus(complex),A) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),hAPP(int,complex,number_number_of(complex),min)). [resolve(5037,a,5012,a)]. 12.60/12.56 Derived: hAPP(rat,rat,uminus_uminus(rat),A) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),hAPP(int,rat,number_number_of(rat),min)). [resolve(5037,a,5030,a)]. 12.60/12.56 5038 -number_ring(A) | -hBOOL(hAPP(A,bool,iszero(A),hAPP(int,A,number_number_of(A),min))) # label(fact_2158_nonzero__number__of__Min) # label(axiom). [clausify(2803)]. 12.60/12.56 Derived: -hBOOL(hAPP(real,bool,iszero(real),hAPP(int,real,number_number_of(real),min))). [resolve(5038,a,4976,a)]. 12.60/12.56 Derived: -hBOOL(hAPP(int,bool,iszero(int),hAPP(int,int,number_number_of(int),min))). [resolve(5038,a,5000,a)]. 12.60/12.56 Derived: -hBOOL(hAPP(complex,bool,iszero(complex),hAPP(int,complex,number_number_of(complex),min))). [resolve(5038,a,5012,a)]. 12.60/12.56 Derived: -hBOOL(hAPP(rat,bool,iszero(rat),hAPP(int,rat,number_number_of(rat),min))). [resolve(5038,a,5030,a)]. 12.60/12.56 5039 -number_ring(A) | -linord1117847801e_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),zero_zero(A)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),hAPP(int,A,number_number_of(A),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) # label(fact_1808_half__gt__zero) # label(axiom). [clausify(2923)]. 12.74/12.70 5040 -number_ring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),B))) | hAPP(nat,A,semiring_1_of_nat(A),hAPP(int,nat,number_number_of(nat),B)) = hAPP(int,A,number_number_of(A),B) # label(fact_316_of__nat__number__of__lemma) # label(axiom). [clausify(3030)]. 12.74/12.70 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | hAPP(nat,real,semiring_1_of_nat(real),hAPP(int,nat,number_number_of(nat),A)) = hAPP(int,real,number_number_of(real),A). [resolve(5040,a,4976,a)]. 12.74/12.70 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,number_number_of(nat),A)) = hAPP(int,int,number_number_of(int),A). [resolve(5040,a,5000,a)]. 12.74/12.70 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | hAPP(nat,complex,semiring_1_of_nat(complex),hAPP(int,nat,number_number_of(nat),A)) = hAPP(int,complex,number_number_of(complex),A). [resolve(5040,a,5012,a)]. 12.74/12.70 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | hAPP(nat,rat,semiring_1_of_nat(rat),hAPP(int,nat,number_number_of(nat),A)) = hAPP(int,rat,number_number_of(rat),A). [resolve(5040,a,5030,a)]. 12.74/12.70 5041 -number_ring(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),B))) | zero_zero(A) = hAPP(nat,A,semiring_1_of_nat(A),hAPP(int,nat,number_number_of(nat),B)) # label(fact_316_of__nat__number__of__lemma) # label(axiom). [clausify(3030)]. 12.74/12.70 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | zero_zero(real) = hAPP(nat,real,semiring_1_of_nat(real),hAPP(int,nat,number_number_of(nat),A)). [resolve(5041,a,4976,a)]. 12.74/12.70 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | zero_zero(int) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,number_number_of(nat),A)). [resolve(5041,a,5000,a)]. 12.74/12.70 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | zero_zero(complex) = hAPP(nat,complex,semiring_1_of_nat(complex),hAPP(int,nat,number_number_of(nat),A)). [resolve(5041,a,5012,a)]. 12.74/12.70 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | zero_zero(rat) = hAPP(nat,rat,semiring_1_of_nat(rat),hAPP(int,nat,number_number_of(nat),A)). [resolve(5041,a,5030,a)]. 12.74/12.70 5042 -number_ring(A) | hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),hAPP(int,int,uminus_uminus(int),C)))),D) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(int,A,number_number_of(A),B)),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),D),hAPP(int,A,number_number_of(A),C))) # label(fact_2093_add__number__of__diff2) # label(axiom). [clausify(3166)]. 12.74/12.70 Derived: hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,number_number_of(real),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),B)))),C) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,number_number_of(real),A)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),C),hAPP(int,real,number_number_of(real),B))). [resolve(5042,a,4976,a)]. 12.74/12.70 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),B)))),C) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),A)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),C),hAPP(int,int,number_number_of(int),B))). [resolve(5042,a,5000,a)]. 12.81/12.76 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),B)))),C) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(int,complex,number_number_of(complex),A)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),C),hAPP(int,complex,number_number_of(complex),B))). [resolve(5042,a,5012,a)]. 12.81/12.76 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),B)))),C) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,number_number_of(rat),A)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),C),hAPP(int,rat,number_number_of(rat),B))). [resolve(5042,a,5030,a)]. 12.81/12.76 5043 -number_ring(A) | hAPP(int,A,number_number_of(A),hAPP(int,int,bit0,B)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A))),hAPP(int,A,number_number_of(A),B)) # label(fact_539_double__number__of__Bit0) # label(axiom). [clausify(3254)]. 12.81/12.76 Derived: hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,A)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),one_one(real))),hAPP(int,real,number_number_of(real),A)). [resolve(5043,a,4976,a)]. 12.81/12.76 Derived: hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,A)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),one_one(int))),hAPP(int,int,number_number_of(int),A)). [resolve(5043,a,5000,a)]. 12.81/12.76 Derived: hAPP(int,complex,number_number_of(complex),hAPP(int,int,bit0,A)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),one_one(complex)),one_one(complex))),hAPP(int,complex,number_number_of(complex),A)). [resolve(5043,a,5012,a)]. 12.81/12.76 Derived: hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit0,A)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),one_one(rat)),one_one(rat))),hAPP(int,rat,number_number_of(rat),A)). [resolve(5043,a,5030,a)]. 12.81/12.76 5044 -number_ring(A) | -linordered_idom(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(int,A,number_number_of(A),B))) # label(fact_50_less__special_I1_J) # label(axiom). [clausify(3259)]. 12.81/12.76 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)),hAPP(int,real,number_number_of(real),A))). [resolve(5044,a,4976,a)]. 12.81/12.76 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)),hAPP(int,int,number_number_of(int),A))). [resolve(5044,a,5000,a)]. 12.81/12.76 Derived: -linordered_idom(complex) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),A)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less(complex),zero_zero(complex)),hAPP(int,complex,number_number_of(complex),A))). [resolve(5044,a,5012,a)]. 12.81/12.76 Derived: -linordered_idom(rat) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(int,rat,number_number_of(rat),A))). [resolve(5044,a,5030,a)]. 12.81/12.76 5045 -number_ring(A) | -linordered_idom(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(int,A,number_number_of(A),B))) # label(fact_50_less__special_I1_J) # label(axiom). [clausify(3259)]. 12.81/12.82 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)),hAPP(int,real,number_number_of(real),A))). [resolve(5045,a,4976,a)]. 12.81/12.82 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)),hAPP(int,int,number_number_of(int),A))). [resolve(5045,a,5000,a)]. 12.81/12.82 Derived: -linordered_idom(complex) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),A)) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less(complex),zero_zero(complex)),hAPP(int,complex,number_number_of(complex),A))). [resolve(5045,a,5012,a)]. 12.81/12.82 Derived: -linordered_idom(rat) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(int,rat,number_number_of(rat),A))). [resolve(5045,a,5030,a)]. 12.81/12.82 5046 -number_ring(A) | one_one(A) = hAPP(int,A,number_number_of(A),hAPP(int,int,bit1,pls)) # label(fact_12_semiring__norm_I110_J) # label(axiom). [clausify(3267)]. 12.81/12.82 5047 -number_ring(A) | hAPP(int,A,number_number_of(A),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)) # label(fact_8_one__add__one__is__two) # label(axiom). [clausify(3288)]. 12.81/12.82 Derived: hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),one_one(real)). [resolve(5047,a,4976,a)]. 12.81/12.82 Derived: hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),one_one(int)). [resolve(5047,a,5000,a)]. 12.81/12.82 Derived: hAPP(int,complex,number_number_of(complex),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),one_one(complex)),one_one(complex)). [resolve(5047,a,5012,a)]. 12.81/12.82 Derived: hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),one_one(rat)),one_one(rat)). [resolve(5047,a,5030,a)]. 12.81/12.82 5048 -number_ring(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(int,A,number_number_of(A),min)),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))) = hAPP(int,A,number_number_of(A),min) # label(fact_1513_power__m1__odd) # label(axiom). [clausify(3348)]. 12.81/12.82 Derived: hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A))) = hAPP(int,real,number_number_of(real),min). [resolve(5048,a,4976,a)]. 12.81/12.82 Derived: hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A))) = hAPP(int,int,number_number_of(int),min). [resolve(5048,a,5000,a)]. 12.81/12.82 Derived: hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(int,complex,number_number_of(complex),min)),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A))) = hAPP(int,complex,number_number_of(complex),min). [resolve(5048,a,5012,a)]. 12.81/12.82 Derived: hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),hAPP(int,rat,number_number_of(rat),min)),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A))) = hAPP(int,rat,number_number_of(rat),min). [resolve(5048,a,5030,a)]. 12.81/12.82 5049 -number_ring(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C))),D) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(int,A,number_number_of(A),B)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(int,A,number_number_of(A),C)),D)) # label(fact_369_mult__number__of__left) # label(axiom). [clausify(3352)]. 12.91/12.85 Derived: hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B))),C) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),A)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),B)),C)). [resolve(5049,a,4976,a)]. 12.91/12.85 Derived: hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B))),C) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),A)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),B)),C)). [resolve(5049,a,5000,a)]. 12.91/12.85 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B))),C) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(int,complex,number_number_of(complex),A)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(int,complex,number_number_of(complex),B)),C)). [resolve(5049,a,5012,a)]. 12.91/12.85 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B))),C) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(int,rat,number_number_of(rat),A)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(int,rat,number_number_of(rat),B)),C)). [resolve(5049,a,5030,a)]. 12.91/12.85 5050 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(int,A,number_number_of(A),B)),zero_zero(A))) | hAPP(A,A,abs_abs(A),hAPP(int,A,number_number_of(A),B)) = hAPP(A,A,uminus_uminus(A),hAPP(int,A,number_number_of(A),B)) # label(fact_2027_abs__number__of) # label(axiom). [clausify(3384)]. 12.91/12.85 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),A)),zero_zero(int))) | hAPP(int,int,abs_abs(int),hAPP(int,int,number_number_of(int),A)) = hAPP(int,int,uminus_uminus(int),hAPP(int,int,number_number_of(int),A)). [resolve(5050,b,5000,a)]. 12.91/12.85 Derived: -linordered_idom(complex) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less(complex),hAPP(int,complex,number_number_of(complex),A)),zero_zero(complex))) | hAPP(complex,complex,abs_abs(complex),hAPP(int,complex,number_number_of(complex),A)) = hAPP(complex,complex,uminus_uminus(complex),hAPP(int,complex,number_number_of(complex),A)). [resolve(5050,b,5012,a)]. 12.91/12.85 5051 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(int,A,number_number_of(A),B)),zero_zero(A))) | hAPP(int,A,number_number_of(A),B) = hAPP(A,A,abs_abs(A),hAPP(int,A,number_number_of(A),B)) # label(fact_2027_abs__number__of) # label(axiom). [clausify(3384)]. 12.91/12.85 Derived: -linordered_idom(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),A)),zero_zero(real))) | hAPP(int,real,number_number_of(real),A) = hAPP(real,real,abs_abs(real),hAPP(int,real,number_number_of(real),A)). [resolve(5051,b,4976,a)]. 12.91/12.85 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),A)),zero_zero(int))) | hAPP(int,int,number_number_of(int),A) = hAPP(int,int,abs_abs(int),hAPP(int,int,number_number_of(int),A)). [resolve(5051,b,5000,a)]. 12.91/12.85 Derived: -linordered_idom(complex) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less(complex),hAPP(int,complex,number_number_of(complex),A)),zero_zero(complex))) | hAPP(int,complex,number_number_of(complex),A) = hAPP(complex,complex,abs_abs(complex),hAPP(int,complex,number_number_of(complex),A)). [resolve(5051,b,5012,a)]. 13.36/13.28 Derived: -linordered_idom(rat) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,number_number_of(rat),A)),zero_zero(rat))) | hAPP(int,rat,number_number_of(rat),A) = hAPP(rat,rat,abs_abs(rat),hAPP(int,rat,number_number_of(rat),A)). [resolve(5051,b,5030,a)]. 13.36/13.28 5052 -number_ring(A) | hAPP(int,A,number_number_of(A),hAPP(int,int,uminus_uminus(int),B)) = hAPP(A,A,uminus_uminus(A),hAPP(int,A,number_number_of(A),B)) # label(fact_1927_arith__simps_I30_J) # label(axiom). [clausify(3404)]. 13.36/13.28 Derived: hAPP(int,real,number_number_of(real),hAPP(int,int,uminus_uminus(int),A)) = hAPP(real,real,uminus_uminus(real),hAPP(int,real,number_number_of(real),A)). [resolve(5052,a,4976,a)]. 13.36/13.28 Derived: hAPP(int,int,number_number_of(int),hAPP(int,int,uminus_uminus(int),A)) = hAPP(int,int,uminus_uminus(int),hAPP(int,int,number_number_of(int),A)). [resolve(5052,a,5000,a)]. 13.36/13.28 Derived: hAPP(int,complex,number_number_of(complex),hAPP(int,int,uminus_uminus(int),A)) = hAPP(complex,complex,uminus_uminus(complex),hAPP(int,complex,number_number_of(complex),A)). [resolve(5052,a,5012,a)]. 13.36/13.28 Derived: hAPP(int,rat,number_number_of(rat),hAPP(int,int,uminus_uminus(int),A)) = hAPP(rat,rat,uminus_uminus(rat),hAPP(int,rat,number_number_of(rat),A)). [resolve(5052,a,5030,a)]. 13.36/13.28 5053 -number_ring(A) | hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),hAPP(int,int,bit1,pls))) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(int,A,number_number_of(A),B)),one_one(A)) # label(fact_7_add__special_I3_J) # label(axiom). [clausify(3435)]. 13.36/13.28 Derived: hAPP(int,real,number_number_of(real),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,bit1,pls))) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,number_number_of(real),A)),one_one(real)). [resolve(5053,a,4976,a)]. 13.36/13.28 Derived: hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,bit1,pls))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),A)),one_one(int)). [resolve(5053,a,5000,a)]. 13.36/13.28 Derived: hAPP(int,complex,number_number_of(complex),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,bit1,pls))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(int,complex,number_number_of(complex),A)),one_one(complex)). [resolve(5053,a,5012,a)]. 13.36/13.28 Derived: hAPP(int,rat,number_number_of(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,bit1,pls))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,number_number_of(rat),A)),one_one(rat)). [resolve(5053,a,5030,a)]. 13.36/13.28 5054 -field_inverse_zero(A) | -number_ring(A) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),hAPP(int,A,number_number_of(A),pls)) # label(fact_1781_divide__Numeral0) # label(axiom). [clausify(3547)]. 13.36/13.28 Derived: -field_inverse_zero(real) | zero_zero(real) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),hAPP(int,real,number_number_of(real),pls)). [resolve(5054,b,4976,a)]. 13.36/13.28 Derived: -field_inverse_zero(int) | zero_zero(int) = hAPP(int,int,hAPP(int,fun(int,int),inverse_divide(int),A),hAPP(int,int,number_number_of(int),pls)). [resolve(5054,b,5000,a)]. 13.36/13.28 Derived: -field_inverse_zero(complex) | zero_zero(complex) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),A),hAPP(int,complex,number_number_of(complex),pls)). [resolve(5054,b,5012,a)]. 13.36/13.28 Derived: -field_inverse_zero(rat) | zero_zero(rat) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),hAPP(int,rat,number_number_of(rat),pls)). [resolve(5054,b,5030,a)]. 13.36/13.28 5055 -number_ring(A) | zero_zero(A) = hAPP(int,A,number_number_of(A),pls) # label(fact_85_number__of__Pls) # label(axiom). [clausify(3781)]. 13.36/13.28 5056 -number_ring(A) | hAPP(int,A,number_number_of(A),hAPP(int,int,bit1,B)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),one_one(A)),hAPP(int,A,number_number_of(A),B))),hAPP(int,A,number_number_of(A),B)) # label(fact_103_number__of__Bit1) # label(axiom). [clausify(3827)]. 13.50/13.45 Derived: hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,A)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(int,real,number_number_of(real),A))),hAPP(int,real,number_number_of(real),A)). [resolve(5056,a,4976,a)]. 13.50/13.45 Derived: hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,A)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(int,int,number_number_of(int),A))),hAPP(int,int,number_number_of(int),A)). [resolve(5056,a,5000,a)]. 13.50/13.45 Derived: hAPP(int,complex,number_number_of(complex),hAPP(int,int,bit1,A)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),one_one(complex)),hAPP(int,complex,number_number_of(complex),A))),hAPP(int,complex,number_number_of(complex),A)). [resolve(5056,a,5012,a)]. 13.50/13.45 Derived: hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit1,A)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),one_one(rat)),hAPP(int,rat,number_number_of(rat),A))),hAPP(int,rat,number_number_of(rat),A)). [resolve(5056,a,5030,a)]. 13.50/13.45 5057 -number_ring(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),hAPP(int,A,number_number_of(A),pls)) # label(fact_88_add__numeral__0__right) # label(axiom). [clausify(3868)]. 13.50/13.45 Derived: ti(real,A) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),hAPP(int,real,number_number_of(real),pls)). [resolve(5057,a,4976,a)]. 13.50/13.45 Derived: ti(int,A) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,number_number_of(int),pls)). [resolve(5057,a,5000,a)]. 13.50/13.45 Derived: ti(complex,A) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),hAPP(int,complex,number_number_of(complex),pls)). [resolve(5057,a,5012,a)]. 13.50/13.45 Derived: ti(rat,A) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),hAPP(int,rat,number_number_of(rat),pls)). [resolve(5057,a,5030,a)]. 13.50/13.45 5058 -number_ring(A) | -hBOOL(hAPP(nat,bool,even_odd_even(nat),B)) | one_one(A) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(int,A,number_number_of(A),min)),B) # label(fact_2513_Parity_Oneg__one__even__power) # label(axiom). [clausify(3996)]. 13.50/13.45 5059 -number_ring(A) | -linordered_idom(A) | one_one(A) = hAPP(A,A,abs_abs(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(int,A,number_number_of(A),min)),B)) # label(fact_1029_abs__power__minus__one) # label(axiom). [clausify(4033)]. 13.50/13.45 Derived: -linordered_idom(real) | one_one(real) = hAPP(real,real,abs_abs(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),A)). [resolve(5059,a,4976,a)]. 13.50/13.45 Derived: -linordered_idom(int) | one_one(int) = hAPP(int,int,abs_abs(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),A)). [resolve(5059,a,5000,a)]. 13.50/13.45 Derived: -linordered_idom(complex) | one_one(complex) = hAPP(complex,complex,abs_abs(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(int,complex,number_number_of(complex),min)),A)). [resolve(5059,a,5012,a)]. 13.50/13.45 Derived: -linordered_idom(rat) | one_one(rat) = hAPP(rat,rat,abs_abs(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),hAPP(int,rat,number_number_of(rat),min)),A)). [resolve(5059,a,5030,a)]. 13.50/13.45 5060 -number_ring(A) | hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),C))),D) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(int,A,number_number_of(A),B)),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(int,A,number_number_of(A),C)),D)) # label(fact_777_add__number__of__diff1) # label(axiom). [clausify(4035)]. 13.50/13.45 Derived: hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,number_number_of(real),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B))),C) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,number_number_of(real),A)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,number_number_of(real),B)),C)). [resolve(5060,a,4976,a)]. 13.79/13.78 Derived: hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B))),C) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),A)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,number_number_of(int),B)),C)). [resolve(5060,a,5000,a)]. 13.79/13.78 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B))),C) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(int,complex,number_number_of(complex),A)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(int,complex,number_number_of(complex),B)),C)). [resolve(5060,a,5012,a)]. 13.79/13.78 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B))),C) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,number_number_of(rat),A)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(int,rat,number_number_of(rat),B)),C)). [resolve(5060,a,5030,a)]. 13.79/13.78 5061 -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,hAPP(A,fun(nat,A),power_power(A),hAPP(int,A,number_number_of(A),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))) # label(fact_106_of__nat__less__two__power) # label(axiom). [clausify(4376)]. 13.79/13.78 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,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A))). [resolve(5061,b,4976,a)]. 13.79/13.78 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,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A))). [resolve(5061,b,5000,a)]. 13.79/13.78 Derived: -linordered_idom(complex) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less(complex),hAPP(nat,complex,semiring_1_of_nat(complex),A)),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A))). [resolve(5061,b,5012,a)]. 13.79/13.78 Derived: -linordered_idom(rat) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(nat,rat,semiring_1_of_nat(rat),A)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A))). [resolve(5061,b,5030,a)]. 13.79/13.78 5062 -number_ring(A) | hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit1,pls)))) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(int,A,number_number_of(A),B)),one_one(A)) # label(fact_2103_diff__special_I2_J) # label(axiom). [clausify(4400)]. 13.79/13.78 Derived: hAPP(int,real,number_number_of(real),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit1,pls)))) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,number_number_of(real),A)),one_one(real)). [resolve(5062,a,4976,a)]. 13.79/13.78 Derived: hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit1,pls)))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,number_number_of(int),A)),one_one(int)). [resolve(5062,a,5000,a)]. 13.79/13.78 Derived: hAPP(int,complex,number_number_of(complex),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit1,pls)))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(int,complex,number_number_of(complex),A)),one_one(complex)). [resolve(5062,a,5012,a)]. 13.88/13.82 Derived: hAPP(int,rat,number_number_of(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit1,pls)))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(int,rat,number_number_of(rat),A)),one_one(rat)). [resolve(5062,a,5030,a)]. 13.88/13.82 5063 -number_ring(A) | hBOOL(hAPP(nat,bool,even_odd_even(nat),B)) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(int,A,number_number_of(A),min)),B) = hAPP(int,A,number_number_of(A),min) # label(fact_2498_Parity_Oneg__one__odd__power) # label(axiom). [clausify(4405)]. 13.88/13.82 5064 -number_ring(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(int,A,number_number_of(A),hAPP(int,int,uminus_uminus(int),B))),C) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,uminus_uminus(A),hAPP(int,A,number_number_of(A),B))),C) # label(fact_1926_minus__number__of__mult) # label(axiom). [clausify(4417)]. 13.88/13.82 Derived: hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,uminus_uminus(int),A))),B) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,uminus_uminus(real),hAPP(int,real,number_number_of(real),A))),B). [resolve(5064,a,4976,a)]. 13.88/13.82 Derived: hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,uminus_uminus(int),A))),B) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,uminus_uminus(int),hAPP(int,int,number_number_of(int),A))),B). [resolve(5064,a,5000,a)]. 13.88/13.82 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,uminus_uminus(int),A))),B) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,uminus_uminus(complex),hAPP(int,complex,number_number_of(complex),A))),B). [resolve(5064,a,5012,a)]. 13.88/13.82 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,uminus_uminus(int),A))),B) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,uminus_uminus(rat),hAPP(int,rat,number_number_of(rat),A))),B). [resolve(5064,a,5030,a)]. 13.88/13.82 5065 -number_ring(A) | hAPP(int,A,ring_1_of_int(A),hAPP(int,int,number_number_of(int),min)) = hAPP(int,A,number_number_of(A),min) # label(fact_3616_of__int__m1) # label(axiom). [clausify(4449)]. 13.88/13.82 Derived: hAPP(int,real,ring_1_of_int(real),hAPP(int,int,number_number_of(int),min)) = hAPP(int,real,number_number_of(real),min). [resolve(5065,a,4976,a)]. 13.88/13.82 Derived: hAPP(int,int,ring_1_of_int(int),hAPP(int,int,number_number_of(int),min)) = hAPP(int,int,number_number_of(int),min). [resolve(5065,a,5000,a)]. 13.88/13.82 Derived: hAPP(int,complex,ring_1_of_int(complex),hAPP(int,int,number_number_of(int),min)) = hAPP(int,complex,number_number_of(complex),min). [resolve(5065,a,5012,a)]. 13.88/13.82 Derived: hAPP(int,rat,ring_1_of_int(rat),hAPP(int,int,number_number_of(int),min)) = hAPP(int,rat,number_number_of(rat),min). [resolve(5065,a,5030,a)]. 13.88/13.82 5066 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(int,A,number_number_of(A),B)),hAPP(int,A,number_number_of(A),C))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)) # label(fact_207_le__number__of) # label(axiom). [clausify(4450)]. 13.88/13.82 Derived: -linordered_idom(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),A)),hAPP(int,real,number_number_of(real),B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(5066,b,4976,a)]. 13.88/13.82 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),A)),hAPP(int,int,number_number_of(int),B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(5066,b,5000,a)]. 13.88/13.82 Derived: -linordered_idom(complex) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less_eq(complex),hAPP(int,complex,number_number_of(complex),A)),hAPP(int,complex,number_number_of(complex),B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(5066,b,5012,a)]. 13.88/13.88 Derived: -linordered_idom(rat) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,number_number_of(rat),A)),hAPP(int,rat,number_number_of(rat),B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(5066,b,5030,a)]. 13.88/13.88 5067 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(int,A,number_number_of(A),B)),hAPP(int,A,number_number_of(A),C))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)) # label(fact_207_le__number__of) # label(axiom). [clausify(4450)]. 13.88/13.88 Derived: -linordered_idom(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),A)),hAPP(int,real,number_number_of(real),B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(5067,b,4976,a)]. 13.88/13.88 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),A)),hAPP(int,int,number_number_of(int),B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(5067,b,5000,a)]. 13.88/13.88 Derived: -linordered_idom(complex) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less_eq(complex),hAPP(int,complex,number_number_of(complex),A)),hAPP(int,complex,number_number_of(complex),B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(5067,b,5012,a)]. 13.88/13.88 Derived: -linordered_idom(rat) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,number_number_of(rat),A)),hAPP(int,rat,number_number_of(rat),B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(5067,b,5030,a)]. 13.88/13.88 5068 -number_ring(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(int,A,number_number_of(A),hAPP(int,int,bit1,pls))),B) # label(fact_495_mult__numeral__1) # label(axiom). [clausify(4461)]. 13.88/13.88 Derived: ti(real,A) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,pls))),A). [resolve(5068,a,4976,a)]. 13.88/13.88 Derived: ti(int,A) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,pls))),A). [resolve(5068,a,5000,a)]. 13.88/13.88 Derived: ti(complex,A) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,bit1,pls))),A). [resolve(5068,a,5012,a)]. 13.88/13.88 Derived: ti(rat,A) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit1,pls))),A). [resolve(5068,a,5030,a)]. 13.88/13.88 5069 -number_ring(A) | hAPP(int,A,number_number_of(A),hAPP(int,int,pred,B)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,uminus_uminus(A),one_one(A))),hAPP(int,A,number_number_of(A),B)) # label(fact_2935_number__of__pred) # label(axiom). [clausify(4492)]. 13.88/13.88 Derived: hAPP(int,real,number_number_of(real),hAPP(int,int,pred,A)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,uminus_uminus(real),one_one(real))),hAPP(int,real,number_number_of(real),A)). [resolve(5069,a,4976,a)]. 13.88/13.88 Derived: hAPP(int,int,number_number_of(int),hAPP(int,int,pred,A)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,uminus_uminus(int),one_one(int))),hAPP(int,int,number_number_of(int),A)). [resolve(5069,a,5000,a)]. 13.88/13.88 Derived: hAPP(int,complex,number_number_of(complex),hAPP(int,int,pred,A)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,uminus_uminus(complex),one_one(complex))),hAPP(int,complex,number_number_of(complex),A)). [resolve(5069,a,5012,a)]. 13.88/13.88 Derived: hAPP(int,rat,number_number_of(rat),hAPP(int,int,pred,A)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,uminus_uminus(rat),one_one(rat))),hAPP(int,rat,number_number_of(rat),A)). [resolve(5069,a,5030,a)]. 13.88/13.88 5070 -number_ring(A) | zero_zero(A) != hAPP(int,A,number_number_of(A),B) | hBOOL(hAPP(A,bool,iszero(A),hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),pls),hAPP(int,int,uminus_uminus(int),B))))) # label(fact_2182_eq__special_I1_J) # label(axiom). [clausify(4517)]. 13.96/13.97 Derived: zero_zero(real) != hAPP(int,real,number_number_of(real),A) | hBOOL(hAPP(real,bool,iszero(real),hAPP(int,real,number_number_of(real),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),pls),hAPP(int,int,uminus_uminus(int),A))))). [resolve(5070,a,4976,a)]. 13.96/13.97 Derived: zero_zero(int) != hAPP(int,int,number_number_of(int),A) | hBOOL(hAPP(int,bool,iszero(int),hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),pls),hAPP(int,int,uminus_uminus(int),A))))). [resolve(5070,a,5000,a)]. 13.96/13.97 Derived: zero_zero(complex) != hAPP(int,complex,number_number_of(complex),A) | hBOOL(hAPP(complex,bool,iszero(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),pls),hAPP(int,int,uminus_uminus(int),A))))). [resolve(5070,a,5012,a)]. 13.96/13.97 Derived: zero_zero(rat) != hAPP(int,rat,number_number_of(rat),A) | hBOOL(hAPP(rat,bool,iszero(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),pls),hAPP(int,int,uminus_uminus(int),A))))). [resolve(5070,a,5030,a)]. 13.96/13.97 5071 -number_ring(A) | zero_zero(A) = hAPP(int,A,number_number_of(A),B) | -hBOOL(hAPP(A,bool,iszero(A),hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),pls),hAPP(int,int,uminus_uminus(int),B))))) # label(fact_2182_eq__special_I1_J) # label(axiom). [clausify(4517)]. 13.96/13.97 Derived: zero_zero(real) = hAPP(int,real,number_number_of(real),A) | -hBOOL(hAPP(real,bool,iszero(real),hAPP(int,real,number_number_of(real),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),pls),hAPP(int,int,uminus_uminus(int),A))))). [resolve(5071,a,4976,a)]. 13.96/13.97 Derived: zero_zero(int) = hAPP(int,int,number_number_of(int),A) | -hBOOL(hAPP(int,bool,iszero(int),hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),pls),hAPP(int,int,uminus_uminus(int),A))))). [resolve(5071,a,5000,a)]. 13.96/13.97 Derived: zero_zero(complex) = hAPP(int,complex,number_number_of(complex),A) | -hBOOL(hAPP(complex,bool,iszero(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),pls),hAPP(int,int,uminus_uminus(int),A))))). [resolve(5071,a,5012,a)]. 13.96/13.97 Derived: zero_zero(rat) = hAPP(int,rat,number_number_of(rat),A) | -hBOOL(hAPP(rat,bool,iszero(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),pls),hAPP(int,int,uminus_uminus(int),A))))). [resolve(5071,a,5030,a)]. 13.96/13.97 5072 -field(A) | -number_ring(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),hAPP(int,A,number_number_of(A),hAPP(int,int,bit1,pls))) # label(fact_1783_divide__Numeral1) # label(axiom). [clausify(4531)]. 13.96/13.97 5073 -number_ring(A) | -field_char_0(A) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),hAPP(int,A,number_number_of(A),B)),field_char_0_Rats(A))) # label(fact_3710_Rats__number__of) # label(axiom). [clausify(4549)]. 13.96/13.97 Derived: -field_char_0(real) | hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),hAPP(int,real,number_number_of(real),A)),field_char_0_Rats(real))). [resolve(5073,a,4976,a)]. 13.96/13.97 Derived: -field_char_0(int) | hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,number_number_of(int),A)),field_char_0_Rats(int))). [resolve(5073,a,5000,a)]. 13.96/13.97 Derived: -field_char_0(complex) | hBOOL(hAPP(fun(complex,bool),bool,hAPP(complex,fun(fun(complex,bool),bool),member(complex),hAPP(int,complex,number_number_of(complex),A)),field_char_0_Rats(complex))). [resolve(5073,a,5012,a)]. 13.96/13.97 Derived: -field_char_0(rat) | hBOOL(hAPP(fun(rat,bool),bool,hAPP(rat,fun(fun(rat,bool),bool),member(rat),hAPP(int,rat,number_number_of(rat),A)),field_char_0_Rats(rat))). [resolve(5073,a,5030,a)]. 13.96/13.97 5074 -number_ring(A) | -linordered_idom(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,pls)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),hAPP(int,A,number_number_of(A),B))) # label(fact_241_le__special_I2_J) # label(axiom). [clausify(4626)]. 14.07/14.01 Derived: -linordered_idom(real) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,pls)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(int,real,number_number_of(real),A))). [resolve(5074,a,4976,a)]. 14.07/14.01 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,pls)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),hAPP(int,int,number_number_of(int),A))). [resolve(5074,a,5000,a)]. 14.07/14.01 Derived: -linordered_idom(complex) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,pls)),A)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less_eq(complex),one_one(complex)),hAPP(int,complex,number_number_of(complex),A))). [resolve(5074,a,5012,a)]. 14.07/14.01 Derived: -linordered_idom(rat) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,pls)),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),hAPP(int,rat,number_number_of(rat),A))). [resolve(5074,a,5030,a)]. 14.07/14.01 5075 -number_ring(A) | -linordered_idom(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,pls)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),hAPP(int,A,number_number_of(A),B))) # label(fact_241_le__special_I2_J) # label(axiom). [clausify(4626)]. 14.07/14.01 Derived: -linordered_idom(real) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,pls)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(int,real,number_number_of(real),A))). [resolve(5075,a,4976,a)]. 14.07/14.01 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,pls)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),hAPP(int,int,number_number_of(int),A))). [resolve(5075,a,5000,a)]. 14.07/14.01 Derived: -linordered_idom(complex) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,pls)),A)) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less_eq(complex),one_one(complex)),hAPP(int,complex,number_number_of(complex),A))). [resolve(5075,a,5012,a)]. 14.07/14.01 Derived: -linordered_idom(rat) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,pls)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),hAPP(int,rat,number_number_of(rat),A))). [resolve(5075,a,5030,a)]. 14.07/14.01 5076 -number_ring(A) | zero_zero(A) != hAPP(int,A,number_number_of(A),B) | hBOOL(hAPP(A,bool,iszero(A),hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),hAPP(int,int,uminus_uminus(int),pls))))) # label(fact_2183_eq__special_I3_J) # label(axiom). [clausify(4665)]. 14.07/14.01 Derived: zero_zero(real) != hAPP(int,real,number_number_of(real),A) | hBOOL(hAPP(real,bool,iszero(real),hAPP(int,real,number_number_of(real),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),pls))))). [resolve(5076,a,4976,a)]. 14.07/14.01 Derived: zero_zero(int) != hAPP(int,int,number_number_of(int),A) | hBOOL(hAPP(int,bool,iszero(int),hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),pls))))). [resolve(5076,a,5000,a)]. 14.07/14.01 Derived: zero_zero(complex) != hAPP(int,complex,number_number_of(complex),A) | hBOOL(hAPP(complex,bool,iszero(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),pls))))). [resolve(5076,a,5012,a)]. 14.07/14.01 Derived: zero_zero(rat) != hAPP(int,rat,number_number_of(rat),A) | hBOOL(hAPP(rat,bool,iszero(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),pls))))). [resolve(5076,a,5030,a)]. 14.07/14.01 5077 -number_ring(A) | zero_zero(A) = hAPP(int,A,number_number_of(A),B) | -hBOOL(hAPP(A,bool,iszero(A),hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),hAPP(int,int,uminus_uminus(int),pls))))) # label(fact_2183_eq__special_I3_J) # label(axiom). [clausify(4665)]. 14.11/14.06 Derived: zero_zero(real) = hAPP(int,real,number_number_of(real),A) | -hBOOL(hAPP(real,bool,iszero(real),hAPP(int,real,number_number_of(real),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),pls))))). [resolve(5077,a,4976,a)]. 14.11/14.06 Derived: zero_zero(int) = hAPP(int,int,number_number_of(int),A) | -hBOOL(hAPP(int,bool,iszero(int),hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),pls))))). [resolve(5077,a,5000,a)]. 14.11/14.06 Derived: zero_zero(complex) = hAPP(int,complex,number_number_of(complex),A) | -hBOOL(hAPP(complex,bool,iszero(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),pls))))). [resolve(5077,a,5012,a)]. 14.11/14.06 Derived: zero_zero(rat) = hAPP(int,rat,number_number_of(rat),A) | -hBOOL(hAPP(rat,bool,iszero(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),pls))))). [resolve(5077,a,5030,a)]. 14.11/14.06 5078 -number_ring(A) | -hBOOL(hAPP(A,bool,iszero(A),hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit1,pls)))))) | one_one(A) = hAPP(int,A,number_number_of(A),B) # label(fact_2118_eq__special_I4_J) # label(axiom). [clausify(4694)]. 14.11/14.06 Derived: -hBOOL(hAPP(real,bool,iszero(real),hAPP(int,real,number_number_of(real),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit1,pls)))))) | one_one(real) = hAPP(int,real,number_number_of(real),A). [resolve(5078,a,4976,a)]. 14.11/14.06 Derived: -hBOOL(hAPP(int,bool,iszero(int),hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit1,pls)))))) | one_one(int) = hAPP(int,int,number_number_of(int),A). [resolve(5078,a,5000,a)]. 14.11/14.06 Derived: -hBOOL(hAPP(complex,bool,iszero(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit1,pls)))))) | one_one(complex) = hAPP(int,complex,number_number_of(complex),A). [resolve(5078,a,5012,a)]. 14.11/14.06 Derived: -hBOOL(hAPP(rat,bool,iszero(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit1,pls)))))) | one_one(rat) = hAPP(int,rat,number_number_of(rat),A). [resolve(5078,a,5030,a)]. 14.11/14.06 5079 -number_ring(A) | hBOOL(hAPP(A,bool,iszero(A),hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit1,pls)))))) | one_one(A) != hAPP(int,A,number_number_of(A),B) # label(fact_2118_eq__special_I4_J) # label(axiom). [clausify(4694)]. 14.11/14.06 Derived: hBOOL(hAPP(real,bool,iszero(real),hAPP(int,real,number_number_of(real),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit1,pls)))))) | one_one(real) != hAPP(int,real,number_number_of(real),A). [resolve(5079,a,4976,a)]. 14.11/14.06 Derived: hBOOL(hAPP(int,bool,iszero(int),hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit1,pls)))))) | one_one(int) != hAPP(int,int,number_number_of(int),A). [resolve(5079,a,5000,a)]. 14.11/14.06 Derived: hBOOL(hAPP(complex,bool,iszero(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit1,pls)))))) | one_one(complex) != hAPP(int,complex,number_number_of(complex),A). [resolve(5079,a,5012,a)]. 14.11/14.06 Derived: hBOOL(hAPP(rat,bool,iszero(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit1,pls)))))) | one_one(rat) != hAPP(int,rat,number_number_of(rat),A). [resolve(5079,a,5030,a)]. 14.11/14.06 5080 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(int,A,number_number_of(A),B)),one_one(A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),hAPP(int,int,bit1,pls))) # label(fact_240_le__special_I4_J) # label(axiom). [clausify(4714)]. 14.11/14.11 Derived: -linordered_idom(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),A)),one_one(real))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(int,int,bit1,pls))). [resolve(5080,b,4976,a)]. 14.11/14.11 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),A)),one_one(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(int,int,bit1,pls))). [resolve(5080,b,5000,a)]. 14.11/14.11 Derived: -linordered_idom(complex) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less_eq(complex),hAPP(int,complex,number_number_of(complex),A)),one_one(complex))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(int,int,bit1,pls))). [resolve(5080,b,5012,a)]. 14.11/14.11 Derived: -linordered_idom(rat) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,number_number_of(rat),A)),one_one(rat))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(int,int,bit1,pls))). [resolve(5080,b,5030,a)]. 14.11/14.11 5081 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(int,A,number_number_of(A),B)),one_one(A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),hAPP(int,int,bit1,pls))) # label(fact_240_le__special_I4_J) # label(axiom). [clausify(4714)]. 14.11/14.11 Derived: -linordered_idom(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),A)),one_one(real))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(int,int,bit1,pls))). [resolve(5081,b,4976,a)]. 14.11/14.11 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),A)),one_one(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(int,int,bit1,pls))). [resolve(5081,b,5000,a)]. 14.11/14.11 Derived: -linordered_idom(complex) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less_eq(complex),hAPP(int,complex,number_number_of(complex),A)),one_one(complex))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(int,int,bit1,pls))). [resolve(5081,b,5012,a)]. 14.11/14.11 Derived: -linordered_idom(rat) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,number_number_of(rat),A)),one_one(rat))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(int,int,bit1,pls))). [resolve(5081,b,5030,a)]. 14.11/14.11 5082 -number_ring(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(int,A,number_number_of(A),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),B) # label(fact_543_mult__2__right) # label(axiom). [clausify(4752)]. 14.11/14.11 Derived: hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),A). [resolve(5082,a,4976,a)]. 14.11/14.11 Derived: hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),A). [resolve(5082,a,5000,a)]. 14.11/14.11 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),hAPP(int,complex,number_number_of(complex),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),A). [resolve(5082,a,5012,a)]. 14.11/14.11 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),A). [resolve(5082,a,5030,a)]. 14.11/14.11 5083 -number_ring(A) | hAPP(int,A,number_number_of(A),hAPP(int,int,uminus_uminus(int),B)) = hAPP(A,A,uminus_uminus(A),hAPP(int,A,number_number_of(A),B)) # label(fact_1928_number__of__minus) # label(axiom). [clausify(4759)]. 14.26/14.22 5084 -number_ring(A) | -field_inverse_zero(A) | hAPP(A,A,uminus_uminus(A),B) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),hAPP(int,A,number_number_of(A),min)) # label(fact_2018_divide__minus1) # label(axiom). [clausify(4763)]. 14.26/14.22 Derived: -field_inverse_zero(real) | hAPP(real,real,uminus_uminus(real),A) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),hAPP(int,real,number_number_of(real),min)). [resolve(5084,a,4976,a)]. 14.26/14.22 Derived: -field_inverse_zero(int) | hAPP(int,int,uminus_uminus(int),A) = hAPP(int,int,hAPP(int,fun(int,int),inverse_divide(int),A),hAPP(int,int,number_number_of(int),min)). [resolve(5084,a,5000,a)]. 14.26/14.22 Derived: -field_inverse_zero(complex) | hAPP(complex,complex,uminus_uminus(complex),A) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),A),hAPP(int,complex,number_number_of(complex),min)). [resolve(5084,a,5012,a)]. 14.26/14.22 Derived: -field_inverse_zero(rat) | hAPP(rat,rat,uminus_uminus(rat),A) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),hAPP(int,rat,number_number_of(rat),min)). [resolve(5084,a,5030,a)]. 14.26/14.22 5085 -number_ring(A) | hAPP(int,A,ring_1_of_int(A),hAPP(int,int,number_number_of(int),B)) = hAPP(int,A,number_number_of(A),B) # label(fact_3606_of__int__number__of__eq) # label(axiom). [clausify(4807)]. 14.26/14.22 Derived: hAPP(int,real,ring_1_of_int(real),hAPP(int,int,number_number_of(int),A)) = hAPP(int,real,number_number_of(real),A). [resolve(5085,a,4976,a)]. 14.26/14.22 Derived: hAPP(int,int,ring_1_of_int(int),hAPP(int,int,number_number_of(int),A)) = hAPP(int,int,number_number_of(int),A). [resolve(5085,a,5000,a)]. 14.26/14.22 Derived: hAPP(int,complex,ring_1_of_int(complex),hAPP(int,int,number_number_of(int),A)) = hAPP(int,complex,number_number_of(complex),A). [resolve(5085,a,5012,a)]. 14.26/14.22 Derived: hAPP(int,rat,ring_1_of_int(rat),hAPP(int,int,number_number_of(int),A)) = hAPP(int,rat,number_number_of(rat),A). [resolve(5085,a,5030,a)]. 14.26/14.22 5086 -number_ring(A) | -hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),B))) | zero_zero(A) = hAPP(nat,A,semiring_1_of_nat(A),hAPP(int,nat,number_number_of(nat),B)) # label(fact_1624_of__nat__number__of__eq) # label(axiom). [clausify(4889)]. 14.26/14.22 Derived: -hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),A))) | zero_zero(real) = hAPP(nat,real,semiring_1_of_nat(real),hAPP(int,nat,number_number_of(nat),A)). [resolve(5086,a,4976,a)]. 14.26/14.22 Derived: -hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),A))) | zero_zero(int) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,number_number_of(nat),A)). [resolve(5086,a,5000,a)]. 14.26/14.22 Derived: -hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),A))) | zero_zero(complex) = hAPP(nat,complex,semiring_1_of_nat(complex),hAPP(int,nat,number_number_of(nat),A)). [resolve(5086,a,5012,a)]. 14.26/14.22 Derived: -hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),A))) | zero_zero(rat) = hAPP(nat,rat,semiring_1_of_nat(rat),hAPP(int,nat,number_number_of(nat),A)). [resolve(5086,a,5030,a)]. 14.26/14.22 5087 -number_ring(A) | hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),B))) | hAPP(nat,A,semiring_1_of_nat(A),hAPP(int,nat,number_number_of(nat),B)) = hAPP(int,A,number_number_of(A),B) # label(fact_1624_of__nat__number__of__eq) # label(axiom). [clausify(4889)]. 14.26/14.22 Derived: hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),A))) | hAPP(nat,real,semiring_1_of_nat(real),hAPP(int,nat,number_number_of(nat),A)) = hAPP(int,real,number_number_of(real),A). [resolve(5087,a,4976,a)]. 14.26/14.22 Derived: hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),A))) | hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,number_number_of(nat),A)) = hAPP(int,int,number_number_of(int),A). [resolve(5087,a,5000,a)]. 14.26/14.22 Derived: hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),A))) | hAPP(nat,complex,semiring_1_of_nat(complex),hAPP(int,nat,number_number_of(nat),A)) = hAPP(int,complex,number_number_of(complex),A). [resolve(5087,a,5012,a)]. 16.40/16.32 Derived: hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),A))) | hAPP(nat,rat,semiring_1_of_nat(rat),hAPP(int,nat,number_number_of(nat),A)) = hAPP(int,rat,number_number_of(rat),A). [resolve(5087,a,5030,a)]. 16.40/16.32 5088 archim1804426504_field(rat) # label(arity_Rat_Orat___Archimedean__Field_Oarchimedean__field) # label(axiom). [assumption]. 16.40/16.32 5089 -archim1804426504_field(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(int,A,ring_1_of_int(A),f1(A,B))),B)) # label(fact_3642_floor__exists) # label(axiom). [clausify(3)]. 16.40/16.32 5090 -archim1804426504_field(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(int,A,ring_1_of_int(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),f1(A,B)),one_one(int))))) # label(fact_3642_floor__exists) # label(axiom). [clausify(3)]. 16.40/16.32 5091 -archim1804426504_field(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(nat,A,semiring_1_of_nat(A),f2(A,B)))) # label(fact_2835_ex__less__of__nat) # label(axiom). [clausify(17)]. 16.40/16.32 5092 -archim1804426504_field(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),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(nat,A,semiring_1_of_nat(A),f56(A,C,B))),B))) # label(fact_2667_ex__less__of__nat__mult) # label(axiom). [clausify(263)]. 16.40/16.32 5093 -archim1804426504_field(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(int,A,ring_1_of_int(A),f297(A,B)))) # label(fact_3651_ex__le__of__int) # label(axiom). [clausify(1783)]. 16.40/16.32 5094 -archim1804426504_field(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(int,A,ring_1_of_int(A),f345(A,B)))) # label(fact_3650_ex__less__of__int) # label(axiom). [clausify(2092)]. 16.40/16.32 5095 -archim1804426504_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),f366(A,B))) # label(fact_2644_ex__inverse__of__nat__less) # label(axiom). [clausify(2193)]. 16.40/16.32 5096 -archim1804426504_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,inverse_inverse(A),hAPP(nat,A,semiring_1_of_nat(A),f366(A,B)))),B)) # label(fact_2644_ex__inverse__of__nat__less) # label(axiom). [clausify(2193)]. 16.40/16.32 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,ring_1_of_int(rat),f1(rat,A))),A)). [resolve(5088,a,5089,a)]. 16.40/16.32 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(int,rat,ring_1_of_int(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),f1(rat,A)),one_one(int))))). [resolve(5088,a,5090,a)]. 16.40/16.32 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(nat,rat,semiring_1_of_nat(rat),f2(rat,A)))). [resolve(5088,a,5091,a)]. 16.40/16.32 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(nat,rat,semiring_1_of_nat(rat),f56(rat,B,A))),A))). [resolve(5088,a,5092,a)]. 16.40/16.32 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),hAPP(int,rat,ring_1_of_int(rat),f297(rat,A)))). [resolve(5088,a,5093,a)]. 16.40/16.32 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(int,rat,ring_1_of_int(rat),f345(rat,A)))). [resolve(5088,a,5094,a)]. 16.40/16.32 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),f366(rat,A))). [resolve(5088,a,5095,a)]. 16.40/16.32 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,inverse_inverse(rat),hAPP(nat,rat,semiring_1_of_nat(rat),f366(rat,A)))),A)). [resolve(5088,a,5096,a)]. 16.40/16.32 5097 -archim1804426504_field(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(int,A,ring_1_of_int(A),f502(A,B))),B)) # label(fact_3644_ex__of__int__less) # label(axiom). [clausify(2987)]. 16.40/16.32 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,ring_1_of_int(rat),f502(rat,A))),A)). [resolve(5097,a,5088,a)]. 17.30/17.26 5098 archim1804426504_field(real) # label(arity_RealDef_Oreal___Archimedean__Field_Oarchimedean__field) # label(axiom). [assumption]. 17.30/17.26 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,ring_1_of_int(real),f1(real,A))),A)). [resolve(5098,a,5089,a)]. 17.30/17.26 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(int,real,ring_1_of_int(real),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),f1(real,A)),one_one(int))))). [resolve(5098,a,5090,a)]. 17.30/17.26 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(nat,real,semiring_1_of_nat(real),f2(real,A)))). [resolve(5098,a,5091,a)]. 17.30/17.26 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,semiring_1_of_nat(real),f56(real,B,A))),A))). [resolve(5098,a,5092,a)]. 17.30/17.26 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),hAPP(int,real,ring_1_of_int(real),f297(real,A)))). [resolve(5098,a,5093,a)]. 17.30/17.26 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(int,real,ring_1_of_int(real),f345(real,A)))). [resolve(5098,a,5094,a)]. 17.30/17.26 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),f366(real,A))). [resolve(5098,a,5095,a)]. 17.30/17.26 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,semiring_1_of_nat(real),f366(real,A)))),A)). [resolve(5098,a,5096,a)]. 17.30/17.26 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,ring_1_of_int(real),f502(real,A))),A)). [resolve(5098,a,5097,a)]. 17.30/17.26 5099 -archim1804426504_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(int,A,ring_1_of_int(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),C),one_one(int))))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(int,A,ring_1_of_int(A),C)),B)) | ti(int,C) = ti(int,f680(A,B)) # label(fact_3641_floor__exists1) # label(axiom). [clausify(3993)]. 17.30/17.26 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(int,rat,ring_1_of_int(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),one_one(int))))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,ring_1_of_int(rat),B)),A)) | ti(int,B) = ti(int,f680(rat,A)). [resolve(5099,a,5088,a)]. 17.30/17.26 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(int,real,ring_1_of_int(real),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),one_one(int))))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,ring_1_of_int(real),B)),A)) | ti(int,B) = ti(int,f680(real,A)). [resolve(5099,a,5098,a)]. 17.30/17.26 5100 -archim1804426504_field(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(int,A,ring_1_of_int(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),f680(A,B)),one_one(int))))) # label(fact_3641_floor__exists1) # label(axiom). [clausify(3993)]. 17.30/17.26 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(int,rat,ring_1_of_int(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),f680(rat,A)),one_one(int))))). [resolve(5100,a,5088,a)]. 17.30/17.26 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(int,real,ring_1_of_int(real),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),f680(real,A)),one_one(int))))). [resolve(5100,a,5098,a)]. 17.30/17.26 5101 -archim1804426504_field(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(int,A,ring_1_of_int(A),f680(A,B))),B)) # label(fact_3641_floor__exists1) # label(axiom). [clausify(3993)]. 17.30/17.26 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,ring_1_of_int(rat),f680(rat,A))),A)). [resolve(5101,a,5088,a)]. 19.53/19.48 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,ring_1_of_int(real),f680(real,A))),A)). [resolve(5101,a,5098,a)]. 19.53/19.48 5102 -archim1804426504_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,inverse_inverse(A),hAPP(nat,A,semiring_1_of_nat(A),hAPP(nat,nat,suc,f723(A,B))))),B)) # label(fact_2662_ex__inverse__of__nat__Suc__less) # label(axiom). [clausify(4211)]. 19.53/19.48 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,inverse_inverse(rat),hAPP(nat,rat,semiring_1_of_nat(rat),hAPP(nat,nat,suc,f723(rat,A))))),A)). [resolve(5102,a,5088,a)]. 19.53/19.48 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,semiring_1_of_nat(real),hAPP(nat,nat,suc,f723(real,A))))),A)). [resolve(5102,a,5098,a)]. 19.53/19.48 5103 -archim1804426504_field(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(nat,A,semiring_1_of_nat(A),f731(A,B)))) # label(fact_2836_ex__le__of__nat) # label(axiom). [clausify(4269)]. 19.53/19.48 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),hAPP(nat,rat,semiring_1_of_nat(rat),f731(rat,A)))). [resolve(5103,a,5088,a)]. 19.53/19.48 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),hAPP(nat,real,semiring_1_of_nat(real),f731(real,A)))). [resolve(5103,a,5098,a)]. 19.53/19.48 5104 -zero_neq_one(A) | zero_zero(A) != one_one(A) # label(fact_167_zero__neq__one) # label(axiom). [clausify(1979)]. 19.53/19.48 5105 zero_neq_one(complex) # label(arity_Complex_Ocomplex___Rings_Ozero__neq__one) # label(axiom). [assumption]. 19.53/19.48 5106 zero_neq_one(nat) # label(arity_Nat_Onat___Rings_Ozero__neq__one) # label(axiom). [assumption]. 19.53/19.48 5107 zero_neq_one(int) # label(arity_Int_Oint___Rings_Ozero__neq__one) # label(axiom). [assumption]. 19.53/19.48 5108 zero_neq_one(quickcheck_code_int) # label(arity_Quickcheck__Narrowing_Ocode__int___Rings_Ozero__neq__one) # label(axiom). [assumption]. 19.53/19.48 Derived: zero_zero(complex) != one_one(complex). [resolve(5104,a,5105,a)]. 19.53/19.48 Derived: zero_zero(nat) != one_one(nat). [resolve(5104,a,5106,a)]. 19.53/19.48 Derived: zero_zero(quickcheck_code_int) != one_one(quickcheck_code_int). [resolve(5104,a,5108,a)]. 19.53/19.48 5109 -zero_neq_one(A) | -no_zero_divisors(A) | -mult_zero(A) | -power(A) | ti(A,B) != zero_zero(A) | zero_zero(nat) = C | zero_zero(A) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C) # label(fact_188_power__eq__0__iff) # label(axiom). [clausify(2101)]. 19.53/19.48 Derived: -no_zero_divisors(complex) | -mult_zero(complex) | -power(complex) | ti(complex,A) != zero_zero(complex) | zero_zero(nat) = B | zero_zero(complex) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),B). [resolve(5109,a,5105,a)]. 19.53/19.48 Derived: -no_zero_divisors(nat) | -mult_zero(nat) | -power(nat) | ti(nat,A) != zero_zero(nat) | zero_zero(nat) = B | zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),B). [resolve(5109,a,5106,a)]. 19.53/19.48 Derived: -no_zero_divisors(int) | -mult_zero(int) | -power(int) | ti(int,A) != zero_zero(int) | zero_zero(nat) = B | zero_zero(int) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),B). [resolve(5109,a,5107,a)]. 19.53/19.48 Derived: -no_zero_divisors(quickcheck_code_int) | -mult_zero(quickcheck_code_int) | -power(quickcheck_code_int) | ti(quickcheck_code_int,A) != zero_zero(quickcheck_code_int) | zero_zero(nat) = B | zero_zero(quickcheck_code_int) = hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B). [resolve(5109,a,5108,a)]. 19.53/19.48 5110 -zero_neq_one(A) | -no_zero_divisors(A) | -mult_zero(A) | -power(A) | ti(A,B) = zero_zero(A) | zero_zero(A) != hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C) # label(fact_188_power__eq__0__iff) # label(axiom). [clausify(2101)]. 19.53/19.48 Derived: -no_zero_divisors(complex) | -mult_zero(complex) | -power(complex) | ti(complex,A) = zero_zero(complex) | zero_zero(complex) != hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),B). [resolve(5110,a,5105,a)]. 20.50/20.50 Derived: -no_zero_divisors(nat) | -mult_zero(nat) | -power(nat) | ti(nat,A) = zero_zero(nat) | zero_zero(nat) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),B). [resolve(5110,a,5106,a)]. 20.50/20.50 Derived: -no_zero_divisors(int) | -mult_zero(int) | -power(int) | ti(int,A) = zero_zero(int) | zero_zero(int) != hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),B). [resolve(5110,a,5107,a)]. 20.50/20.50 Derived: -no_zero_divisors(quickcheck_code_int) | -mult_zero(quickcheck_code_int) | -power(quickcheck_code_int) | ti(quickcheck_code_int,A) = zero_zero(quickcheck_code_int) | zero_zero(quickcheck_code_int) != hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B). [resolve(5110,a,5108,a)]. 20.50/20.50 5111 -zero_neq_one(A) | -no_zero_divisors(A) | -mult_zero(A) | -power(A) | zero_zero(nat) != B | zero_zero(A) != hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),B) # label(fact_188_power__eq__0__iff) # label(axiom). [clausify(2101)]. 20.50/20.50 Derived: -no_zero_divisors(complex) | -mult_zero(complex) | -power(complex) | zero_zero(nat) != A | zero_zero(complex) != hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),B),A). [resolve(5111,a,5105,a)]. 20.50/20.50 Derived: -no_zero_divisors(nat) | -mult_zero(nat) | -power(nat) | zero_zero(nat) != A | zero_zero(nat) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),A). [resolve(5111,a,5106,a)]. 20.50/20.50 Derived: -no_zero_divisors(int) | -mult_zero(int) | -power(int) | zero_zero(nat) != A | zero_zero(int) != hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),A). [resolve(5111,a,5107,a)]. 20.50/20.50 Derived: -no_zero_divisors(quickcheck_code_int) | -mult_zero(quickcheck_code_int) | -power(quickcheck_code_int) | zero_zero(nat) != A | zero_zero(quickcheck_code_int) != hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),B),A). [resolve(5111,a,5108,a)]. 20.50/20.50 5112 -zero_neq_one(A) | zero_zero(A) != one_one(A) # label(fact_166_one__neq__zero) # label(axiom). [clausify(2369)]. 20.50/20.50 5113 zero_neq_one(real) # label(arity_RealDef_Oreal___Rings_Ozero__neq__one) # label(axiom). [assumption]. 20.50/20.50 Derived: zero_zero(real) != one_one(real). [resolve(5113,a,5104,a)]. 20.50/20.50 Derived: -no_zero_divisors(real) | -mult_zero(real) | -power(real) | ti(real,A) != zero_zero(real) | zero_zero(nat) = B | zero_zero(real) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B). [resolve(5113,a,5109,a)]. 20.50/20.50 Derived: -no_zero_divisors(real) | -mult_zero(real) | -power(real) | ti(real,A) = zero_zero(real) | zero_zero(real) != hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B). [resolve(5113,a,5110,a)]. 20.50/20.50 Derived: -no_zero_divisors(real) | -mult_zero(real) | -power(real) | zero_zero(nat) != A | zero_zero(real) != hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),A). [resolve(5113,a,5111,a)]. 20.50/20.50 5114 zero_neq_one(rat) # label(arity_Rat_Orat___Rings_Ozero__neq__one) # label(axiom). [assumption]. 20.50/20.50 Derived: zero_zero(rat) != one_one(rat). [resolve(5114,a,5104,a)]. 20.50/20.50 Derived: -no_zero_divisors(rat) | -mult_zero(rat) | -power(rat) | ti(rat,A) != zero_zero(rat) | zero_zero(nat) = B | zero_zero(rat) = hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B). [resolve(5114,a,5109,a)]. 20.50/20.50 Derived: -no_zero_divisors(rat) | -mult_zero(rat) | -power(rat) | ti(rat,A) = zero_zero(rat) | zero_zero(rat) != hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B). [resolve(5114,a,5110,a)]. 20.50/20.50 Derived: -no_zero_divisors(rat) | -mult_zero(rat) | -power(rat) | zero_zero(nat) != A | zero_zero(rat) != hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),B),A). [resolve(5114,a,5111,a)]. 20.50/20.50 5115 -power(A) | -no_zero_divisors(A) | -zero_neq_one(A) | -mult_zero(A) | zero_zero(nat) = hAPP(int,nat,number_number_of(nat),B) | ti(A,C) != zero_zero(A) | zero_zero(A) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),hAPP(int,nat,number_number_of(nat),B)) # label(fact_89_power__eq__0__iff__number__of) # label(axiom). [clausify(3660)]. 23.17/23.10 5116 -power(A) | -no_zero_divisors(A) | -zero_neq_one(A) | -mult_zero(A) | zero_zero(nat) != hAPP(int,nat,number_number_of(nat),B) | zero_zero(A) != hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),hAPP(int,nat,number_number_of(nat),B)) # label(fact_89_power__eq__0__iff__number__of) # label(axiom). [clausify(3660)]. 23.17/23.10 5117 -power(A) | -no_zero_divisors(A) | -zero_neq_one(A) | -mult_zero(A) | ti(A,B) = zero_zero(A) | zero_zero(A) != hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(int,nat,number_number_of(nat),C)) # label(fact_89_power__eq__0__iff__number__of) # label(axiom). [clausify(3660)]. 23.17/23.10 5118 zero_neq_one(code_code_numeral) # label(arity_Code__Numeral_Ocode__numeral___Rings_Ozero__neq__one) # label(axiom). [assumption]. 23.17/23.10 Derived: zero_zero(code_code_numeral) != one_one(code_code_numeral). [resolve(5118,a,5104,a)]. 23.17/23.10 Derived: -no_zero_divisors(code_code_numeral) | -mult_zero(code_code_numeral) | -power(code_code_numeral) | ti(code_code_numeral,A) != zero_zero(code_code_numeral) | zero_zero(nat) = B | zero_zero(code_code_numeral) = hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B). [resolve(5118,a,5109,a)]. 23.17/23.10 Derived: -no_zero_divisors(code_code_numeral) | -mult_zero(code_code_numeral) | -power(code_code_numeral) | ti(code_code_numeral,A) = zero_zero(code_code_numeral) | zero_zero(code_code_numeral) != hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B). [resolve(5118,a,5110,a)]. 23.17/23.10 Derived: -no_zero_divisors(code_code_numeral) | -mult_zero(code_code_numeral) | -power(code_code_numeral) | zero_zero(nat) != A | zero_zero(code_code_numeral) != hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),B),A). [resolve(5118,a,5111,a)]. 23.17/23.10 5119 linordered_field(rat) # label(arity_Rat_Orat___Fields_Olinordered__field) # label(axiom). [assumption]. 23.17/23.10 5120 -linordered_field(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),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),B)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),D),B))) # label(fact_1892_mult__imp__le__div__pos) # label(axiom). [clausify(11)]. 23.17/23.10 5121 -linordered_field(A) | ti(A,B) = zero_zero(A) | hAPP(A,A,abs_abs(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),B)) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),hAPP(A,A,abs_abs(A),C)),hAPP(A,A,abs_abs(A),B)) # label(fact_1858_nonzero__abs__divide) # label(axiom). [clausify(68)]. 23.17/23.10 5122 -linordered_field(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_eq(A),zero_zero(A)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),C))) # label(fact_1866_divide__nonpos__neg) # label(axiom). [clausify(310)]. 23.17/23.10 5123 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))),C)) # label(fact_1885_gt__half__sum) # label(axiom). [clausify(314)]. 23.17/23.10 5124 -linordered_field(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),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),B))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),B)),D)) # label(fact_1893_mult__imp__div__pos__le) # label(axiom). [clausify(588)]. 23.17/23.10 5125 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),B)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),D),B))) # label(fact_1872_mult__imp__less__div__pos) # label(axiom). [clausify(767)]. 23.17/23.10 5126 -linordered_field(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),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),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),E)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),D))) # label(fact_1862_frac__less2) # label(axiom). [clausify(802)]. 23.17/23.10 5127 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),D))) # label(fact_1838_divide__strict__right__mono) # label(axiom). [clausify(854)]. 23.17/23.10 5128 -linordered_field(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),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),C)),zero_zero(A))) # label(fact_1865_divide__nonpos__pos) # label(axiom). [clausify(1026)]. 23.17/23.10 5129 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),B)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),D),B))) # label(fact_1869_pos__less__divide__eq) # label(axiom). [clausify(1142)]. 23.17/23.10 5130 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),B)),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),D),B))) # label(fact_1869_pos__less__divide__eq) # label(axiom). [clausify(1142)]. 23.17/23.10 5131 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),B)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),B))) # label(fact_1870_pos__divide__less__eq) # label(axiom). [clausify(1194)]. 23.17/23.10 5132 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),B)),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),B))) # label(fact_1870_pos__divide__less__eq) # label(axiom). [clausify(1194)]. 23.17/23.10 5133 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),B)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),B)),C)) # label(fact_1890_neg__divide__le__eq) # label(axiom). [clausify(1302)]. 23.17/23.10 5134 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),B)),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),B)),C)) # label(fact_1890_neg__divide__le__eq) # label(axiom). [clausify(1302)]. 23.17/23.10 5135 -linordered_field(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),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_eq(A),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),E)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),D))) # label(fact_1864_frac__le) # label(axiom). [clausify(1598)]. 23.17/23.10 5136 -linordered_field(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(A),zero_zero(A)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),B))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),D),C)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),D),B))) # label(fact_1889_divide__left__mono) # label(axiom). [clausify(1701)]. 23.17/23.10 5137 -linordered_field(A) | ti(A,B) = zero_zero(A) | hAPP(A,A,abs_abs(A),hAPP(A,A,inverse_inverse(A),B)) = hAPP(A,A,inverse_inverse(A),hAPP(A,A,abs_abs(A),B)) # label(fact_2601_nonzero__abs__inverse) # label(axiom). [clausify(1806)]. 23.17/23.10 5138 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,inverse_inverse(A),B)),zero_zero(A))) | ti(A,B) = zero_zero(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) # label(fact_2587_inverse__negative__imp__negative) # label(axiom). [clausify(1809)]. 23.17/23.10 5139 -linordered_field(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),B),one_one(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),hAPP(A,A,inverse_inverse(A),B))) # label(fact_2628_one__le__inverse) # label(axiom). [clausify(1828)]. 23.17/23.10 5140 -linordered_field(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,inverse_inverse(A),hAPP(nat,A,semiring_1_of_nat(A),hAPP(nat,nat,fact_fact(nat),B))))) # label(fact_2632_inv__of__nat__fact__ge__zero) # label(axiom). [clausify(1995)]. 23.17/23.10 5141 -linordered_field(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),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_eq(A),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),E)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),D))) # label(fact_1863_frac__less) # label(axiom). [clausify(2038)]. 23.17/23.10 5142 -linordered_field(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),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),C)),zero_zero(A))) # label(fact_1861_divide__nonneg__neg) # label(axiom). [clausify(2080)]. 23.17/23.10 5143 -linordered_field(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),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),B))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),B))) # label(fact_1891_neg__le__divide__eq) # label(axiom). [clausify(2112)]. 23.17/23.10 5144 -linordered_field(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),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),B))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),B))) # label(fact_1891_neg__le__divide__eq) # label(axiom). [clausify(2112)]. 23.17/23.10 5145 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,inverse_inverse(A),B))) | ti(A,B) = zero_zero(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) # label(fact_2591_inverse__positive__imp__positive) # label(axiom). [clausify(2241)]. 23.17/23.10 5146 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),B)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),D),B)),C)) # label(fact_1874_neg__divide__less__eq) # label(axiom). [clausify(2269)]. 23.17/23.10 5147 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),B)),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),D),B)),C)) # label(fact_1874_neg__divide__less__eq) # label(axiom). [clausify(2269)]. 23.17/23.11 5148 -linordered_field(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),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),D),B))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),B)),D)) # label(fact_1823_pos__le__divide__eq) # label(axiom). [clausify(2305)]. 23.17/23.11 5149 -linordered_field(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),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),D),B))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),B)),D)) # label(fact_1823_pos__le__divide__eq) # label(axiom). [clausify(2305)]. 23.17/23.11 5150 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),C)),zero_zero(A))) # label(fact_1840_divide__neg__pos) # label(axiom). [clausify(2476)]. 23.17/23.11 5151 -linordered_field(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),one_one(A)),hAPP(A,A,inverse_inverse(A),B))) # label(fact_2614_one__less__inverse) # label(axiom). [clausify(2484)]. 23.17/23.11 5152 -linordered_field(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),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),B)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),B))) # label(fact_1824_pos__divide__le__eq) # label(axiom). [clausify(2515)]. 23.17/23.11 5153 -linordered_field(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),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),B)),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),B))) # label(fact_1824_pos__divide__le__eq) # label(axiom). [clausify(2515)]. 23.17/23.11 5154 -linordered_field(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,inverse_inverse(A),hAPP(nat,A,semiring_1_of_nat(A),hAPP(nat,nat,fact_fact(nat),B))))) # label(fact_2633_inv__of__nat__fact__gt__zero) # label(axiom). [clausify(2519)]. 23.17/23.11 5155 -linordered_field(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),zero_zero(A)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),D),B)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),D),C))) # label(fact_1876_divide__strict__left__mono__neg) # label(axiom). [clausify(2532)]. 23.17/23.11 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),A)),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),C),A))). [resolve(5119,a,5120,a)]. 23.17/23.11 Derived: ti(rat,A) = zero_zero(rat) | hAPP(rat,rat,abs_abs(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),B),A)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),hAPP(rat,rat,abs_abs(rat),B)),hAPP(rat,rat,abs_abs(rat),A)). [resolve(5119,a,5121,a)]. 23.17/23.11 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),zero_zero(rat))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),B))). [resolve(5119,a,5122,a)]. 23.17/23.12 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),one_one(rat)),one_one(rat)))),B)). [resolve(5119,a,5123,a)]. 23.17/23.12 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),A))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),B),A)),C)). [resolve(5119,a,5124,a)]. 23.17/23.12 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),A)),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),C),A))). [resolve(5119,a,5125,a)]. 23.17/23.12 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),C)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),C),D)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),D)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),B),C))). [resolve(5119,a,5126,a)]. 23.17/23.12 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),C)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),B),C))). [resolve(5119,a,5127,a)]. 23.17/23.12 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),zero_zero(rat))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),B)),zero_zero(rat))). [resolve(5119,a,5128,a)]. 23.17/23.12 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),A)),C)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),C),A))). [resolve(5119,a,5130,a)]. 23.17/23.12 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),B),A)),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),A))). [resolve(5119,a,5131,a)]. 23.17/23.12 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),B),A)),C)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),A))). [resolve(5119,a,5132,a)]. 23.17/23.12 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),zero_zero(rat))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),B),A)),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),A)),B)). [resolve(5119,a,5133,a)]. 23.17/23.12 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),B),A)),C)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),A)),B)). [resolve(5119,a,5134,a)]. 23.17/23.13 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),C)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),C),D)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),D)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),B),C))). [resolve(5119,a,5135,a)]. 23.17/23.13 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),C)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),A))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),C),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),C),A))). [resolve(5119,a,5136,a)]. 23.17/23.13 Derived: ti(rat,A) = zero_zero(rat) | hAPP(rat,rat,abs_abs(rat),hAPP(rat,rat,inverse_inverse(rat),A)) = hAPP(rat,rat,inverse_inverse(rat),hAPP(rat,rat,abs_abs(rat),A)). [resolve(5119,a,5137,a)]. 23.17/23.13 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,inverse_inverse(rat),A)),zero_zero(rat))) | ti(rat,A) = zero_zero(rat) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),zero_zero(rat))). [resolve(5119,a,5138,a)]. 23.17/23.13 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),one_one(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),hAPP(rat,rat,inverse_inverse(rat),A))). [resolve(5119,a,5139,a)]. 23.17/23.13 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(rat,rat,inverse_inverse(rat),hAPP(nat,rat,semiring_1_of_nat(rat),hAPP(nat,nat,fact_fact(nat),A))))). [resolve(5119,a,5140,a)]. 23.17/23.13 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),C)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),C),D)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),D)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),B),C))). [resolve(5119,a,5141,a)]. 23.17/23.13 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),B)),zero_zero(rat))). [resolve(5119,a,5142,a)]. 23.17/23.13 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),zero_zero(rat))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),A))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),C),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),B),A))). [resolve(5119,a,5143,a)]. 23.17/23.13 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),A))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),C),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),B),A))). [resolve(5119,a,5144,a)]. 23.17/23.13 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(rat,rat,inverse_inverse(rat),A))) | ti(rat,A) = zero_zero(rat) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)). [resolve(5119,a,5145,a)]. 23.17/23.13 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),zero_zero(rat))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),A)),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),C),A)),B)). [resolve(5119,a,5146,a)]. 23.29/23.21 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),A)),C)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),C),A)),B)). [resolve(5119,a,5147,a)]. 23.29/23.21 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),C),A))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),A)),C)). [resolve(5119,a,5148,a)]. 23.29/23.21 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),zero_zero(rat))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),B)),zero_zero(rat))). [resolve(5119,a,5150,a)]. 23.29/23.21 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),one_one(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),hAPP(rat,rat,inverse_inverse(rat),A))). [resolve(5119,a,5151,a)]. 23.29/23.21 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),B),A)),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),A))). [resolve(5119,a,5152,a)]. 23.29/23.21 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(rat,rat,inverse_inverse(rat),hAPP(nat,rat,semiring_1_of_nat(rat),hAPP(nat,nat,fact_fact(nat),A))))). [resolve(5119,a,5154,a)]. 23.29/23.21 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),C),zero_zero(rat))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),C),A)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),C),B))). [resolve(5119,a,5155,a)]. 23.29/23.21 5156 -linordered_field(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),B),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A))))) # label(fact_1886_less__half__sum) # label(axiom). [clausify(2664)]. 23.29/23.21 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),one_one(rat)),one_one(rat))))). [resolve(5156,a,5119,a)]. 23.29/23.21 5157 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),C))) # label(fact_1839_divide__neg__neg) # label(axiom). [clausify(2692)]. 23.29/23.21 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),zero_zero(rat))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),B))). [resolve(5157,a,5119,a)]. 23.60/23.54 5158 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),D)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),D))) # label(fact_1837_divide__strict__right__mono__neg) # label(axiom). [clausify(2738)]. 23.60/23.54 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),C),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),B),C)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),C))). [resolve(5158,a,5119,a)]. 23.60/23.54 5159 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,inverse_inverse(A),B))) # label(fact_2592_positive__imp__inverse__positive) # label(axiom). [clausify(2746)]. 23.60/23.54 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(rat,rat,inverse_inverse(rat),A))). [resolve(5159,a,5119,a)]. 23.60/23.54 5160 -linordered_field(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),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,inverse_inverse(A),C)),hAPP(A,A,inverse_inverse(A),B))) # label(fact_2608_le__imp__inverse__le__neg) # label(axiom). [clausify(2957)]. 23.60/23.54 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,inverse_inverse(rat),B)),hAPP(rat,rat,inverse_inverse(rat),A))). [resolve(5160,a,5119,a)]. 23.60/23.54 5161 -linordered_field(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),f527(A,B,C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_1897_field__le__epsilon) # label(axiom). [clausify(3097)]. 23.60/23.54 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),f527(rat,A,B))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)). [resolve(5161,a,5119,a)]. 23.60/23.54 5162 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),f527(A,B,C)))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_1897_field__le__epsilon) # label(axiom). [clausify(3097)]. 23.60/23.54 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),B),f527(rat,A,B)))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)). [resolve(5162,a,5119,a)]. 23.60/23.54 5163 -linordered_field(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),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),D),B))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),B))) # label(fact_1873_neg__less__divide__eq) # label(axiom). [clausify(3262)]. 23.60/23.54 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),zero_zero(rat))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),C),A))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),C),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),A))). [resolve(5163,a,5119,a)]. 23.60/23.54 5164 -linordered_field(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),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),D),B))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),B))) # label(fact_1873_neg__less__divide__eq) # label(axiom). [clausify(3262)]. 23.77/23.77 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),C),A))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),C),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),A))). [resolve(5164,a,5119,a)]. 23.77/23.77 5165 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,inverse_inverse(A),B)),hAPP(A,A,inverse_inverse(A),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),B)) # label(fact_2585_inverse__less__imp__less__neg) # label(axiom). [clausify(3298)]. 23.77/23.77 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,inverse_inverse(rat),A)),hAPP(rat,rat,inverse_inverse(rat),B))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),A)). [resolve(5165,a,5119,a)]. 23.77/23.77 5166 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),C)),zero_zero(A))) # label(fact_1841_divide__pos__neg) # label(axiom). [clausify(3428)]. 23.77/23.77 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),B)),zero_zero(rat))). [resolve(5166,a,5119,a)]. 23.77/23.77 5167 linordered_field(real) # label(arity_RealDef_Oreal___Fields_Olinordered__field) # label(axiom). [assumption]. 23.77/23.77 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),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),A)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),C),A))). [resolve(5167,a,5120,a)]. 23.77/23.77 Derived: ti(real,A) = zero_zero(real) | hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),A)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,abs_abs(real),B)),hAPP(real,real,abs_abs(real),A)). [resolve(5167,a,5121,a)]. 23.77/23.77 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_eq(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),B))). [resolve(5167,a,5122,a)]. 23.77/23.77 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),one_one(real)))),B)). [resolve(5167,a,5123,a)]. 23.77/23.77 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),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),A))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),A)),C)). [resolve(5167,a,5124,a)]. 23.77/23.77 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),A)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),C),A))). [resolve(5167,a,5125,a)]. 23.77/23.77 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),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),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),D)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),C))). [resolve(5167,a,5126,a)]. 23.77/23.78 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),C)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),C))). [resolve(5167,a,5127,a)]. 23.77/23.78 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),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),B)),zero_zero(real))). [resolve(5167,a,5128,a)]. 23.77/23.78 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),A)),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),C),A))). [resolve(5167,a,5130,a)]. 23.77/23.78 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),A)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),A))). [resolve(5167,a,5131,a)]. 23.77/23.78 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),A)),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),A))). [resolve(5167,a,5132,a)]. 23.77/23.78 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),A)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),A)),B)). [resolve(5167,a,5133,a)]. 23.77/23.78 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),A)),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),A)),B)). [resolve(5167,a,5134,a)]. 23.77/23.78 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),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_eq(real),C),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),D)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),C))). [resolve(5167,a,5135,a)]. 23.77/23.78 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(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),A))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),C),B)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),C),A))). [resolve(5167,a,5136,a)]. 23.87/23.79 Derived: ti(real,A) = zero_zero(real) | hAPP(real,real,abs_abs(real),hAPP(real,real,inverse_inverse(real),A)) = hAPP(real,real,inverse_inverse(real),hAPP(real,real,abs_abs(real),A)). [resolve(5167,a,5137,a)]. 23.87/23.79 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,inverse_inverse(real),A)),zero_zero(real))) | ti(real,A) = zero_zero(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))). [resolve(5167,a,5138,a)]. 23.87/23.79 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),A),one_one(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(real,real,inverse_inverse(real),A))). [resolve(5167,a,5139,a)]. 23.87/23.79 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,semiring_1_of_nat(real),hAPP(nat,nat,fact_fact(nat),A))))). [resolve(5167,a,5140,a)]. 23.87/23.79 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),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_eq(real),C),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),D)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),C))). [resolve(5167,a,5141,a)]. 23.87/23.79 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),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),B)),zero_zero(real))). [resolve(5167,a,5142,a)]. 23.87/23.79 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),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),A))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),A))). [resolve(5167,a,5143,a)]. 23.87/23.79 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),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),A))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),A))). [resolve(5167,a,5144,a)]. 23.87/23.79 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,inverse_inverse(real),A))) | ti(real,A) = zero_zero(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)). [resolve(5167,a,5145,a)]. 23.87/23.79 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),A)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),C),A)),B)). [resolve(5167,a,5146,a)]. 23.87/23.79 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),A)),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),C),A)),B)). [resolve(5167,a,5147,a)]. 23.87/23.79 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),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),C),A))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),A)),C)). [resolve(5167,a,5148,a)]. 23.89/23.80 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),B)),zero_zero(real))). [resolve(5167,a,5150,a)]. 23.89/23.80 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),one_one(real)),hAPP(real,real,inverse_inverse(real),A))). [resolve(5167,a,5151,a)]. 23.89/23.80 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),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),A)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),A))). [resolve(5167,a,5152,a)]. 23.89/23.80 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,semiring_1_of_nat(real),hAPP(nat,nat,fact_fact(nat),A))))). [resolve(5167,a,5154,a)]. 23.89/23.80 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),C),A)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),C),B))). [resolve(5167,a,5155,a)]. 23.89/23.80 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),one_one(real))))). [resolve(5167,a,5156,a)]. 23.89/23.80 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),B))). [resolve(5167,a,5157,a)]. 23.89/23.80 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),C)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),C))). [resolve(5167,a,5158,a)]. 23.89/23.80 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,inverse_inverse(real),A))). [resolve(5167,a,5159,a)]. 23.89/23.80 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),B),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,inverse_inverse(real),B)),hAPP(real,real,inverse_inverse(real),A))). [resolve(5167,a,5160,a)]. 23.89/23.80 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),f527(real,A,B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)). [resolve(5167,a,5161,a)]. 23.89/23.80 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),f527(real,A,B)))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)). [resolve(5167,a,5162,a)]. 24.19/24.15 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),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),C),A))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),A))). [resolve(5167,a,5163,a)]. 24.19/24.15 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),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),C),A))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),A))). [resolve(5167,a,5164,a)]. 24.19/24.15 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,inverse_inverse(real),A)),hAPP(real,real,inverse_inverse(real),B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),A)). [resolve(5167,a,5165,a)]. 24.19/24.15 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),B)),zero_zero(real))). [resolve(5167,a,5166,a)]. 24.19/24.15 5168 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,inverse_inverse(A),B)),zero_zero(A))) # label(fact_2590_negative__imp__inverse__negative) # label(axiom). [clausify(3738)]. 24.19/24.15 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,inverse_inverse(rat),A)),zero_zero(rat))). [resolve(5168,a,5119,a)]. 24.19/24.15 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,inverse_inverse(real),A)),zero_zero(real))). [resolve(5168,a,5167,a)]. 24.19/24.15 5169 -linordered_field(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),zero_zero(A)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),B))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),D),C)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),D),B))) # label(fact_1875_divide__strict__left__mono) # label(axiom). [clausify(3858)]. 24.19/24.15 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),C)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),A))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),C),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),C),A))). [resolve(5169,a,5119,a)]. 24.19/24.15 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),A))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),C),B)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),C),A))). [resolve(5169,a,5167,a)]. 24.19/24.15 5170 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,inverse_inverse(A),B)),hAPP(A,A,inverse_inverse(A),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_eq(A),C),B)) # label(fact_2609_inverse__le__imp__le) # label(axiom). [clausify(3873)]. 24.71/24.67 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,inverse_inverse(rat),A)),hAPP(rat,rat,inverse_inverse(rat),B))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),A)). [resolve(5170,a,5119,a)]. 24.71/24.67 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,inverse_inverse(real),A)),hAPP(real,real,inverse_inverse(real),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_eq(real),B),A)). [resolve(5170,a,5167,a)]. 24.71/24.67 5171 -linordered_field(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),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,inverse_inverse(A),C)),hAPP(A,A,inverse_inverse(A),B))) # label(fact_2588_less__imp__inverse__less__neg) # label(axiom). [clausify(4353)]. 24.71/24.67 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,inverse_inverse(rat),B)),hAPP(rat,rat,inverse_inverse(rat),A))). [resolve(5171,a,5119,a)]. 24.71/24.67 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),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,inverse_inverse(real),B)),hAPP(real,real,inverse_inverse(real),A))). [resolve(5171,a,5167,a)]. 24.71/24.67 5172 -linordered_field(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)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,inverse_inverse(A),C)),hAPP(A,A,inverse_inverse(A),B))) # label(fact_2589_less__imp__inverse__less) # label(axiom). [clausify(4395)]. 24.71/24.67 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,inverse_inverse(rat),B)),hAPP(rat,rat,inverse_inverse(rat),A))). [resolve(5172,a,5119,a)]. 24.71/24.67 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)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,inverse_inverse(real),B)),hAPP(real,real,inverse_inverse(real),A))). [resolve(5172,a,5167,a)]. 24.71/24.67 5173 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,inverse_inverse(A),B)),hAPP(A,A,inverse_inverse(A),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),B)) # label(fact_2610_inverse__le__imp__le__neg) # label(axiom). [clausify(4415)]. 24.71/24.67 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,inverse_inverse(rat),A)),hAPP(rat,rat,inverse_inverse(rat),B))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),A)). [resolve(5173,a,5119,a)]. 24.71/24.67 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,inverse_inverse(real),A)),hAPP(real,real,inverse_inverse(real),B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),A)). [resolve(5173,a,5167,a)]. 24.71/24.67 5174 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),C))) # label(fact_1842_divide__pos__pos) # label(axiom). [clausify(4422)]. 24.99/24.91 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),B))). [resolve(5174,a,5119,a)]. 24.99/24.91 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),B))). [resolve(5174,a,5167,a)]. 24.99/24.91 5175 -linordered_field(A) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),hAPP(fun(B,A),fun(B,A),hAPP(fun(A,A),fun(fun(B,A),fun(B,A)),combb(A,A,B),abs_abs(A)),C)),D) = hAPP(A,A,abs_abs(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),C),D)) # label(fact_3975_abs__setprod) # label(axiom). [clausify(4474)]. 24.99/24.91 Derived: hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),hAPP(fun(A,rat),fun(A,rat),hAPP(fun(rat,rat),fun(fun(A,rat),fun(A,rat)),combb(rat,rat,A),abs_abs(rat)),B)),C) = hAPP(rat,rat,abs_abs(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),B),C)). [resolve(5175,a,5119,a)]. 24.99/24.91 Derived: hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),hAPP(fun(A,real),fun(A,real),hAPP(fun(real,real),fun(fun(A,real),fun(A,real)),combb(real,real,A),abs_abs(real)),B)),C) = hAPP(real,real,abs_abs(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),B),C)). [resolve(5175,a,5167,a)]. 24.99/24.91 5176 -linordered_field(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),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,inverse_inverse(A),C)),hAPP(A,A,inverse_inverse(A),B))) # label(fact_2607_le__imp__inverse__le) # label(axiom). [clausify(4519)]. 24.99/24.91 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,inverse_inverse(rat),B)),hAPP(rat,rat,inverse_inverse(rat),A))). [resolve(5176,a,5119,a)]. 24.99/24.91 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,inverse_inverse(real),B)),hAPP(real,real,inverse_inverse(real),A))). [resolve(5176,a,5167,a)]. 24.99/24.91 5177 -linordered_field(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),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),B))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),B)),D)) # label(fact_1871_mult__imp__div__pos__less) # label(axiom). [clausify(4560)]. 24.99/24.91 5178 -linordered_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,inverse_inverse(A),B)),hAPP(A,A,inverse_inverse(A),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),C),B)) # label(fact_2586_inverse__less__imp__less) # label(axiom). [clausify(4681)]. 24.99/24.91 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,inverse_inverse(rat),A)),hAPP(rat,rat,inverse_inverse(rat),B))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),A)). [resolve(5178,a,5119,a)]. 24.99/24.91 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,inverse_inverse(real),A)),hAPP(real,real,inverse_inverse(real),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),B),A)). [resolve(5178,a,5167,a)]. 26.19/26.13 5179 -linordered_field(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_eq(A),zero_zero(A)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),C))) # label(fact_1860_divide__nonneg__pos) # label(axiom). [clausify(4688)]. 26.19/26.13 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),B))). [resolve(5179,a,5119,a)]. 26.19/26.13 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),B))). [resolve(5179,a,5167,a)]. 26.19/26.13 5180 real_normed_field(real) # label(arity_RealDef_Oreal___RealVector_Oreal__normed__field) # label(axiom). [assumption]. 26.19/26.13 5181 -real_normed_field(A) | -hBOOL(hAPP(fun(nat,A),bool,cauchy(A),B)) | hBOOL(hAPP(fun(nat,A),bool,cauchy(A),hAPP(A,fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(A,fun(nat,A)),combc(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),inverse_divide(A)),B)),C))) # label(fact_4129_divide_OCauchy) # label(axiom). [clausify(13)]. 26.19/26.13 5182 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | zero_zero(A) = hAPP(A,A,B,C) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,A),fun(fun(A,A),fun(A,A)),combb(A,A,A),inverse_inverse(A)),B)),C),hAPP(A,A,uminus_uminus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),hAPP(A,A,inverse_inverse(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(A,A,B,C)),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))))))) # label(fact_3930_DERIV__inverse__fun) # label(axiom). [clausify(51)]. 26.19/26.13 5183 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),hAPP(nat,fun(A,A),hAPP(fun(A,fun(nat,A)),fun(nat,fun(A,A)),combc(A,nat,A),hAPP(fun(A,A),fun(A,fun(nat,A)),hAPP(fun(A,fun(nat,A)),fun(fun(A,A),fun(A,fun(nat,A))),combb(A,fun(nat,A),A),power_power(A)),B)),E)),C),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(nat,A,semiring_1_of_nat(A),E)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(A,A,B,C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),E),hAPP(nat,nat,suc,zero_zero(nat)))))))) # label(fact_3918_DERIV__power) # label(axiom). [clausify(63)]. 26.19/26.13 5184 -banach(A) | -real_normed_field(A) | hAPP(A,A,exp(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),hAPP(A,A,exp(A),B)),hAPP(A,A,exp(A),C)) # label(fact_1748_exp__diff) # label(axiom). [clausify(85)]. 26.19/26.13 5185 -real_normed_field(A) | -banach(A) | hAPP(A,A,exp(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,exp(A),B)),hAPP(A,A,exp(A),C)) # label(fact_1774_mult__exp__exp) # label(axiom). [clausify(124)]. 26.19/26.13 5186 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),hAPP(A,fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(A,fun(nat,A)),combc(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),inverse_divide(A)),B)),D)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),D))) # label(fact_4112_sums__divide) # label(axiom). [clausify(189)]. 26.22/26.13 5187 -real_normed_field(A) | ti(A,B) = zero_zero(A) | hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(A,real,norm_norm(A),C)),hAPP(A,real,norm_norm(A),B)) = hAPP(A,real,norm_norm(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),B)) # label(fact_2006_nonzero__norm__divide) # label(axiom). [clausify(349)]. 26.22/26.13 5188 -real_normed_field(A) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),hAPP(A,fun(B,A),hAPP(fun(B,fun(A,A)),fun(A,fun(B,A)),combc(B,A,A),hAPP(fun(B,A),fun(B,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),combb(A,fun(A,A),B),inverse_divide(A)),C)),D)),E) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),C),E)),D) # label(fact_4208_divide_Osetsum) # label(axiom). [clausify(379)]. 26.22/26.13 5189 -real_normed_field(A) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),zero_zero(A)),B) # label(fact_1750_divide_Ozero) # label(axiom). [clausify(462)]. 26.22/26.13 5190 -real_normed_field(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),B)) | hAPP(fun(nat,A),A,suminf(A),hAPP(A,fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(A,fun(nat,A)),combc(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),inverse_divide(A)),B)),C)) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),hAPP(fun(nat,A),A,suminf(A),B)),C) # label(fact_4011_divide_Osuminf) # label(axiom). [clausify(575)]. 26.22/26.13 5191 -real_normed_field(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f122(B,A,D,E,F,C)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),hAPP(fun(B,bool),fun(A,A),hAPP(fun(A,fun(fun(B,bool),A)),fun(fun(B,bool),fun(A,A)),combc(A,fun(B,bool),A),hAPP(fun(A,fun(B,A)),fun(A,fun(fun(B,bool),A)),hAPP(fun(fun(B,A),fun(fun(B,bool),A)),fun(fun(A,fun(B,A)),fun(A,fun(fun(B,bool),A))),combb(fun(B,A),fun(fun(B,bool),A),A),big_co1399186613setsum(B,A)),D)),C)),E),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),hAPP(A,fun(B,A),F,E)),C))) # label(fact_4331_DERIV__setsum) # label(axiom). [clausify(605)]. 26.22/26.13 5192 -real_normed_field(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),hAPP(B,fun(A,A),hAPP(fun(A,fun(B,A)),fun(B,fun(A,A)),combc(A,B,A),D),f122(B,A,D,E,F,C))),E),hAPP(B,A,hAPP(A,fun(B,A),F,E),f122(B,A,D,E,F,C)))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),hAPP(fun(B,bool),fun(A,A),hAPP(fun(A,fun(fun(B,bool),A)),fun(fun(B,bool),fun(A,A)),combc(A,fun(B,bool),A),hAPP(fun(A,fun(B,A)),fun(A,fun(fun(B,bool),A)),hAPP(fun(fun(B,A),fun(fun(B,bool),A)),fun(fun(A,fun(B,A)),fun(A,fun(fun(B,bool),A))),combb(fun(B,A),fun(fun(B,bool),A),A),big_co1399186613setsum(B,A)),D)),C)),E),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),hAPP(A,fun(B,A),F,E)),C))) # label(fact_4331_DERIV__setsum) # label(axiom). [clausify(605)]. 26.22/26.13 5193 -real_normed_field(A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(A,real,norm_norm(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),C))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(A,real,norm_norm(A),B)),f123(A,C)))) # label(fact_2049_divide_Opos__bounded) # label(axiom). [clausify(607)]. 26.22/26.13 5194 -real_normed_field(A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),f123(A,B))) # label(fact_2049_divide_Opos__bounded) # label(axiom). [clausify(607)]. 26.22/26.13 Derived: -hBOOL(hAPP(fun(nat,real),bool,cauchy(real),A)) | hBOOL(hAPP(fun(nat,real),bool,cauchy(real),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),A)),B))). [resolve(5180,a,5181,a)]. 26.23/26.14 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | zero_zero(real) = hAPP(real,real,A,B) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),inverse_inverse(real)),A)),B),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,A,B)),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))))))). [resolve(5180,a,5182,a)]. 26.23/26.14 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),A)),D)),B),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,semiring_1_of_nat(real),D)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,A,B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),D),hAPP(nat,nat,suc,zero_zero(nat)))))))). [resolve(5180,a,5183,a)]. 26.23/26.14 Derived: -banach(real) | hAPP(real,real,exp(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,exp(real),A)),hAPP(real,real,exp(real),B)). [resolve(5180,a,5184,b)]. 26.23/26.14 Derived: -banach(real) | hAPP(real,real,exp(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,exp(real),A)),hAPP(real,real,exp(real),B)). [resolve(5180,a,5185,a)]. 26.23/26.14 Derived: -hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),A)),C)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),C))). [resolve(5180,a,5186,a)]. 26.23/26.14 Derived: ti(real,A) = zero_zero(real) | hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,norm_norm(real),B)),hAPP(real,real,norm_norm(real),A)) = hAPP(real,real,norm_norm(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),A)). [resolve(5180,a,5187,a)]. 26.23/26.14 Derived: hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),hAPP(real,fun(A,real),hAPP(fun(A,fun(real,real)),fun(real,fun(A,real)),combc(A,real,real),hAPP(fun(A,real),fun(A,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(A,real),fun(A,fun(real,real))),combb(real,fun(real,real),A),inverse_divide(real)),B)),C)),D) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),B),D)),C). [resolve(5180,a,5188,a)]. 26.23/26.14 Derived: zero_zero(real) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),zero_zero(real)),A). [resolve(5180,a,5189,a)]. 26.23/26.14 Derived: -hBOOL(hAPP(fun(nat,real),bool,summable(real),A)) | hAPP(fun(nat,real),real,suminf(real),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),A)),B)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(fun(nat,real),real,suminf(real),A)),B). [resolve(5180,a,5190,a)]. 26.23/26.17 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f122(A,real,C,D,E,B)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(A,bool),fun(real,real),hAPP(fun(real,fun(fun(A,bool),real)),fun(fun(A,bool),fun(real,real)),combc(real,fun(A,bool),real),hAPP(fun(real,fun(A,real)),fun(real,fun(fun(A,bool),real)),hAPP(fun(fun(A,real),fun(fun(A,bool),real)),fun(fun(real,fun(A,real)),fun(real,fun(fun(A,bool),real))),combb(fun(A,real),fun(fun(A,bool),real),real),big_co1399186613setsum(A,real)),C)),B)),D),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),hAPP(real,fun(A,real),E,D)),B))). [resolve(5180,a,5191,a)]. 26.23/26.17 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(A,fun(real,real),hAPP(fun(real,fun(A,real)),fun(A,fun(real,real)),combc(real,A,real),C),f122(A,real,C,D,E,B))),D),hAPP(A,real,hAPP(real,fun(A,real),E,D),f122(A,real,C,D,E,B)))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(A,bool),fun(real,real),hAPP(fun(real,fun(fun(A,bool),real)),fun(fun(A,bool),fun(real,real)),combc(real,fun(A,bool),real),hAPP(fun(real,fun(A,real)),fun(real,fun(fun(A,bool),real)),hAPP(fun(fun(A,real),fun(fun(A,bool),real)),fun(fun(real,fun(A,real)),fun(real,fun(fun(A,bool),real))),combb(fun(A,real),fun(fun(A,bool),real),real),big_co1399186613setsum(A,real)),C)),B)),D),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),hAPP(real,fun(A,real),E,D)),B))). [resolve(5180,a,5192,a)]. 26.23/26.17 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),A)),f123(real,B)))). [resolve(5180,a,5193,a)]. 26.23/26.17 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),f123(real,A))). [resolve(5180,a,5194,a)]. 26.23/26.17 5195 -topological_space(A) | -real_normed_field(B) | -hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),C),D),hAPP(A,filter(A),at(A),E))) | hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),hAPP(B,fun(A,B),hAPP(fun(A,fun(B,B)),fun(B,fun(A,B)),combc(A,B,B),hAPP(fun(A,B),fun(A,fun(B,B)),hAPP(fun(B,fun(B,B)),fun(fun(A,B),fun(A,fun(B,B))),combb(B,fun(B,B),A),inverse_divide(B)),C)),F)),hAPP(B,B,hAPP(B,fun(B,B),inverse_divide(B),D),F)),hAPP(A,filter(A),at(A),E))) # label(fact_4579_divide_OLIM) # label(axiom). [clausify(697)]. 26.23/26.17 Derived: -topological_space(A) | -hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),B),C),hAPP(A,filter(A),at(A),D))) | hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),hAPP(real,fun(A,real),hAPP(fun(A,fun(real,real)),fun(real,fun(A,real)),combc(A,real,real),hAPP(fun(A,real),fun(A,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(A,real),fun(A,fun(real,real))),combb(real,fun(real,real),A),inverse_divide(real)),B)),E)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),C),E)),hAPP(A,filter(A),at(A),D))). [resolve(5195,b,5180,a)]. 26.23/26.17 5196 -real_normed_field(A) | ti(A,B) = zero_zero(A) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(A,real,norm_norm(A),C)),D)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(A,real,norm_norm(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),B))),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(A,real,norm_norm(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),B)),E)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),E))),B)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(nat,A,semiring_1_of_nat(A),E)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),E),hAPP(nat,nat,suc,zero_zero(nat)))))))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,semiring_1_of_nat(real),E)),hAPP(nat,real,semiring_1_of_nat(real),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),E),hAPP(nat,nat,suc,zero_zero(nat)))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),D),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),E),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(A,real,norm_norm(A),B)))) # label(fact_1895_lemma__termdiff3) # label(axiom). [clausify(811)]. 26.23/26.18 Derived: ti(real,A) = zero_zero(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),B)),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),A))),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),A)),D)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),D))),A)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,semiring_1_of_nat(real),D)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),D),hAPP(nat,nat,suc,zero_zero(nat)))))))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,semiring_1_of_nat(real),D)),hAPP(nat,real,semiring_1_of_nat(real),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),D),hAPP(nat,nat,suc,zero_zero(nat)))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),D),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,real,norm_norm(real),A)))). [resolve(5196,a,5180,a)]. 26.23/26.18 5197 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),hAPP(A,fun(A,A),hAPP(fun(A,fun(A,A)),fun(A,fun(A,A)),combc(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),inverse_divide(A)),B)),E)),C),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),D),E))) # label(fact_3872_DERIV__cdivide) # label(axiom). [clausify(833)]. 26.23/26.18 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real)),A)),D)),B),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),C),D))). [resolve(5197,a,5180,a)]. 26.23/26.18 5198 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | hBOOL(hAPP(filter(A),bool,hAPP(A,fun(filter(A),bool),hAPP(fun(A,A),fun(A,fun(filter(A),bool)),tendsto(A,A),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,A)),combs(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),inverse_divide(A)),hAPP(A,fun(A,A),hAPP(fun(A,fun(A,A)),fun(A,fun(A,A)),combc(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),minus_minus(A)),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,A),fun(fun(A,A),fun(A,A)),combb(A,A,A),B),hAPP(A,fun(A,A),plus_plus(A),C)))),hAPP(A,A,B,C)))),combi(A))),D),hAPP(A,filter(A),at(A),zero_zero(A)))) # label(fact_4644_DERIV__D) # label(axiom). [clausify(857)]. 26.38/26.32 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | hBOOL(hAPP(filter(real),bool,hAPP(real,fun(filter(real),bool),hAPP(fun(real,real),fun(real,fun(filter(real),bool)),tendsto(real,real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real)),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),minus_minus(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),A),hAPP(real,fun(real,real),plus_plus(real),B)))),hAPP(real,real,A,B)))),combi(real))),C),hAPP(real,filter(real),at(real),zero_zero(real)))). [resolve(5198,a,5180,a)]. 26.38/26.32 5199 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),hAPP(A,fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(A,fun(nat,A)),combc(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),inverse_divide(A)),B)),D)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),D))) # label(fact_4113_divide_Osums) # label(axiom). [clausify(981)]. 26.38/26.32 5200 -real_normed_field(A) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(A,real,norm_norm(A),B)),one_one(real))) | hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),hAPP(A,fun(nat,A),power_power(A),B)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),one_one(A)),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),one_one(A)),B)))) # label(fact_4125_geometric__sums) # label(axiom). [clausify(1015)]. 26.38/26.32 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,norm_norm(real),A)),one_one(real))) | hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(real,fun(nat,real),power_power(real),A)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),A)))). [resolve(5200,a,5180,a)]. 26.38/26.32 5201 -real_normed_field(A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(A,real,norm_norm(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),C))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(A,real,norm_norm(A),B)),f194(A,C)))) # label(fact_2109_divide_Obounded) # label(axiom). [clausify(1107)]. 26.38/26.32 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),A)),f194(real,B)))). [resolve(5201,a,5180,a)]. 26.38/26.32 5202 -real_normed_field(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),hAPP(A,fun(A,A),combk(A,A),B)),C),zero_zero(A))) # label(fact_3889_DERIV__const) # label(axiom). [clausify(1221)]. 26.38/26.32 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(real,fun(real,real),combk(real,real),A)),B),zero_zero(real))). [resolve(5202,a,5180,a)]. 26.38/26.32 5203 -banach(A) | -real_normed_field(A) | hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,real),fun(nat,fun(A,A)),hAPP(fun(real,fun(A,A)),fun(fun(nat,real),fun(nat,fun(A,A))),combb(real,fun(A,A),nat),scaleR_scaleR(A)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(A,fun(nat,A),power_power(A),B))),hAPP(A,A,exp(A),B))) # label(fact_4777_exp__converges) # label(axiom). [clausify(1408)]. 26.38/26.35 Derived: -banach(real) | hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),scaleR_scaleR(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),A))),hAPP(real,real,exp(real),A))). [resolve(5203,b,5180,a)]. 26.38/26.35 5204 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),D),E)) | hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),minus_minus(A)),B)),D)),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),C),E))) # label(fact_4114_sums__diff) # label(axiom). [clausify(1425)]. 26.38/26.35 Derived: -hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),C),D)) | hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),minus_minus(real)),A)),C)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),D))). [resolve(5204,a,5180,a)]. 26.38/26.35 5205 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | ti(A,D) = hAPP(A,A,f255(A,B,C,D),C) # label(fact_4903_CARAT__DERIV) # label(axiom). [clausify(1483)]. 26.38/26.35 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | ti(real,C) = hAPP(real,real,f255(real,A,B,C),B). [resolve(5205,a,5180,a)]. 26.38/26.35 5206 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | hBOOL(hAPP(A,bool,hAPP(fun(A,A),fun(A,bool),isCont(A,A),f255(A,B,C,D)),C)) # label(fact_4903_CARAT__DERIV) # label(axiom). [clausify(1483)]. 26.38/26.35 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),f255(real,A,B,C)),B)). [resolve(5206,a,5180,a)]. 26.38/26.35 5207 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,f255(A,B,C,D),E)),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),E),C)) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,B,E)),hAPP(A,A,B,C)) # label(fact_4903_CARAT__DERIV) # label(axiom). [clausify(1483)]. 26.38/26.35 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,f255(real,A,B,C),D)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),D),B)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,A,D)),hAPP(real,real,A,B)). [resolve(5207,a,5180,a)]. 26.38/26.37 5208 -real_normed_field(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | ti(A,D) != hAPP(A,A,E,C) | -hBOOL(hAPP(A,bool,hAPP(fun(A,A),fun(A,bool),isCont(A,A),E),C)) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,E,f256(A,B,C,D,E))),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),f256(A,B,C,D,E)),C)) != hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,B,f256(A,B,C,D,E))),hAPP(A,A,B,C)) # label(fact_4903_CARAT__DERIV) # label(axiom). [clausify(1483)]. 26.38/26.37 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | ti(real,C) != hAPP(real,real,D,B) | -hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),D),B)) | hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,D,f256(real,A,B,C,D))),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),f256(real,A,B,C,D)),B)) != hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,A,f256(real,A,B,C,D))),hAPP(real,real,A,B)). [resolve(5208,a,5180,a)]. 26.38/26.37 5209 -banach(A) | -real_normed_field(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),exp(A)),B),hAPP(A,A,exp(A),B))) # label(fact_2645_DERIV__exp) # label(axiom). [clausify(1522)]. 26.38/26.37 Derived: -banach(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),exp(real)),A),hAPP(real,real,exp(real),A))). [resolve(5209,b,5180,a)]. 26.38/26.37 5210 -real_normed_field(A) | hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),D)) # label(fact_1751_divide_Oadd) # label(axiom). [clausify(1523)]. 26.38/26.37 Derived: hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B)),C) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),C)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),C)). [resolve(5210,a,5180,a)]. 26.38/26.37 5211 -real_normed_field(A) | ti(A,B) = zero_zero(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),inverse_inverse(A)),B),hAPP(A,A,uminus_uminus(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(A,A,inverse_inverse(A),B)),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))))) # label(fact_3917_DERIV__inverse) # label(axiom). [clausify(1540)]. 26.38/26.37 Derived: ti(real,A) = zero_zero(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),inverse_inverse(real)),A),hAPP(real,real,uminus_uminus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,inverse_inverse(real),A)),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))))). [resolve(5211,a,5180,a)]. 26.38/26.37 5212 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | ti(A,E) != ti(A,D) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),E)) # label(fact_2647_DERIV__cong) # label(axiom). [clausify(1553)]. 26.38/26.37 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | ti(real,D) != ti(real,C) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),D)). [resolve(5212,a,5180,a)]. 26.38/26.37 5213 -real_normed_field(A) | hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),C)),D) # label(fact_4704_divide_OscaleR) # label(axiom). [clausify(1558)]. 26.38/26.37 Derived: hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),C)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),B)),C). [resolve(5213,a,5180,a)]. 26.52/26.48 5214 -real_normed_field(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),B)) | hBOOL(hAPP(fun(nat,A),bool,summable(A),hAPP(A,fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(A,fun(nat,A)),combc(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),inverse_divide(A)),B)),C))) # label(fact_4036_divide_Osummable) # label(axiom). [clausify(1815)]. 26.52/26.48 Derived: -hBOOL(hAPP(fun(nat,real),bool,summable(real),A)) | hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),A)),B))). [resolve(5214,a,5180,a)]. 26.52/26.48 5215 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),E),C),F)) | zero_zero(A) = hAPP(A,A,E,C) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,A)),combs(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),inverse_divide(A)),B)),E)),C),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),hAPP(A,A,E,C))),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),F),hAPP(A,A,B,C)))),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(A,A,E,C)),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))))) # label(fact_3928_DERIV__quotient) # label(axiom). [clausify(1821)]. 26.52/26.48 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),D),B),E)) | zero_zero(real) = hAPP(real,real,D,B) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real)),A)),D)),B),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),hAPP(real,real,D,B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),E),hAPP(real,real,A,B)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,D,B)),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))))). [resolve(5215,a,5180,a)]. 26.52/26.48 5216 -real_normed_field(A) | hBOOL(hAPP(A,bool,hAPP(fun(A,A),fun(A,bool),isCont(A,A),hAPP(A,fun(A,A),hAPP(fun(A,fun(A,A)),fun(A,fun(A,A)),combc(A,A,A),inverse_divide(A)),B)),C)) # label(fact_4922_divide_OisCont) # label(axiom). [clausify(1865)]. 26.52/26.48 Derived: hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),inverse_divide(real)),A)),B)). [resolve(5216,a,5180,a)]. 26.52/26.48 5217 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,A),fun(fun(A,A),fun(A,A)),combb(A,A,A),hAPP(A,fun(A,A),times_times(A),E)),B)),C),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),E),D))) # label(fact_3887_DERIV__cmult) # label(axiom). [clausify(1888)]. 26.52/26.48 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),hAPP(real,fun(real,real),times_times(real),D)),A)),B),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),D),C))). [resolve(5217,a,5180,a)]. 26.52/26.50 5218 -real_normed_field(A) | -hBOOL(hAPP(filter(nat),bool,hAPP(A,fun(filter(nat),bool),hAPP(fun(nat,A),fun(A,fun(filter(nat),bool)),tendsto(nat,A),B),C),sequentially)) | -hBOOL(hAPP(filter(nat),bool,hAPP(A,fun(filter(nat),bool),hAPP(fun(nat,A),fun(A,fun(filter(nat),bool)),tendsto(nat,A),D),E),sequentially)) | ti(A,E) = zero_zero(A) | hBOOL(hAPP(filter(nat),bool,hAPP(A,fun(filter(nat),bool),hAPP(fun(nat,A),fun(A,fun(filter(nat),bool)),tendsto(nat,A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),inverse_divide(A)),B)),D)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),E)),sequentially)) # label(fact_4466_LIMSEQ__divide) # label(axiom). [clausify(1897)]. 26.52/26.50 Derived: -hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A),B),sequentially)) | -hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),C),D),sequentially)) | ti(real,D) = zero_zero(real) | hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),A)),C)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),D)),sequentially)). [resolve(5218,a,5180,a)]. 26.52/26.50 5219 -real_normed_field(A) | -hBOOL(hAPP(filter(B),bool,hAPP(A,fun(filter(B),bool),hAPP(fun(B,A),fun(A,fun(filter(B),bool)),tendsto(B,A),C),D),E)) | hBOOL(hAPP(filter(B),bool,hAPP(A,fun(filter(B),bool),hAPP(fun(B,A),fun(A,fun(filter(B),bool)),tendsto(B,A),hAPP(A,fun(B,A),hAPP(fun(B,fun(A,A)),fun(A,fun(B,A)),combc(B,A,A),hAPP(fun(B,A),fun(B,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),combb(A,fun(A,A),B),inverse_divide(A)),C)),F)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),D),F)),E)) # label(fact_4498_divide_Otendsto) # label(axiom). [clausify(1918)]. 26.52/26.50 Derived: -hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),B),C),D)) | hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),hAPP(real,fun(A,real),hAPP(fun(A,fun(real,real)),fun(real,fun(A,real)),combc(A,real,real),hAPP(fun(A,real),fun(A,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(A,real),fun(A,fun(real,real))),combb(real,fun(real,real),A),inverse_divide(real)),B)),E)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),C),E)),D)). [resolve(5219,a,5180,a)]. 26.52/26.50 5220 -real_normed_field(A) | -banach(A) | hAPP(A,A,exp(A),hAPP(A,A,uminus_uminus(A),B)) = hAPP(A,A,inverse_inverse(A),hAPP(A,A,exp(A),B)) # label(fact_2606_exp__minus) # label(axiom). [clausify(1929)]. 26.52/26.50 Derived: -banach(real) | hAPP(real,real,exp(real),hAPP(real,real,uminus_uminus(real),A)) = hAPP(real,real,inverse_inverse(real),hAPP(real,real,exp(real),A)). [resolve(5220,a,5180,a)]. 26.52/26.50 5221 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),E),C),F)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,A)),combs(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),plus_plus(A)),B)),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,A),fun(fun(A,A),fun(A,A)),combb(A,A,A),uminus_uminus(A)),E))),C),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),D),hAPP(A,A,uminus_uminus(A),F)))) # label(fact_3859_DERIV__add__minus) # label(axiom). [clausify(1958)]. 26.61/26.55 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),D),B),E)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),plus_plus(real)),A)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),uminus_uminus(real)),D))),B),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),C),hAPP(real,real,uminus_uminus(real),E)))). [resolve(5221,a,5180,a)]. 26.61/26.55 5222 -real_normed_field(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),hAPP(A,fun(A,A),times_times(A),B)),C),B)) # label(fact_2649_DERIV__cmult__Id) # label(axiom). [clausify(1967)]. 26.61/26.55 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(real,fun(real,real),times_times(real),A)),B),A)). [resolve(5222,a,5180,a)]. 26.61/26.55 5223 -real_normed_field(A) | -hBOOL(hAPP(filter(nat),bool,hAPP(A,fun(filter(nat),bool),hAPP(fun(nat,A),fun(A,fun(filter(nat),bool)),tendsto(nat,A),B),C),sequentially)) | hBOOL(hAPP(filter(nat),bool,hAPP(A,fun(filter(nat),bool),hAPP(fun(nat,A),fun(A,fun(filter(nat),bool)),tendsto(nat,A),hAPP(A,fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(A,fun(nat,A)),combc(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),inverse_divide(A)),B)),D)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),D)),sequentially)) # label(fact_4497_divide_OLIMSEQ) # label(axiom). [clausify(2013)]. 26.61/26.55 5224 real_normed_field(complex) # label(arity_Complex_Ocomplex___RealVector_Oreal__normed__field) # label(axiom). [assumption]. 26.61/26.55 Derived: -hBOOL(hAPP(fun(nat,complex),bool,cauchy(complex),A)) | hBOOL(hAPP(fun(nat,complex),bool,cauchy(complex),hAPP(complex,fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(complex,fun(nat,complex)),combc(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),inverse_divide(complex)),A)),B))). [resolve(5224,a,5181,a)]. 26.61/26.55 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | zero_zero(complex) = hAPP(complex,complex,A,B) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,complex),fun(fun(complex,complex),fun(complex,complex)),combb(complex,complex,complex),inverse_inverse(complex)),A)),B),hAPP(complex,complex,uminus_uminus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),hAPP(complex,complex,inverse_inverse(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(complex,complex,A,B)),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))))))). [resolve(5224,a,5182,a)]. 26.61/26.55 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),hAPP(nat,fun(complex,complex),hAPP(fun(complex,fun(nat,complex)),fun(nat,fun(complex,complex)),combc(complex,nat,complex),hAPP(fun(complex,complex),fun(complex,fun(nat,complex)),hAPP(fun(complex,fun(nat,complex)),fun(fun(complex,complex),fun(complex,fun(nat,complex))),combb(complex,fun(nat,complex),complex),power_power(complex)),A)),D)),B),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(nat,complex,semiring_1_of_nat(complex),D)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(complex,complex,A,B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),D),hAPP(nat,nat,suc,zero_zero(nat)))))))). [resolve(5224,a,5183,a)]. 26.61/26.56 Derived: -banach(complex) | hAPP(complex,complex,exp(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),A),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),hAPP(complex,complex,exp(complex),A)),hAPP(complex,complex,exp(complex),B)). [resolve(5224,a,5184,b)]. 26.61/26.56 Derived: -banach(complex) | hAPP(complex,complex,exp(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,exp(complex),A)),hAPP(complex,complex,exp(complex),B)). [resolve(5224,a,5185,a)]. 26.61/26.56 Derived: -hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),A),B)) | hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),hAPP(complex,fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(complex,fun(nat,complex)),combc(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),inverse_divide(complex)),A)),C)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),B),C))). [resolve(5224,a,5186,a)]. 26.61/26.56 Derived: ti(complex,A) = zero_zero(complex) | hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(complex,real,norm_norm(complex),B)),hAPP(complex,real,norm_norm(complex),A)) = hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),B),A)). [resolve(5224,a,5187,a)]. 26.61/26.56 Derived: hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co1399186613setsum(A,complex),hAPP(complex,fun(A,complex),hAPP(fun(A,fun(complex,complex)),fun(complex,fun(A,complex)),combc(A,complex,complex),hAPP(fun(A,complex),fun(A,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(A,complex),fun(A,fun(complex,complex))),combb(complex,fun(complex,complex),A),inverse_divide(complex)),B)),C)),D) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co1399186613setsum(A,complex),B),D)),C). [resolve(5224,a,5188,a)]. 26.61/26.56 Derived: zero_zero(complex) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),zero_zero(complex)),A). [resolve(5224,a,5189,a)]. 26.61/26.56 Derived: -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),A)) | hAPP(fun(nat,complex),complex,suminf(complex),hAPP(complex,fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(complex,fun(nat,complex)),combc(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),inverse_divide(complex)),A)),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),hAPP(fun(nat,complex),complex,suminf(complex),A)),B). [resolve(5224,a,5190,a)]. 26.61/26.56 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f122(A,complex,C,D,E,B)),B)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),hAPP(fun(A,bool),fun(complex,complex),hAPP(fun(complex,fun(fun(A,bool),complex)),fun(fun(A,bool),fun(complex,complex)),combc(complex,fun(A,bool),complex),hAPP(fun(complex,fun(A,complex)),fun(complex,fun(fun(A,bool),complex)),hAPP(fun(fun(A,complex),fun(fun(A,bool),complex)),fun(fun(complex,fun(A,complex)),fun(complex,fun(fun(A,bool),complex))),combb(fun(A,complex),fun(fun(A,bool),complex),complex),big_co1399186613setsum(A,complex)),C)),B)),D),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co1399186613setsum(A,complex),hAPP(complex,fun(A,complex),E,D)),B))). [resolve(5224,a,5191,a)]. 26.61/26.56 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),hAPP(A,fun(complex,complex),hAPP(fun(complex,fun(A,complex)),fun(A,fun(complex,complex)),combc(complex,A,complex),C),f122(A,complex,C,D,E,B))),D),hAPP(A,complex,hAPP(complex,fun(A,complex),E,D),f122(A,complex,C,D,E,B)))) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),hAPP(fun(A,bool),fun(complex,complex),hAPP(fun(complex,fun(fun(A,bool),complex)),fun(fun(A,bool),fun(complex,complex)),combc(complex,fun(A,bool),complex),hAPP(fun(complex,fun(A,complex)),fun(complex,fun(fun(A,bool),complex)),hAPP(fun(fun(A,complex),fun(fun(A,bool),complex)),fun(fun(complex,fun(A,complex)),fun(complex,fun(fun(A,bool),complex))),combb(fun(A,complex),fun(fun(A,bool),complex),complex),big_co1399186613setsum(A,complex)),C)),B)),D),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co1399186613setsum(A,complex),hAPP(complex,fun(A,complex),E,D)),B))). [resolve(5224,a,5192,a)]. 26.61/26.56 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),A)),f123(complex,B)))). [resolve(5224,a,5193,a)]. 26.61/26.56 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),f123(complex,A))). [resolve(5224,a,5194,a)]. 26.61/26.56 Derived: -topological_space(A) | -hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),B),C),hAPP(A,filter(A),at(A),D))) | hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),hAPP(complex,fun(A,complex),hAPP(fun(A,fun(complex,complex)),fun(complex,fun(A,complex)),combc(A,complex,complex),hAPP(fun(A,complex),fun(A,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(A,complex),fun(A,fun(complex,complex))),combb(complex,fun(complex,complex),A),inverse_divide(complex)),B)),E)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),C),E)),hAPP(A,filter(A),at(A),D))). [resolve(5224,a,5195,b)]. 26.61/26.56 Derived: ti(complex,A) = zero_zero(complex) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),B)),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),B),A))),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),B),A)),D)),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),B),D))),A)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(nat,complex,semiring_1_of_nat(complex),D)),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),D),hAPP(nat,nat,suc,zero_zero(nat)))))))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,semiring_1_of_nat(real),D)),hAPP(nat,real,semiring_1_of_nat(real),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),D),hAPP(nat,nat,suc,zero_zero(nat)))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),D),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(complex,real,norm_norm(complex),A)))). [resolve(5224,a,5196,a)]. 26.61/26.57 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),hAPP(complex,fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(complex,fun(complex,complex)),combc(complex,complex,complex),hAPP(fun(complex,complex),fun(complex,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,fun(complex,complex))),combb(complex,fun(complex,complex),complex),inverse_divide(complex)),A)),D)),B),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),C),D))). [resolve(5224,a,5197,a)]. 26.61/26.57 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | hBOOL(hAPP(filter(complex),bool,hAPP(complex,fun(filter(complex),bool),hAPP(fun(complex,complex),fun(complex,fun(filter(complex),bool)),tendsto(complex,complex),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,complex)),combs(complex,complex,complex),hAPP(fun(complex,complex),fun(complex,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,fun(complex,complex))),combb(complex,fun(complex,complex),complex),inverse_divide(complex)),hAPP(complex,fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(complex,fun(complex,complex)),combc(complex,complex,complex),hAPP(fun(complex,complex),fun(complex,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,fun(complex,complex))),combb(complex,fun(complex,complex),complex),minus_minus(complex)),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,complex),fun(fun(complex,complex),fun(complex,complex)),combb(complex,complex,complex),A),hAPP(complex,fun(complex,complex),plus_plus(complex),B)))),hAPP(complex,complex,A,B)))),combi(complex))),C),hAPP(complex,filter(complex),at(complex),zero_zero(complex)))). [resolve(5224,a,5198,a)]. 26.61/26.57 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(complex,real,norm_norm(complex),A)),one_one(real))) | hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),hAPP(complex,fun(nat,complex),power_power(complex),A)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),one_one(complex)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),one_one(complex)),A)))). [resolve(5224,a,5200,a)]. 26.61/26.57 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),A)),f194(complex,B)))). [resolve(5224,a,5201,a)]. 26.61/26.57 Derived: hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),hAPP(complex,fun(complex,complex),combk(complex,complex),A)),B),zero_zero(complex))). [resolve(5224,a,5202,a)]. 26.61/26.57 Derived: -banach(complex) | hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,real),fun(nat,fun(complex,complex)),hAPP(fun(real,fun(complex,complex)),fun(fun(nat,real),fun(nat,fun(complex,complex))),combb(real,fun(complex,complex),nat),scaleR_scaleR(complex)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(complex,fun(nat,complex),power_power(complex),A))),hAPP(complex,complex,exp(complex),A))). [resolve(5224,a,5203,b)]. 26.61/26.58 Derived: -hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),A),B)) | -hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),C),D)) | hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),minus_minus(complex)),A)),C)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),B),D))). [resolve(5224,a,5204,a)]. 26.61/26.58 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | ti(complex,C) = hAPP(complex,complex,f255(complex,A,B,C),B). [resolve(5224,a,5205,a)]. 26.61/26.58 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | hBOOL(hAPP(complex,bool,hAPP(fun(complex,complex),fun(complex,bool),isCont(complex,complex),f255(complex,A,B,C)),B)). [resolve(5224,a,5206,a)]. 26.61/26.58 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,f255(complex,A,B,C),D)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),D),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(complex,complex,A,D)),hAPP(complex,complex,A,B)). [resolve(5224,a,5207,a)]. 26.61/26.58 Derived: hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | ti(complex,C) != hAPP(complex,complex,D,B) | -hBOOL(hAPP(complex,bool,hAPP(fun(complex,complex),fun(complex,bool),isCont(complex,complex),D),B)) | hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,D,f256(complex,A,B,C,D))),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),f256(complex,A,B,C,D)),B)) != hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(complex,complex,A,f256(complex,A,B,C,D))),hAPP(complex,complex,A,B)). [resolve(5224,a,5208,a)]. 26.61/26.58 Derived: -banach(complex) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),exp(complex)),A),hAPP(complex,complex,exp(complex),A))). [resolve(5224,a,5209,b)]. 26.61/26.58 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),B)),C) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),A),C)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),B),C)). [resolve(5224,a,5210,a)]. 26.61/26.58 Derived: ti(complex,A) = zero_zero(complex) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),inverse_inverse(complex)),A),hAPP(complex,complex,uminus_uminus(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(complex,complex,inverse_inverse(complex),A)),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))))). [resolve(5224,a,5211,a)]. 26.68/26.58 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | ti(complex,D) != ti(complex,C) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),D)). [resolve(5224,a,5212,a)]. 26.68/26.58 Derived: hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),B),C)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),B)),C). [resolve(5224,a,5213,a)]. 26.68/26.58 Derived: -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),A)) | hBOOL(hAPP(fun(nat,complex),bool,summable(complex),hAPP(complex,fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(complex,fun(nat,complex)),combc(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),inverse_divide(complex)),A)),B))). [resolve(5224,a,5214,a)]. 26.68/26.58 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),D),B),E)) | zero_zero(complex) = hAPP(complex,complex,D,B) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,complex)),combs(complex,complex,complex),hAPP(fun(complex,complex),fun(complex,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,fun(complex,complex))),combb(complex,fun(complex,complex),complex),inverse_divide(complex)),A)),D)),B),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),hAPP(complex,complex,D,B))),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),E),hAPP(complex,complex,A,B)))),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(complex,complex,D,B)),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))))). [resolve(5224,a,5215,a)]. 26.68/26.58 Derived: hBOOL(hAPP(complex,bool,hAPP(fun(complex,complex),fun(complex,bool),isCont(complex,complex),hAPP(complex,fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(complex,fun(complex,complex)),combc(complex,complex,complex),inverse_divide(complex)),A)),B)). [resolve(5224,a,5216,a)]. 26.68/26.58 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,complex),fun(fun(complex,complex),fun(complex,complex)),combb(complex,complex,complex),hAPP(complex,fun(complex,complex),times_times(complex),D)),A)),B),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),D),C))). [resolve(5224,a,5217,a)]. 26.68/26.58 Derived: -hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),A),B),sequentially)) | -hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),C),D),sequentially)) | ti(complex,D) = zero_zero(complex) | hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),inverse_divide(complex)),A)),C)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),B),D)),sequentially)). [resolve(5224,a,5218,a)]. 26.68/26.60 Derived: -hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),B),C),D)) | hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),hAPP(complex,fun(A,complex),hAPP(fun(A,fun(complex,complex)),fun(complex,fun(A,complex)),combc(A,complex,complex),hAPP(fun(A,complex),fun(A,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(A,complex),fun(A,fun(complex,complex))),combb(complex,fun(complex,complex),A),inverse_divide(complex)),B)),E)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),C),E)),D)). [resolve(5224,a,5219,a)]. 26.68/26.60 Derived: -banach(complex) | hAPP(complex,complex,exp(complex),hAPP(complex,complex,uminus_uminus(complex),A)) = hAPP(complex,complex,inverse_inverse(complex),hAPP(complex,complex,exp(complex),A)). [resolve(5224,a,5220,a)]. 26.68/26.60 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),D),B),E)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,complex)),combs(complex,complex,complex),hAPP(fun(complex,complex),fun(complex,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,fun(complex,complex))),combb(complex,fun(complex,complex),complex),plus_plus(complex)),A)),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,complex),fun(fun(complex,complex),fun(complex,complex)),combb(complex,complex,complex),uminus_uminus(complex)),D))),B),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),C),hAPP(complex,complex,uminus_uminus(complex),E)))). [resolve(5224,a,5221,a)]. 26.68/26.60 Derived: hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),hAPP(complex,fun(complex,complex),times_times(complex),A)),B),A)). [resolve(5224,a,5222,a)]. 26.68/26.60 5225 -banach(A) | -real_normed_field(A) | hAPP(fun(nat,A),A,suminf(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,real),fun(nat,fun(A,A)),hAPP(fun(real,fun(A,A)),fun(fun(nat,real),fun(nat,fun(A,A))),combb(real,fun(A,A),nat),scaleR_scaleR(A)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(A,fun(nat,A),power_power(A),B))) = hAPP(A,A,exp(A),B) # label(fact_4775_lemma__exp__ext) # label(axiom). [clausify(2076)]. 26.68/26.60 Derived: -banach(real) | hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),scaleR_scaleR(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),A))) = hAPP(real,real,exp(real),A). [resolve(5225,b,5180,a)]. 26.77/26.70 Derived: -banach(complex) | hAPP(fun(nat,complex),complex,suminf(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,real),fun(nat,fun(complex,complex)),hAPP(fun(real,fun(complex,complex)),fun(fun(nat,real),fun(nat,fun(complex,complex))),combb(real,fun(complex,complex),nat),scaleR_scaleR(complex)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(complex,fun(nat,complex),power_power(complex),A))) = hAPP(complex,complex,exp(complex),A). [resolve(5225,b,5224,a)]. 26.77/26.70 5226 -real_normed_field(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),B)) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),C)) | hAPP(fun(nat,A),A,suminf(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),minus_minus(A)),B)),C)) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(fun(nat,A),A,suminf(A),B)),hAPP(fun(nat,A),A,suminf(A),C)) # label(fact_4010_suminf__diff) # label(axiom). [clausify(2165)]. 26.77/26.70 Derived: -hBOOL(hAPP(fun(nat,real),bool,summable(real),A)) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),B)) | hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),minus_minus(real)),A)),B)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(fun(nat,real),real,suminf(real),A)),hAPP(fun(nat,real),real,suminf(real),B)). [resolve(5226,a,5180,a)]. 26.77/26.70 Derived: -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),A)) | -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),B)) | hAPP(fun(nat,complex),complex,suminf(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),minus_minus(complex)),A)),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(fun(nat,complex),complex,suminf(complex),A)),hAPP(fun(nat,complex),complex,suminf(complex),B)). [resolve(5226,a,5224,a)]. 26.77/26.70 5227 -real_normed_field(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),combi(A)),B),one_one(A))) # label(fact_3871_DERIV__ident) # label(axiom). [clausify(2230)]. 26.77/26.70 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),combi(real)),A),one_one(real))). [resolve(5227,a,5180,a)]. 26.77/26.70 Derived: hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),combi(complex)),A),one_one(complex))). [resolve(5227,a,5224,a)]. 26.77/26.70 5228 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(A,A),fun(fun(nat,A),fun(nat,A)),combb(A,A,nat),uminus_uminus(A)),B)),hAPP(A,A,uminus_uminus(A),C))) # label(fact_4106_sums__minus) # label(axiom). [clausify(2291)]. 26.77/26.70 Derived: -hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),uminus_uminus(real)),A)),hAPP(real,real,uminus_uminus(real),B))). [resolve(5228,a,5180,a)]. 26.77/26.71 Derived: -hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),A),B)) | hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(complex,complex),fun(fun(nat,complex),fun(nat,complex)),combb(complex,complex,nat),uminus_uminus(complex)),A)),hAPP(complex,complex,uminus_uminus(complex),B))). [resolve(5228,a,5224,a)]. 26.77/26.71 5229 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),E),C),F)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,A)),combs(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),times_times(A)),B)),E)),C),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,B,C)),F)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),hAPP(A,A,E,C))))) # label(fact_3857_DERIV__mult_H) # label(axiom). [clausify(2302)]. 26.77/26.71 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),D),B),E)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),A)),D)),B),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,A,B)),E)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),hAPP(real,real,D,B))))). [resolve(5229,a,5180,a)]. 26.77/26.71 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),D),B),E)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,complex)),combs(complex,complex,complex),hAPP(fun(complex,complex),fun(complex,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,fun(complex,complex))),combb(complex,fun(complex,complex),complex),times_times(complex)),A)),D)),B),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,A,B)),E)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),hAPP(complex,complex,D,B))))). [resolve(5229,a,5224,a)]. 26.77/26.71 5230 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),E),C),F)) | zero_zero(A) = hAPP(A,A,E,C) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,A)),combs(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),inverse_divide(A)),B)),E)),C),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),hAPP(A,A,E,C))),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,B,C)),F))),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,E,C)),hAPP(A,A,E,C))))) # label(fact_3846_DERIV__divide) # label(axiom). [clausify(2308)]. 26.82/26.74 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),D),B),E)) | zero_zero(real) = hAPP(real,real,D,B) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real)),A)),D)),B),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),hAPP(real,real,D,B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,A,B)),E))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,D,B)),hAPP(real,real,D,B))))). [resolve(5230,a,5180,a)]. 26.82/26.74 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),D),B),E)) | zero_zero(complex) = hAPP(complex,complex,D,B) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,complex)),combs(complex,complex,complex),hAPP(fun(complex,complex),fun(complex,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,fun(complex,complex))),combb(complex,fun(complex,complex),complex),inverse_divide(complex)),A)),D)),B),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),hAPP(complex,complex,D,B))),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,A,B)),E))),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,D,B)),hAPP(complex,complex,D,B))))). [resolve(5230,a,5224,a)]. 26.82/26.74 5231 -real_normed_field(A) | -banach(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),B)),hAPP(A,fun(nat,A),power_power(A),C)))) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),hAPP(fun(nat,A),fun(nat,A),diffs(A),B))),hAPP(A,fun(nat,A),power_power(A),C)))) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),hAPP(fun(nat,A),fun(nat,A),diffs(A),hAPP(fun(nat,A),fun(nat,A),diffs(A),B)))),hAPP(A,fun(nat,A),power_power(A),C)))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(A,real,norm_norm(A),D)),hAPP(A,real,norm_norm(A),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),hAPP(fun(A,fun(nat,A)),fun(A,A),hAPP(fun(fun(nat,A),A),fun(fun(A,fun(nat,A)),fun(A,A)),combb(fun(nat,A),A,A),suminf(A)),hAPP(fun(A,fun(nat,A)),fun(A,fun(nat,A)),hAPP(fun(fun(nat,A),fun(nat,A)),fun(fun(A,fun(nat,A)),fun(A,fun(nat,A))),combb(fun(nat,A),fun(nat,A),A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),B))),power_power(A)))),D),hAPP(fun(nat,A),A,suminf(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),hAPP(fun(nat,A),fun(nat,A),diffs(A),B))),hAPP(A,fun(nat,A),power_power(A),D))))) # label(fact_4151_termdiffs) # label(axiom). [clausify(2363)]. 26.82/26.75 Derived: -banach(real) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),A)),hAPP(real,fun(nat,real),power_power(real),B)))) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),diffs(real),A))),hAPP(real,fun(nat,real),power_power(real),B)))) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),diffs(real),hAPP(fun(nat,real),fun(nat,real),diffs(real),A)))),hAPP(real,fun(nat,real),power_power(real),B)))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,norm_norm(real),C)),hAPP(real,real,norm_norm(real),B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,fun(nat,real)),fun(real,real),hAPP(fun(fun(nat,real),real),fun(fun(real,fun(nat,real)),fun(real,real)),combb(fun(nat,real),real,real),suminf(real)),hAPP(fun(real,fun(nat,real)),fun(real,fun(nat,real)),hAPP(fun(fun(nat,real),fun(nat,real)),fun(fun(real,fun(nat,real)),fun(real,fun(nat,real))),combb(fun(nat,real),fun(nat,real),real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),A))),power_power(real)))),C),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),diffs(real),A))),hAPP(real,fun(nat,real),power_power(real),C))))). [resolve(5231,a,5180,a)]. 26.82/26.75 Derived: -banach(complex) | -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),A)),hAPP(complex,fun(nat,complex),power_power(complex),B)))) | -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),hAPP(fun(nat,complex),fun(nat,complex),diffs(complex),A))),hAPP(complex,fun(nat,complex),power_power(complex),B)))) | -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),hAPP(fun(nat,complex),fun(nat,complex),diffs(complex),hAPP(fun(nat,complex),fun(nat,complex),diffs(complex),A)))),hAPP(complex,fun(nat,complex),power_power(complex),B)))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(complex,real,norm_norm(complex),C)),hAPP(complex,real,norm_norm(complex),B))) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),hAPP(fun(complex,fun(nat,complex)),fun(complex,complex),hAPP(fun(fun(nat,complex),complex),fun(fun(complex,fun(nat,complex)),fun(complex,complex)),combb(fun(nat,complex),complex,complex),suminf(complex)),hAPP(fun(complex,fun(nat,complex)),fun(complex,fun(nat,complex)),hAPP(fun(fun(nat,complex),fun(nat,complex)),fun(fun(complex,fun(nat,complex)),fun(complex,fun(nat,complex))),combb(fun(nat,complex),fun(nat,complex),complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),A))),power_power(complex)))),C),hAPP(fun(nat,complex),complex,suminf(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),hAPP(fun(nat,complex),fun(nat,complex),diffs(complex),A))),hAPP(complex,fun(nat,complex),power_power(complex),C))))). [resolve(5231,a,5224,a)]. 26.82/26.78 5232 -real_normed_field(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),B)) | hAPP(fun(nat,A),A,suminf(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(A,A),fun(fun(nat,A),fun(nat,A)),combb(A,A,nat),uminus_uminus(A)),B)) = hAPP(A,A,uminus_uminus(A),hAPP(fun(nat,A),A,suminf(A),B)) # label(fact_4017_suminf__minus) # label(axiom). [clausify(2414)]. 26.82/26.78 Derived: -hBOOL(hAPP(fun(nat,real),bool,summable(real),A)) | hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),uminus_uminus(real)),A)) = hAPP(real,real,uminus_uminus(real),hAPP(fun(nat,real),real,suminf(real),A)). [resolve(5232,a,5180,a)]. 26.82/26.78 Derived: -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),A)) | hAPP(fun(nat,complex),complex,suminf(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(complex,complex),fun(fun(nat,complex),fun(nat,complex)),combb(complex,complex,nat),uminus_uminus(complex)),A)) = hAPP(complex,complex,uminus_uminus(complex),hAPP(fun(nat,complex),complex,suminf(complex),A)). [resolve(5232,a,5224,a)]. 26.82/26.78 5233 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),E),C),F)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,A)),combs(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),times_times(A)),B)),E)),C),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),hAPP(A,A,E,C))),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),F),hAPP(A,A,B,C))))) # label(fact_3858_DERIV__mult) # label(axiom). [clausify(2444)]. 26.82/26.80 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),D),B),E)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),A)),D)),B),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),hAPP(real,real,D,B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),E),hAPP(real,real,A,B))))). [resolve(5233,a,5180,a)]. 26.82/26.80 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),D),B),E)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,complex)),combs(complex,complex,complex),hAPP(fun(complex,complex),fun(complex,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,fun(complex,complex))),combb(complex,fun(complex,complex),complex),times_times(complex)),A)),D)),B),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),hAPP(complex,complex,D,B))),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),E),hAPP(complex,complex,A,B))))). [resolve(5233,a,5224,a)]. 26.82/26.80 5234 -banach(A) | -real_normed_field(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),hAPP(fun(nat,A),fun(nat,A),diffs(A),hAPP(fun(nat,A),fun(nat,A),diffs(A),B)))),hAPP(A,fun(nat,A),power_power(A),C)))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(A,real,norm_norm(A),D)),hAPP(A,real,norm_norm(A),C))) | hBOOL(hAPP(filter(A),bool,hAPP(A,fun(filter(A),bool),hAPP(fun(A,A),fun(A,fun(filter(A),bool)),tendsto(A,A),hAPP(fun(A,fun(nat,A)),fun(A,A),hAPP(fun(fun(nat,A),A),fun(fun(A,fun(nat,A)),fun(A,A)),combb(fun(nat,A),A,A),suminf(A)),hAPP(fun(A,fun(nat,A)),fun(A,fun(nat,A)),hAPP(fun(fun(nat,A),fun(nat,A)),fun(fun(A,fun(nat,A)),fun(A,fun(nat,A))),combb(fun(nat,A),fun(nat,A),A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),B))),hAPP(fun(nat,A),fun(A,fun(nat,A)),hAPP(fun(A,fun(fun(nat,A),fun(nat,A))),fun(fun(nat,A),fun(A,fun(nat,A))),combc(A,fun(nat,A),fun(nat,A)),hAPP(fun(A,fun(nat,fun(A,A))),fun(A,fun(fun(nat,A),fun(nat,A))),hAPP(fun(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A))),fun(fun(A,fun(nat,fun(A,A))),fun(A,fun(fun(nat,A),fun(nat,A)))),combb(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),A),combs(nat,A,A)),hAPP(fun(A,fun(nat,A)),fun(A,fun(nat,fun(A,A))),hAPP(fun(fun(nat,A),fun(nat,fun(A,A))),fun(fun(A,fun(nat,A)),fun(A,fun(nat,fun(A,A)))),combb(fun(nat,A),fun(nat,fun(A,A)),A),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),minus_minus(A))),hAPP(fun(A,A),fun(A,fun(nat,A)),hAPP(fun(A,fun(A,fun(nat,A))),fun(fun(A,A),fun(A,fun(nat,A))),combs(A,A,fun(nat,A)),hAPP(fun(A,fun(nat,fun(A,A))),fun(A,fun(A,fun(nat,A))),hAPP(fun(fun(nat,fun(A,A)),fun(A,fun(nat,A))),fun(fun(A,fun(nat,fun(A,A))),fun(A,fun(A,fun(nat,A)))),combb(fun(nat,fun(A,A)),fun(A,fun(nat,A)),A),combc(nat,A,A)),hAPP(fun(A,fun(nat,A)),fun(A,fun(nat,fun(A,A))),hAPP(fun(fun(nat,A),fun(nat,fun(A,A))),fun(fun(A,fun(nat,A)),fun(A,fun(nat,fun(A,A)))),combb(fun(nat,A),fun(nat,fun(A,A)),A),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),inverse_divide(A))),hAPP(fun(nat,A),fun(A,fun(nat,A)),hAPP(fun(A,fun(fun(nat,A),fun(nat,A))),fun(fun(nat,A),fun(A,fun(nat,A))),combc(A,fun(nat,A),fun(nat,A)),hAPP(fun(A,fun(nat,fun(A,A))),fun(A,fun(fun(nat,A),fun(nat,A))),hAPP(fun(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A))),fun(fun(A,fun(nat,fun(A,A))),fun(A,fun(fun(nat,A),fun(nat,A)))),combb(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),A),combs(nat,A,A)),hAPP(fun(A,fun(nat,A)),fun(A,fun(nat,fun(A,A))),hAPP(fun(fun(nat,A),fun(nat,fun(A,A))),fun(fun(A,fun(nat,A)),fun(A,fun(nat,fun(A,A)))),combb(fun(nat,A),fun(nat,fun(A,A)),A),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),minus_minus(A))),hAPP(fun(A,A),fun(A,fun(nat,A)),hAPP(fun(A,fun(nat,A)),fun(fun(A,A),fun(A,fun(nat,A))),combb(A,fun(nat,A),A),power_power(A)),hAPP(A,fun(A,A),plus_plus(A),D))))),hAPP(A,fun(nat,A),power_power(A),D))))),combi(A))))),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),semiring_1_of_nat(A))),hAPP(fun(nat,nat),fun(nat,A),hAPP(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),combb(nat,A,nat),hAPP(A,fun(nat,A),power_power(A),D)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),minus_minus(nat)),hAPP(nat,nat,suc,zero_zero(nat))))))))),zero_zero(A)),hAPP(A,filter(A),at(A),zero_zero(A)))) # label(fact_4557_termdiffs__aux) # label(axiom). [clausify(2466)]. 26.82/26.80 Derived: -banach(real) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),diffs(real),hAPP(fun(nat,real),fun(nat,real),diffs(real),A)))),hAPP(real,fun(nat,real),power_power(real),B)))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,norm_norm(real),C)),hAPP(real,real,norm_norm(real),B))) | hBOOL(hAPP(filter(real),bool,hAPP(real,fun(filter(real),bool),hAPP(fun(real,real),fun(real,fun(filter(real),bool)),tendsto(real,real),hAPP(fun(real,fun(nat,real)),fun(real,real),hAPP(fun(fun(nat,real),real),fun(fun(real,fun(nat,real)),fun(real,real)),combb(fun(nat,real),real,real),suminf(real)),hAPP(fun(real,fun(nat,real)),fun(real,fun(nat,real)),hAPP(fun(fun(nat,real),fun(nat,real)),fun(fun(real,fun(nat,real)),fun(real,fun(nat,real))),combb(fun(nat,real),fun(nat,real),real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),A))),hAPP(fun(nat,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(fun(nat,real),fun(nat,real))),fun(fun(nat,real),fun(real,fun(nat,real))),combc(real,fun(nat,real),fun(nat,real)),hAPP(fun(real,fun(nat,fun(real,real))),fun(real,fun(fun(nat,real),fun(nat,real))),hAPP(fun(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real))),fun(fun(real,fun(nat,fun(real,real))),fun(real,fun(fun(nat,real),fun(nat,real)))),combb(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),real),combs(nat,real,real)),hAPP(fun(real,fun(nat,real)),fun(real,fun(nat,fun(real,real))),hAPP(fun(fun(nat,real),fun(nat,fun(real,real))),fun(fun(real,fun(nat,real)),fun(real,fun(nat,fun(real,real)))),combb(fun(nat,real),fun(nat,fun(real,real)),real),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),minus_minus(real))),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(real,fun(nat,real))),fun(fun(real,real),fun(real,fun(nat,real))),combs(real,real,fun(nat,real)),hAPP(fun(real,fun(nat,fun(real,real))),fun(real,fun(real,fun(nat,real))),hAPP(fun(fun(nat,fun(real,real)),fun(real,fun(nat,real))),fun(fun(real,fun(nat,fun(real,real))),fun(real,fun(real,fun(nat,real)))),combb(fun(nat,fun(real,real)),fun(real,fun(nat,real)),real),combc(nat,real,real)),hAPP(fun(real,fun(nat,real)),fun(real,fun(nat,fun(real,real))),hAPP(fun(fun(nat,real),fun(nat,fun(real,real))),fun(fun(real,fun(nat,real)),fun(real,fun(nat,fun(real,real)))),combb(fun(nat,real),fun(nat,fun(real,real)),real),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real))),hAPP(fun(nat,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(fun(nat,real),fun(nat,real))),fun(fun(nat,real),fun(real,fun(nat,real))),combc(real,fun(nat,real),fun(nat,real)),hAPP(fun(real,fun(nat,fun(real,real))),fun(real,fun(fun(nat,real),fun(nat,real))),hAPP(fun(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real))),fun(fun(real,fun(nat,fun(real,real))),fun(real,fun(fun(nat,real),fun(nat,real)))),combb(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),real),combs(nat,real,real)),hAPP(fun(real,fun(nat,real)),fun(real,fun(nat,fun(real,real))),hAPP(fun(fun(nat,real),fun(nat,fun(real,real))),fun(fun(real,fun(nat,real)),fun(real,fun(nat,fun(real,real)))),combb(fun(nat,real),fun(nat,fun(real,real)),real),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),minus_minus(real))),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),hAPP(real,fun(real,real),plus_plus(real),C))))),hAPP(real,fun(nat,real),power_power(real),C))))),combi(real))))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),semiring_1_of_nat(real))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),C)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),minus_minus(nat)),hAPP(nat,nat,suc,zero_zero(nat))))))))),zero_zero(real)),hAPP(real,filter(real),at(real),zero_zero(real)))). [resolve(5234,b,5180,a)]. 26.82/26.80 Derived: -banach(complex) | -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),hAPP(fun(nat,complex),fun(nat,complex),diffs(complex),hAPP(fun(nat,complex),fun(nat,complex),diffs(complex),A)))),hAPP(complex,fun(nat,complex),power_power(complex),B)))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(complex,real,norm_norm(complex),C)),hAPP(complex,real,norm_norm(complex),B))) | hBOOL(hAPP(filter(complex),bool,hAPP(complex,fun(filter(complex),bool),hAPP(fun(complex,complex),fun(complex,fun(filter(complex),bool)),tendsto(complex,complex),hAPP(fun(complex,fun(nat,complex)),fun(complex,complex),hAPP(fun(fun(nat,complex),complex),fun(fun(complex,fun(nat,complex)),fun(complex,complex)),combb(fun(nat,complex),complex,complex),suminf(complex)),hAPP(fun(complex,fun(nat,complex)),fun(complex,fun(nat,complex)),hAPP(fun(fun(nat,complex),fun(nat,complex)),fun(fun(complex,fun(nat,complex)),fun(complex,fun(nat,complex))),combb(fun(nat,complex),fun(nat,complex),complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),A))),hAPP(fun(nat,complex),fun(complex,fun(nat,complex)),hAPP(fun(complex,fun(fun(nat,complex),fun(nat,complex))),fun(fun(nat,complex),fun(complex,fun(nat,complex))),combc(complex,fun(nat,complex),fun(nat,complex)),hAPP(fun(complex,fun(nat,fun(complex,complex))),fun(complex,fun(fun(nat,complex),fun(nat,complex))),hAPP(fun(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex))),fun(fun(complex,fun(nat,fun(complex,complex))),fun(complex,fun(fun(nat,complex),fun(nat,complex)))),combb(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),complex),combs(nat,complex,complex)),hAPP(fun(complex,fun(nat,complex)),fun(complex,fun(nat,fun(complex,complex))),hAPP(fun(fun(nat,complex),fun(nat,fun(complex,complex))),fun(fun(complex,fun(nat,complex)),fun(complex,fun(nat,fun(complex,complex)))),combb(fun(nat,complex),fun(nat,fun(complex,complex)),complex),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),minus_minus(complex))),hAPP(fun(complex,complex),fun(complex,fun(nat,complex)),hAPP(fun(complex,fun(complex,fun(nat,complex))),fun(fun(complex,complex),fun(complex,fun(nat,complex))),combs(complex,complex,fun(nat,complex)),hAPP(fun(complex,fun(nat,fun(complex,complex))),fun(complex,fun(complex,fun(nat,complex))),hAPP(fun(fun(nat,fun(complex,complex)),fun(complex,fun(nat,complex))),fun(fun(complex,fun(nat,fun(complex,complex))),fun(complex,fun(complex,fun(nat,complex)))),combb(fun(nat,fun(complex,complex)),fun(complex,fun(nat,complex)),complex),combc(nat,complex,complex)),hAPP(fun(complex,fun(nat,complex)),fun(complex,fun(nat,fun(complex,complex))),hAPP(fun(fun(nat,complex),fun(nat,fun(complex,complex))),fun(fun(complex,fun(nat,complex)),fun(complex,fun(nat,fun(complex,complex)))),combb(fun(nat,complex),fun(nat,fun(complex,complex)),complex),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),inverse_divide(complex))),hAPP(fun(nat,complex),fun(complex,fun(nat,complex)),hAPP(fun(complex,fun(fun(nat,complex),fun(nat,complex))),fun(fun(nat,complex),fun(complex,fun(nat,complex))),combc(complex,fun(nat,complex),fun(nat,complex)),hAPP(fun(complex,fun(nat,fun(complex,complex))),fun(complex,fun(fun(nat,complex),fun(nat,complex))),hAPP(fun(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex))),fun(fun(complex,fun(nat,fun(complex,complex))),fun(complex,fun(fun(nat,complex),fun(nat,complex)))),combb(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),complex),combs(nat,complex,complex)),hAPP(fun(complex,fun(nat,complex)),fun(complex,fun(nat,fun(complex,complex))),hAPP(fun(fun(nat,complex),fun(nat,fun(complex,complex))),fun(fun(complex,fun(nat,complex)),fun(complex,fun(nat,fun(complex,complex)))),combb(fun(nat,complex),fun(nat,fun(complex,complex)),complex),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),minus_minus(complex))),hAPP(fun(complex,complex),fun(complex,fun(nat,complex)),hAPP(fun(complex,fun(nat,complex)),fun(fun(complex,complex),fun(complex,fun(nat,complex))),combb(complex,fun(nat,complex),complex),power_power(complex)),hAPP(complex,fun(complex,complex),plus_plus(complex),C))))),hAPP(complex,fun(nat,complex),power_power(complex),C))))),combi(complex))))),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),semiring_1_of_nat(complex))),hAPP(fun(nat,nat),fun(nat,complex),hAPP(fun(nat,complex),fun(fun(nat,nat),fun(nat,complex)),combb(nat,complex,nat),hAPP(complex,fun(nat,complex),power_power(complex),C)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),minus_minus(nat)),hAPP(nat,nat,suc,zero_zero(nat))))))))),zero_zero(complex)),hAPP(complex,filter(complex),at(complex),zero_zero(complex)))). [resolve(5234,b,5224,a)]. 26.91/26.84 5235 -real_normed_field(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C)) | hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),hAPP(fun(nat,fun(nat,bool)),fun(nat,A),hAPP(fun(fun(nat,bool),A),fun(fun(nat,fun(nat,bool)),fun(nat,A)),combb(fun(nat,bool),A,nat),hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),B)),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,fun(nat,bool))),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combs(nat,nat,fun(nat,bool)),hAPP(fun(nat,nat),fun(nat,fun(nat,fun(nat,bool))),hAPP(fun(nat,fun(nat,fun(nat,bool))),fun(fun(nat,nat),fun(nat,fun(nat,fun(nat,bool)))),combb(nat,fun(nat,fun(nat,bool)),nat),ord_atLeastLessThan(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),C))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),C))),C)))),hAPP(fun(nat,A),A,suminf(A),B))) # label(fact_4258_sums__group) # label(axiom). [clausify(2556)]. 26.91/26.84 Derived: -hBOOL(hAPP(fun(nat,real),bool,summable(real),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,fun(nat,bool)),fun(nat,real),hAPP(fun(fun(nat,bool),real),fun(fun(nat,fun(nat,bool)),fun(nat,real)),combb(fun(nat,bool),real,nat),hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),A)),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,fun(nat,bool))),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combs(nat,nat,fun(nat,bool)),hAPP(fun(nat,nat),fun(nat,fun(nat,fun(nat,bool))),hAPP(fun(nat,fun(nat,fun(nat,bool))),fun(fun(nat,nat),fun(nat,fun(nat,fun(nat,bool)))),combb(nat,fun(nat,fun(nat,bool)),nat),ord_atLeastLessThan(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),B))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),B))),B)))),hAPP(fun(nat,real),real,suminf(real),A))). [resolve(5235,a,5180,a)]. 26.91/26.84 Derived: -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),hAPP(fun(nat,fun(nat,bool)),fun(nat,complex),hAPP(fun(fun(nat,bool),complex),fun(fun(nat,fun(nat,bool)),fun(nat,complex)),combb(fun(nat,bool),complex,nat),hAPP(fun(nat,complex),fun(fun(nat,bool),complex),big_co1399186613setsum(nat,complex),A)),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,fun(nat,bool))),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combs(nat,nat,fun(nat,bool)),hAPP(fun(nat,nat),fun(nat,fun(nat,fun(nat,bool))),hAPP(fun(nat,fun(nat,fun(nat,bool))),fun(fun(nat,nat),fun(nat,fun(nat,fun(nat,bool)))),combb(nat,fun(nat,fun(nat,bool)),nat),ord_atLeastLessThan(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),B))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),B))),B)))),hAPP(fun(nat,complex),complex,suminf(complex),A))). [resolve(5235,a,5224,a)]. 26.99/26.92 5236 -real_normed_field(A) | -banach(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,A),fun(fun(A,A),fun(A,A)),combb(A,A,A),exp(A)),B)),C),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,exp(A),hAPP(A,A,B,C))),D))) # label(fact_3884_DERIV__fun__exp) # label(axiom). [clausify(2681)]. 26.99/26.92 Derived: -banach(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),exp(real)),A)),B),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,exp(real),hAPP(real,real,A,B))),C))). [resolve(5236,a,5180,a)]. 26.99/26.92 Derived: -banach(complex) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,complex),fun(fun(complex,complex),fun(complex,complex)),combb(complex,complex,complex),exp(complex)),A)),B),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,exp(complex),hAPP(complex,complex,A,B))),C))). [resolve(5236,a,5224,a)]. 26.99/26.92 5237 -real_normed_field(A) | hAPP(A,A,uminus_uminus(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),hAPP(A,A,uminus_uminus(A),B)),C) # label(fact_1939_divide_Ominus) # label(axiom). [clausify(2683)]. 26.99/26.92 Derived: hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),B)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,uminus_uminus(real),A)),B). [resolve(5237,a,5180,a)]. 26.99/26.92 Derived: hAPP(complex,complex,uminus_uminus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),A),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),hAPP(complex,complex,uminus_uminus(complex),A)),B). [resolve(5237,a,5224,a)]. 26.99/26.92 5238 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),E)) | ti(A,E) = ti(A,D) # label(fact_2648_DERIV__unique) # label(axiom). [clausify(2701)]. 26.99/26.92 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),D)) | ti(real,D) = ti(real,C). [resolve(5238,a,5180,a)]. 26.99/26.92 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),D)) | ti(complex,D) = ti(complex,C). [resolve(5238,a,5224,a)]. 26.99/26.92 5239 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),D),E)) | hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),plus_plus(A)),B)),D)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),E))) # label(fact_4115_sums__add) # label(axiom). [clausify(2724)]. 26.99/26.92 Derived: -hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),C),D)) | hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),plus_plus(real)),A)),C)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),D))). [resolve(5239,a,5180,a)]. 26.99/26.94 Derived: -hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),A),B)) | -hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),C),D)) | hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),plus_plus(complex)),A)),C)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),B),D))). [resolve(5239,a,5224,a)]. 26.99/26.94 5240 -real_normed_field(A) | ti(A,B) = zero_zero(A) | ti(A,C) = zero_zero(A) | hAPP(A,A,uminus_uminus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,inverse_inverse(A),B)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C)),D))),hAPP(A,A,inverse_inverse(A),C))) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,inverse_inverse(A),B)),hAPP(A,A,inverse_inverse(A),C))),D) # label(fact_2638_DERIV__inverse__lemma) # label(axiom). [clausify(2745)]. 26.99/26.94 Derived: ti(real,A) = zero_zero(real) | ti(real,B) = zero_zero(real) | hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,inverse_inverse(real),A)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B)),C))),hAPP(real,real,inverse_inverse(real),B))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,inverse_inverse(real),A)),hAPP(real,real,inverse_inverse(real),B))),C). [resolve(5240,a,5180,a)]. 26.99/26.94 Derived: ti(complex,A) = zero_zero(complex) | ti(complex,B) = zero_zero(complex) | hAPP(complex,complex,uminus_uminus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,inverse_inverse(complex),A)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),A),B)),C))),hAPP(complex,complex,inverse_inverse(complex),B))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(complex,complex,inverse_inverse(complex),A)),hAPP(complex,complex,inverse_inverse(complex),B))),C). [resolve(5240,a,5224,a)]. 26.99/26.94 5241 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,A),fun(fun(A,A),fun(A,A)),combb(A,A,A),uminus_uminus(A)),B)),C),hAPP(A,A,uminus_uminus(A),D))) # label(fact_3891_DERIV__minus) # label(axiom). [clausify(2751)]. 26.99/26.94 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),uminus_uminus(real)),A)),B),hAPP(real,real,uminus_uminus(real),C))). [resolve(5241,a,5180,a)]. 26.99/26.96 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,complex),fun(fun(complex,complex),fun(complex,complex)),combb(complex,complex,complex),uminus_uminus(complex)),A)),B),hAPP(complex,complex,uminus_uminus(complex),C))). [resolve(5241,a,5224,a)]. 26.99/26.96 5242 -banach(A) | -real_normed_field(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),B)),hAPP(A,fun(nat,A),power_power(A),C)))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(A,real,norm_norm(A),D)),hAPP(A,real,norm_norm(A),C))) | hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,A),fun(nat,real),hAPP(fun(A,real),fun(fun(nat,A),fun(nat,real)),combb(A,real,nat),norm_norm(A)),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),B)),hAPP(A,fun(nat,A),power_power(A),D))))) # label(fact_4024_powser__insidea) # label(axiom). [clausify(2760)]. 26.99/26.96 Derived: -banach(real) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),A)),hAPP(real,fun(nat,real),power_power(real),B)))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,norm_norm(real),C)),hAPP(real,real,norm_norm(real),B))) | hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),norm_norm(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),A)),hAPP(real,fun(nat,real),power_power(real),C))))). [resolve(5242,b,5180,a)]. 26.99/26.96 Derived: -banach(complex) | -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),A)),hAPP(complex,fun(nat,complex),power_power(complex),B)))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(complex,real,norm_norm(complex),C)),hAPP(complex,real,norm_norm(complex),B))) | hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),norm_norm(complex)),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),A)),hAPP(complex,fun(nat,complex),power_power(complex),C))))). [resolve(5242,b,5224,a)]. 26.99/26.96 5243 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | hBOOL(hAPP(A,bool,hAPP(fun(A,A),fun(A,bool),isCont(A,A),B),C)) # label(fact_4926_DERIV__isCont) # label(axiom). [clausify(2791)]. 27.09/26.98 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),A),B)). [resolve(5243,a,5180,a)]. 27.09/26.98 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | hBOOL(hAPP(complex,bool,hAPP(fun(complex,complex),fun(complex,bool),isCont(complex,complex),A),B)). [resolve(5243,a,5224,a)]. 27.09/26.98 5244 -real_normed_field(A) | -banach(B) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),C)) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),D)) | ti(A,f472(B,A,E,D,C)) != zero_zero(A) | hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),hAPP(fun(A,fun(nat,B)),fun(A,B),hAPP(fun(fun(nat,B),B),fun(fun(A,fun(nat,B)),fun(A,B)),combb(fun(nat,B),B,A),suminf(B)),E)),zero_zero(B)),hAPP(A,filter(A),at(A),zero_zero(A)))) # label(fact_4650_lemma__termdiff5) # label(axiom). [clausify(2830)]. 27.09/26.98 Derived: -banach(A) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),C)) | ti(real,f472(A,real,D,C,B)) != zero_zero(real) | hBOOL(hAPP(filter(real),bool,hAPP(A,fun(filter(real),bool),hAPP(fun(real,A),fun(A,fun(filter(real),bool)),tendsto(real,A),hAPP(fun(real,fun(nat,A)),fun(real,A),hAPP(fun(fun(nat,A),A),fun(fun(real,fun(nat,A)),fun(real,A)),combb(fun(nat,A),A,real),suminf(A)),D)),zero_zero(A)),hAPP(real,filter(real),at(real),zero_zero(real)))). [resolve(5244,a,5180,a)]. 27.09/26.98 Derived: -banach(A) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),C)) | ti(complex,f472(A,complex,D,C,B)) != zero_zero(complex) | hBOOL(hAPP(filter(complex),bool,hAPP(A,fun(filter(complex),bool),hAPP(fun(complex,A),fun(A,fun(filter(complex),bool)),tendsto(complex,A),hAPP(fun(complex,fun(nat,A)),fun(complex,A),hAPP(fun(fun(nat,A),A),fun(fun(complex,fun(nat,A)),fun(complex,A)),combb(fun(nat,A),A,complex),suminf(A)),D)),zero_zero(A)),hAPP(complex,filter(complex),at(complex),zero_zero(complex)))). [resolve(5244,a,5224,a)]. 27.09/26.98 5245 -real_normed_field(A) | -banach(B) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),C)) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(A,real,norm_norm(A),f472(B,A,E,D,C))),C)) | hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),hAPP(fun(A,fun(nat,B)),fun(A,B),hAPP(fun(fun(nat,B),B),fun(fun(A,fun(nat,B)),fun(A,B)),combb(fun(nat,B),B,A),suminf(B)),E)),zero_zero(B)),hAPP(A,filter(A),at(A),zero_zero(A)))) # label(fact_4650_lemma__termdiff5) # label(axiom). [clausify(2830)]. 27.09/26.98 Derived: -banach(A) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,norm_norm(real),f472(A,real,D,C,B))),B)) | hBOOL(hAPP(filter(real),bool,hAPP(A,fun(filter(real),bool),hAPP(fun(real,A),fun(A,fun(filter(real),bool)),tendsto(real,A),hAPP(fun(real,fun(nat,A)),fun(real,A),hAPP(fun(fun(nat,A),A),fun(fun(real,fun(nat,A)),fun(real,A)),combb(fun(nat,A),A,real),suminf(A)),D)),zero_zero(A)),hAPP(real,filter(real),at(real),zero_zero(real)))). [resolve(5245,a,5180,a)]. 27.09/26.98 Derived: -banach(A) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(complex,real,norm_norm(complex),f472(A,complex,D,C,B))),B)) | hBOOL(hAPP(filter(complex),bool,hAPP(A,fun(filter(complex),bool),hAPP(fun(complex,A),fun(A,fun(filter(complex),bool)),tendsto(complex,A),hAPP(fun(complex,fun(nat,A)),fun(complex,A),hAPP(fun(fun(nat,A),A),fun(fun(complex,fun(nat,A)),fun(complex,A)),combb(fun(nat,A),A,complex),suminf(A)),D)),zero_zero(A)),hAPP(complex,filter(complex),at(complex),zero_zero(complex)))). [resolve(5245,a,5224,a)]. 27.11/27.03 5246 -real_normed_field(A) | -banach(B) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),C)) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),D)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(B,real,norm_norm(B),hAPP(nat,B,hAPP(A,fun(nat,B),E,f472(B,A,E,D,C)),f473(B,A,E,D,C)))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,D,f473(B,A,E,D,C))),hAPP(A,real,norm_norm(A),f472(B,A,E,D,C))))) | hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),hAPP(fun(A,fun(nat,B)),fun(A,B),hAPP(fun(fun(nat,B),B),fun(fun(A,fun(nat,B)),fun(A,B)),combb(fun(nat,B),B,A),suminf(B)),E)),zero_zero(B)),hAPP(A,filter(A),at(A),zero_zero(A)))) # label(fact_4650_lemma__termdiff5) # label(axiom). [clausify(2830)]. 27.11/27.03 Derived: -banach(A) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(A,real,norm_norm(A),hAPP(nat,A,hAPP(real,fun(nat,A),D,f472(A,real,D,C,B)),f473(A,real,D,C,B)))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,C,f473(A,real,D,C,B))),hAPP(real,real,norm_norm(real),f472(A,real,D,C,B))))) | hBOOL(hAPP(filter(real),bool,hAPP(A,fun(filter(real),bool),hAPP(fun(real,A),fun(A,fun(filter(real),bool)),tendsto(real,A),hAPP(fun(real,fun(nat,A)),fun(real,A),hAPP(fun(fun(nat,A),A),fun(fun(real,fun(nat,A)),fun(real,A)),combb(fun(nat,A),A,real),suminf(A)),D)),zero_zero(A)),hAPP(real,filter(real),at(real),zero_zero(real)))). [resolve(5246,a,5180,a)]. 27.11/27.03 Derived: -banach(A) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(A,real,norm_norm(A),hAPP(nat,A,hAPP(complex,fun(nat,A),D,f472(A,complex,D,C,B)),f473(A,complex,D,C,B)))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,C,f473(A,complex,D,C,B))),hAPP(complex,real,norm_norm(complex),f472(A,complex,D,C,B))))) | hBOOL(hAPP(filter(complex),bool,hAPP(A,fun(filter(complex),bool),hAPP(fun(complex,A),fun(A,fun(filter(complex),bool)),tendsto(complex,A),hAPP(fun(complex,fun(nat,A)),fun(complex,A),hAPP(fun(fun(nat,A),A),fun(fun(complex,fun(nat,A)),fun(complex,A)),combb(fun(nat,A),A,complex),suminf(A)),D)),zero_zero(A)),hAPP(complex,filter(complex),at(complex),zero_zero(complex)))). [resolve(5246,a,5224,a)]. 27.11/27.03 5247 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | zero_zero(A) = hAPP(A,A,B,C) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,A),fun(fun(A,A),fun(A,A)),combb(A,A,A),inverse_inverse(A)),B)),C),hAPP(A,A,uminus_uminus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,inverse_inverse(A),hAPP(A,A,B,C))),D)),hAPP(A,A,inverse_inverse(A),hAPP(A,A,B,C)))))) # label(fact_3888_DERIV__inverse_H) # label(axiom). [clausify(2878)]. 27.11/27.03 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | zero_zero(real) = hAPP(real,real,A,B) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),inverse_inverse(real)),A)),B),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,inverse_inverse(real),hAPP(real,real,A,B))),C)),hAPP(real,real,inverse_inverse(real),hAPP(real,real,A,B)))))). [resolve(5247,a,5180,a)]. 27.11/27.03 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | zero_zero(complex) = hAPP(complex,complex,A,B) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,complex),fun(fun(complex,complex),fun(complex,complex)),combb(complex,complex,complex),inverse_inverse(complex)),A)),B),hAPP(complex,complex,uminus_uminus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,inverse_inverse(complex),hAPP(complex,complex,A,B))),C)),hAPP(complex,complex,inverse_inverse(complex),hAPP(complex,complex,A,B)))))). [resolve(5247,a,5224,a)]. 27.11/27.11 5248 -banach(A) | -real_normed_field(A) | hAPP(fun(nat,A),A,suminf(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,real),fun(nat,fun(A,A)),hAPP(fun(real,fun(A,A)),fun(fun(nat,real),fun(nat,fun(A,A))),combb(real,fun(A,A),nat),scaleR_scaleR(A)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(A,fun(nat,A),power_power(A),B))) = hAPP(A,A,exp(A),B) # label(fact_4776_exp__def) # label(axiom). [clausify(2968)]. 27.11/27.11 5249 -real_normed_field(A) | -hBOOL(hAPP(filter(B),bool,hAPP(A,fun(filter(B),bool),hAPP(fun(B,A),fun(A,fun(filter(B),bool)),tendsto(B,A),C),D),E)) | -hBOOL(hAPP(filter(B),bool,hAPP(A,fun(filter(B),bool),hAPP(fun(B,A),fun(A,fun(filter(B),bool)),tendsto(B,A),F),V6),E)) | ti(A,V6) = zero_zero(A) | hBOOL(hAPP(filter(B),bool,hAPP(A,fun(filter(B),bool),hAPP(fun(B,A),fun(A,fun(filter(B),bool)),tendsto(B,A),hAPP(fun(B,A),fun(B,A),hAPP(fun(B,fun(A,A)),fun(fun(B,A),fun(B,A)),combs(B,A,A),hAPP(fun(B,A),fun(B,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),combb(A,fun(A,A),B),inverse_divide(A)),C)),F)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),D),V6)),E)) # label(fact_4465_tendsto__divide) # label(axiom). [clausify(2976)]. 27.11/27.11 Derived: -hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),B),C),D)) | -hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),E),F),D)) | ti(real,F) = zero_zero(real) | hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),hAPP(fun(A,real),fun(A,real),hAPP(fun(A,fun(real,real)),fun(fun(A,real),fun(A,real)),combs(A,real,real),hAPP(fun(A,real),fun(A,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(A,real),fun(A,fun(real,real))),combb(real,fun(real,real),A),inverse_divide(real)),B)),E)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),C),F)),D)). [resolve(5249,a,5180,a)]. 27.11/27.11 Derived: -hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),B),C),D)) | -hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),E),F),D)) | ti(complex,F) = zero_zero(complex) | hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),hAPP(fun(A,complex),fun(A,complex),hAPP(fun(A,fun(complex,complex)),fun(fun(A,complex),fun(A,complex)),combs(A,complex,complex),hAPP(fun(A,complex),fun(A,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(A,complex),fun(A,fun(complex,complex))),combb(complex,fun(complex,complex),A),inverse_divide(complex)),B)),E)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),C),F)),D)). [resolve(5249,a,5224,a)]. 27.11/27.11 5250 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),E),C),F)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,A)),combs(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),plus_plus(A)),B)),E)),C),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),D),F))) # label(fact_3865_DERIV__add) # label(axiom). [clausify(3003)]. 27.21/27.19 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),D),B),E)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),plus_plus(real)),A)),D)),B),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),C),E))). [resolve(5250,a,5180,a)]. 27.21/27.19 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),D),B),E)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,complex)),combs(complex,complex,complex),hAPP(fun(complex,complex),fun(complex,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,fun(complex,complex))),combb(complex,fun(complex,complex),complex),plus_plus(complex)),A)),D)),B),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),C),E))). [resolve(5250,a,5224,a)]. 27.21/27.19 5251 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),E),hAPP(A,A,B,C)),F)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,A),fun(fun(A,A),fun(A,A)),combb(A,A,A),E),B)),C),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),F),D))) # label(fact_3886_DERIV__chain_H) # label(axiom). [clausify(3058)]. 27.21/27.19 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),D),hAPP(real,real,A,B)),E)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),D),A)),B),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),E),C))). [resolve(5251,a,5180,a)]. 27.21/27.19 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),D),hAPP(complex,complex,A,B)),E)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,complex),fun(fun(complex,complex),fun(complex,complex)),combb(complex,complex,complex),D),A)),B),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),E),C))). [resolve(5251,a,5224,a)]. 27.21/27.19 5252 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | hBOOL(hAPP(filter(A),bool,hAPP(A,fun(filter(A),bool),hAPP(fun(A,A),fun(A,fun(filter(A),bool)),tendsto(A,A),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,A)),combs(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),inverse_divide(A)),hAPP(A,fun(A,A),hAPP(fun(A,fun(A,A)),fun(A,fun(A,A)),combc(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),minus_minus(A)),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,A),fun(fun(A,A),fun(A,A)),combb(A,A,A),B),hAPP(A,fun(A,A),plus_plus(A),C)))),hAPP(A,A,B,C)))),combi(A))),D),hAPP(A,filter(A),at(A),zero_zero(A)))) # label(fact_4645_DERIV__iff) # label(axiom). [clausify(3154)]. 27.21/27.19 5253 -real_normed_field(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | -hBOOL(hAPP(filter(A),bool,hAPP(A,fun(filter(A),bool),hAPP(fun(A,A),fun(A,fun(filter(A),bool)),tendsto(A,A),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,A)),combs(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),inverse_divide(A)),hAPP(A,fun(A,A),hAPP(fun(A,fun(A,A)),fun(A,fun(A,A)),combc(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),minus_minus(A)),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,A),fun(fun(A,A),fun(A,A)),combb(A,A,A),B),hAPP(A,fun(A,A),plus_plus(A),C)))),hAPP(A,A,B,C)))),combi(A))),D),hAPP(A,filter(A),at(A),zero_zero(A)))) # label(fact_4645_DERIV__iff) # label(axiom). [clausify(3154)]. 27.21/27.19 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | -hBOOL(hAPP(filter(real),bool,hAPP(real,fun(filter(real),bool),hAPP(fun(real,real),fun(real,fun(filter(real),bool)),tendsto(real,real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real)),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),minus_minus(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),A),hAPP(real,fun(real,real),plus_plus(real),B)))),hAPP(real,real,A,B)))),combi(real))),C),hAPP(real,filter(real),at(real),zero_zero(real)))). [resolve(5253,a,5180,a)]. 27.21/27.19 Derived: hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | -hBOOL(hAPP(filter(complex),bool,hAPP(complex,fun(filter(complex),bool),hAPP(fun(complex,complex),fun(complex,fun(filter(complex),bool)),tendsto(complex,complex),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,complex)),combs(complex,complex,complex),hAPP(fun(complex,complex),fun(complex,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,fun(complex,complex))),combb(complex,fun(complex,complex),complex),inverse_divide(complex)),hAPP(complex,fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(complex,fun(complex,complex)),combc(complex,complex,complex),hAPP(fun(complex,complex),fun(complex,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,fun(complex,complex))),combb(complex,fun(complex,complex),complex),minus_minus(complex)),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,complex),fun(fun(complex,complex),fun(complex,complex)),combb(complex,complex,complex),A),hAPP(complex,fun(complex,complex),plus_plus(complex),B)))),hAPP(complex,complex,A,B)))),combi(complex))),C),hAPP(complex,filter(complex),at(complex),zero_zero(complex)))). [resolve(5253,a,5224,a)]. 27.39/27.34 5254 -real_normed_field(A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),f551(A,B))) # label(fact_2053_divide_Ononneg__bounded) # label(axiom). [clausify(3281)]. 27.39/27.34 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),f551(real,A))). [resolve(5254,a,5180,a)]. 27.39/27.34 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),f551(complex,A))). [resolve(5254,a,5224,a)]. 27.39/27.34 5255 -real_normed_field(A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(A,real,norm_norm(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),C))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(A,real,norm_norm(A),B)),f551(A,C)))) # label(fact_2053_divide_Ononneg__bounded) # label(axiom). [clausify(3281)]. 27.39/27.34 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),A)),f551(real,B)))). [resolve(5255,a,5180,a)]. 27.39/27.34 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),A)),f551(complex,B)))). [resolve(5255,a,5224,a)]. 27.39/27.34 5256 -real_normed_field(A) | -banach(A) | one_one(A) = hAPP(A,A,exp(A),zero_zero(A)) # label(fact_1773_exp__zero) # label(axiom). [clausify(3301)]. 27.39/27.34 Derived: -banach(real) | one_one(real) = hAPP(real,real,exp(real),zero_zero(real)). [resolve(5256,a,5180,a)]. 27.39/27.34 Derived: -banach(complex) | one_one(complex) = hAPP(complex,complex,exp(complex),zero_zero(complex)). [resolve(5256,a,5224,a)]. 27.39/27.34 5257 -real_normed_field(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),B)) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),C)) | hAPP(fun(nat,A),A,suminf(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),plus_plus(A)),B)),C)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(fun(nat,A),A,suminf(A),B)),hAPP(fun(nat,A),A,suminf(A),C)) # label(fact_4009_suminf__add) # label(axiom). [clausify(3310)]. 27.39/27.34 Derived: -hBOOL(hAPP(fun(nat,real),bool,summable(real),A)) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),B)) | hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),plus_plus(real)),A)),B)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,real),real,suminf(real),A)),hAPP(fun(nat,real),real,suminf(real),B)). [resolve(5257,a,5180,a)]. 27.39/27.34 Derived: -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),A)) | -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),B)) | hAPP(fun(nat,complex),complex,suminf(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),plus_plus(complex)),A)),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(fun(nat,complex),complex,suminf(complex),A)),hAPP(fun(nat,complex),complex,suminf(complex),B)). [resolve(5257,a,5224,a)]. 27.39/27.34 5258 -real_normed_field(A) | deriv(A) = ti(fun(fun(A,A),fun(A,fun(A,bool))),deriv(A)) # label(tsy_c_Deriv_Oderiv_res) # label(axiom). [clausify(3403)]. 27.39/27.34 Derived: deriv(real) = ti(fun(fun(real,real),fun(real,fun(real,bool))),deriv(real)). [resolve(5258,a,5180,a)]. 27.39/27.34 Derived: deriv(complex) = ti(fun(fun(complex,complex),fun(complex,fun(complex,bool))),deriv(complex)). [resolve(5258,a,5224,a)]. 27.61/27.57 5259 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | ti(A,E) != ti(A,D) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),E)) # label(fact_2646_Deriv_Olemma__DERIV__subst) # label(axiom). [clausify(3415)]. 27.61/27.57 5260 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),E),C),F)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,A)),combs(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),minus_minus(A)),B)),E)),C),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),D),F))) # label(fact_3866_DERIV__diff) # label(axiom). [clausify(3424)]. 27.61/27.57 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),D),B),E)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),minus_minus(real)),A)),D)),B),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),C),E))). [resolve(5260,a,5180,a)]. 27.61/27.57 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),D),B),E)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,complex)),combs(complex,complex,complex),hAPP(fun(complex,complex),fun(complex,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,fun(complex,complex))),combb(complex,fun(complex,complex),complex),minus_minus(complex)),A)),D)),B),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),C),E))). [resolve(5260,a,5224,a)]. 27.61/27.57 5261 -real_normed_field(A) | -hBOOL(hAPP(filter(A),bool,hAPP(A,fun(filter(A),bool),hAPP(fun(A,A),fun(A,fun(filter(A),bool)),tendsto(A,A),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,A)),combs(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),inverse_divide(A)),hAPP(A,fun(A,A),hAPP(fun(A,fun(A,A)),fun(A,fun(A,A)),combc(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),minus_minus(A)),B)),hAPP(A,A,B,C)))),hAPP(A,fun(A,A),hAPP(fun(A,fun(A,A)),fun(A,fun(A,A)),combc(A,A,A),minus_minus(A)),C))),D),hAPP(A,filter(A),at(A),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) # label(fact_4626_DERIV__iff2) # label(axiom). [clausify(3545)]. 27.61/27.57 Derived: -hBOOL(hAPP(filter(real),bool,hAPP(real,fun(filter(real),bool),hAPP(fun(real,real),fun(real,fun(filter(real),bool)),tendsto(real,real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real)),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),minus_minus(real)),A)),hAPP(real,real,A,B)))),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),minus_minus(real)),B))),C),hAPP(real,filter(real),at(real),B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)). [resolve(5261,a,5180,a)]. 27.61/27.57 Derived: -hBOOL(hAPP(filter(complex),bool,hAPP(complex,fun(filter(complex),bool),hAPP(fun(complex,complex),fun(complex,fun(filter(complex),bool)),tendsto(complex,complex),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,complex)),combs(complex,complex,complex),hAPP(fun(complex,complex),fun(complex,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,fun(complex,complex))),combb(complex,fun(complex,complex),complex),inverse_divide(complex)),hAPP(complex,fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(complex,fun(complex,complex)),combc(complex,complex,complex),hAPP(fun(complex,complex),fun(complex,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,fun(complex,complex))),combb(complex,fun(complex,complex),complex),minus_minus(complex)),A)),hAPP(complex,complex,A,B)))),hAPP(complex,fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(complex,fun(complex,complex)),combc(complex,complex,complex),minus_minus(complex)),B))),C),hAPP(complex,filter(complex),at(complex),B))) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)). [resolve(5261,a,5224,a)]. 27.61/27.57 5262 -real_normed_field(A) | hBOOL(hAPP(filter(A),bool,hAPP(A,fun(filter(A),bool),hAPP(fun(A,A),fun(A,fun(filter(A),bool)),tendsto(A,A),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,A)),combs(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),inverse_divide(A)),hAPP(A,fun(A,A),hAPP(fun(A,fun(A,A)),fun(A,fun(A,A)),combc(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),minus_minus(A)),B)),hAPP(A,A,B,C)))),hAPP(A,fun(A,A),hAPP(fun(A,fun(A,A)),fun(A,fun(A,A)),combc(A,A,A),minus_minus(A)),C))),D),hAPP(A,filter(A),at(A),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) # label(fact_4626_DERIV__iff2) # label(axiom). [clausify(3545)]. 27.61/27.57 Derived: hBOOL(hAPP(filter(real),bool,hAPP(real,fun(filter(real),bool),hAPP(fun(real,real),fun(real,fun(filter(real),bool)),tendsto(real,real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real)),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),minus_minus(real)),A)),hAPP(real,real,A,B)))),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),minus_minus(real)),B))),C),hAPP(real,filter(real),at(real),B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)). [resolve(5262,a,5180,a)]. 27.61/27.57 Derived: hBOOL(hAPP(filter(complex),bool,hAPP(complex,fun(filter(complex),bool),hAPP(fun(complex,complex),fun(complex,fun(filter(complex),bool)),tendsto(complex,complex),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,complex)),combs(complex,complex,complex),hAPP(fun(complex,complex),fun(complex,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,fun(complex,complex))),combb(complex,fun(complex,complex),complex),inverse_divide(complex)),hAPP(complex,fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(complex,fun(complex,complex)),combc(complex,complex,complex),hAPP(fun(complex,complex),fun(complex,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,fun(complex,complex))),combb(complex,fun(complex,complex),complex),minus_minus(complex)),A)),hAPP(complex,complex,A,B)))),hAPP(complex,fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(complex,fun(complex,complex)),combc(complex,complex,complex),minus_minus(complex)),B))),C),hAPP(complex,filter(complex),at(complex),B))) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)). [resolve(5262,a,5224,a)]. 28.00/27.97 5263 -real_normed_field(A) | -banach(A) | hAPP(A,A,exp(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,exp(A),B)),hAPP(A,A,exp(A),C)) # label(fact_1775_exp__add) # label(axiom). [clausify(3857)]. 28.00/27.97 5264 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),hAPP(A,A,C,D)),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),C),D),F)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,A),fun(fun(A,A),fun(A,A)),combb(A,A,A),B),C)),D),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),E),F))) # label(fact_3885_DERIV__chain2) # label(axiom). [clausify(3874)]. 28.00/27.97 5265 -real_normed_field(A) | -banach(A) | hBOOL(hAPP(A,bool,hAPP(fun(A,A),fun(A,bool),isCont(A,A),exp(A)),B)) # label(fact_4929_isCont__exp) # label(axiom). [clausify(3983)]. 28.00/27.97 Derived: -banach(real) | hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),exp(real)),A)). [resolve(5265,a,5180,a)]. 28.00/27.97 Derived: -banach(complex) | hBOOL(hAPP(complex,bool,hAPP(fun(complex,complex),fun(complex,bool),isCont(complex,complex),exp(complex)),A)). [resolve(5265,a,5224,a)]. 28.00/27.97 5266 -real_normed_field(A) | hBOOL(hAPP(filter(A),bool,hAPP(A,fun(filter(A),bool),hAPP(fun(A,A),fun(A,fun(filter(A),bool)),tendsto(A,A),hAPP(A,fun(A,A),hAPP(fun(A,fun(A,A)),fun(A,fun(A,A)),combc(A,A,A),inverse_divide(A)),B)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),B)),hAPP(A,filter(A),at(A),C))) # label(fact_4578_divide_Ocont) # label(axiom). [clausify(4017)]. 28.00/27.97 Derived: hBOOL(hAPP(filter(real),bool,hAPP(real,fun(filter(real),bool),hAPP(fun(real,real),fun(real,fun(filter(real),bool)),tendsto(real,real),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),inverse_divide(real)),A)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),A)),hAPP(real,filter(real),at(real),B))). [resolve(5266,a,5180,a)]. 28.00/27.97 Derived: hBOOL(hAPP(filter(complex),bool,hAPP(complex,fun(filter(complex),bool),hAPP(fun(complex,complex),fun(complex,fun(filter(complex),bool)),tendsto(complex,complex),hAPP(complex,fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(complex,fun(complex,complex)),combc(complex,complex,complex),inverse_divide(complex)),A)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),B),A)),hAPP(complex,filter(complex),at(complex),B))). [resolve(5266,a,5224,a)]. 28.00/27.97 5267 -topological_space(A) | -real_normed_field(B) | -hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),C),zero_zero(B)),hAPP(A,filter(A),at(A),D))) | hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),hAPP(B,fun(A,B),hAPP(fun(A,fun(B,B)),fun(B,fun(A,B)),combc(A,B,B),hAPP(fun(A,B),fun(A,fun(B,B)),hAPP(fun(B,fun(B,B)),fun(fun(A,B),fun(A,fun(B,B))),combb(B,fun(B,B),A),inverse_divide(B)),C)),E)),zero_zero(B)),hAPP(A,filter(A),at(A),D))) # label(fact_4611_divide_OLIM__zero) # label(axiom). [clausify(4075)]. 28.00/27.97 Derived: -topological_space(A) | -hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),B),zero_zero(real)),hAPP(A,filter(A),at(A),C))) | hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),hAPP(real,fun(A,real),hAPP(fun(A,fun(real,real)),fun(real,fun(A,real)),combc(A,real,real),hAPP(fun(A,real),fun(A,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(A,real),fun(A,fun(real,real))),combb(real,fun(real,real),A),inverse_divide(real)),B)),D)),zero_zero(real)),hAPP(A,filter(A),at(A),C))). [resolve(5267,b,5180,a)]. 28.00/27.98 Derived: -topological_space(A) | -hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),B),zero_zero(complex)),hAPP(A,filter(A),at(A),C))) | hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),hAPP(complex,fun(A,complex),hAPP(fun(A,fun(complex,complex)),fun(complex,fun(A,complex)),combc(A,complex,complex),hAPP(fun(A,complex),fun(A,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(A,complex),fun(A,fun(complex,complex))),combb(complex,fun(complex,complex),A),inverse_divide(complex)),B)),D)),zero_zero(complex)),hAPP(A,filter(A),at(A),C))). [resolve(5267,b,5224,a)]. 28.00/27.98 5268 -real_normed_field(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),B),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),hAPP(fun(A,A),fun(A,fun(A,bool)),deriv(A),hAPP(nat,fun(A,A),hAPP(fun(A,fun(nat,A)),fun(nat,fun(A,A)),combc(A,nat,A),hAPP(fun(A,A),fun(A,fun(nat,A)),hAPP(fun(A,fun(nat,A)),fun(fun(A,A),fun(A,fun(nat,A))),combb(A,fun(nat,A),A),power_power(A)),B)),hAPP(nat,nat,suc,E))),C),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),one_one(A)),hAPP(nat,A,semiring_1_of_nat(A),E))),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(A,A,B,C)),E))))) # label(fact_3916_DERIV__power__Suc) # label(axiom). [clausify(4078)]. 28.00/27.98 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),A),B),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),A)),hAPP(nat,nat,suc,D))),B),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(nat,real,semiring_1_of_nat(real),D))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,A,B)),D))))). [resolve(5268,a,5180,a)]. 28.00/27.98 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),A),B),C)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),hAPP(fun(complex,complex),fun(complex,fun(complex,bool)),deriv(complex),hAPP(nat,fun(complex,complex),hAPP(fun(complex,fun(nat,complex)),fun(nat,fun(complex,complex)),combc(complex,nat,complex),hAPP(fun(complex,complex),fun(complex,fun(nat,complex)),hAPP(fun(complex,fun(nat,complex)),fun(fun(complex,complex),fun(complex,fun(nat,complex))),combb(complex,fun(nat,complex),complex),power_power(complex)),A)),hAPP(nat,nat,suc,D))),B),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),one_one(complex)),hAPP(nat,complex,semiring_1_of_nat(complex),D))),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(complex,complex,A,B)),D))))). [resolve(5268,a,5224,a)]. 28.00/27.98 5269 -real_normed_field(A) | -real_normed_vector(B) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),C)) | ti(A,f703(B,A,D,E,C)) != zero_zero(A) | hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),D),zero_zero(B)),hAPP(A,filter(A),at(A),zero_zero(A)))) # label(fact_4651_lemma__termdiff4) # label(axiom). [clausify(4087)]. 28.00/27.99 Derived: -real_normed_vector(A) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)) | ti(real,f703(A,real,C,D,B)) != zero_zero(real) | hBOOL(hAPP(filter(real),bool,hAPP(A,fun(filter(real),bool),hAPP(fun(real,A),fun(A,fun(filter(real),bool)),tendsto(real,A),C),zero_zero(A)),hAPP(real,filter(real),at(real),zero_zero(real)))). [resolve(5269,a,5180,a)]. 28.00/27.99 Derived: -real_normed_vector(A) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)) | ti(complex,f703(A,complex,C,D,B)) != zero_zero(complex) | hBOOL(hAPP(filter(complex),bool,hAPP(A,fun(filter(complex),bool),hAPP(fun(complex,A),fun(A,fun(filter(complex),bool)),tendsto(complex,A),C),zero_zero(A)),hAPP(complex,filter(complex),at(complex),zero_zero(complex)))). [resolve(5269,a,5224,a)]. 28.00/27.99 5270 -real_normed_field(A) | -real_normed_vector(B) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(A,real,norm_norm(A),f703(B,A,D,E,C))),C)) | hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),D),zero_zero(B)),hAPP(A,filter(A),at(A),zero_zero(A)))) # label(fact_4651_lemma__termdiff4) # label(axiom). [clausify(4087)]. 28.00/27.99 Derived: -real_normed_vector(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),hAPP(real,real,norm_norm(real),f703(A,real,C,D,B))),B)) | hBOOL(hAPP(filter(real),bool,hAPP(A,fun(filter(real),bool),hAPP(fun(real,A),fun(A,fun(filter(real),bool)),tendsto(real,A),C),zero_zero(A)),hAPP(real,filter(real),at(real),zero_zero(real)))). [resolve(5270,a,5180,a)]. 28.00/27.99 Derived: -real_normed_vector(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),hAPP(complex,real,norm_norm(complex),f703(A,complex,C,D,B))),B)) | hBOOL(hAPP(filter(complex),bool,hAPP(A,fun(filter(complex),bool),hAPP(fun(complex,A),fun(A,fun(filter(complex),bool)),tendsto(complex,A),C),zero_zero(A)),hAPP(complex,filter(complex),at(complex),zero_zero(complex)))). [resolve(5270,a,5224,a)]. 28.00/27.99 5271 -real_normed_field(A) | -real_normed_vector(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_eq(real),hAPP(B,real,norm_norm(B),hAPP(A,B,D,f703(B,A,D,E,C)))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),E),hAPP(A,real,norm_norm(A),f703(B,A,D,E,C))))) | hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),D),zero_zero(B)),hAPP(A,filter(A),at(A),zero_zero(A)))) # label(fact_4651_lemma__termdiff4) # label(axiom). [clausify(4087)]. 28.00/27.99 Derived: -real_normed_vector(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_eq(real),hAPP(A,real,norm_norm(A),hAPP(real,A,C,f703(A,real,C,D,B)))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),D),hAPP(real,real,norm_norm(real),f703(A,real,C,D,B))))) | hBOOL(hAPP(filter(real),bool,hAPP(A,fun(filter(real),bool),hAPP(fun(real,A),fun(A,fun(filter(real),bool)),tendsto(real,A),C),zero_zero(A)),hAPP(real,filter(real),at(real),zero_zero(real)))). [resolve(5271,a,5180,a)]. 28.00/27.99 Derived: -real_normed_vector(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_eq(real),hAPP(A,real,norm_norm(A),hAPP(complex,A,C,f703(A,complex,C,D,B)))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),D),hAPP(complex,real,norm_norm(complex),f703(A,complex,C,D,B))))) | hBOOL(hAPP(filter(complex),bool,hAPP(A,fun(filter(complex),bool),hAPP(fun(complex,A),fun(A,fun(filter(complex),bool)),tendsto(complex,A),C),zero_zero(A)),hAPP(complex,filter(complex),at(complex),zero_zero(complex)))). [resolve(5271,a,5224,a)]. 28.21/28.20 5272 -real_normed_field(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),B)) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),C)) | hBOOL(hAPP(fun(nat,A),bool,summable(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),plus_plus(A)),B)),C))) # label(fact_4039_summable__add) # label(axiom). [clausify(4090)]. 28.21/28.20 Derived: -hBOOL(hAPP(fun(nat,real),bool,summable(real),A)) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),B)) | hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),plus_plus(real)),A)),B))). [resolve(5272,a,5180,a)]. 28.21/28.20 Derived: -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),A)) | -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),B)) | hBOOL(hAPP(fun(nat,complex),bool,summable(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),plus_plus(complex)),A)),B))). [resolve(5272,a,5224,a)]. 28.21/28.20 5273 -real_normed_field(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),B)) | hAPP(fun(nat,A),A,suminf(A),hAPP(A,fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(A,fun(nat,A)),combc(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),inverse_divide(A)),B)),C)) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),hAPP(fun(nat,A),A,suminf(A),B)),C) # label(fact_4012_suminf__divide) # label(axiom). [clausify(4138)]. 28.21/28.20 5274 -real_normed_field(A) | -banach(A) | hAPP(real,A,of_real(A),hAPP(real,real,exp(real),B)) = hAPP(A,A,exp(A),hAPP(real,A,of_real(A),B)) # label(fact_2810_exp__of__real) # label(axiom). [clausify(4160)]. 28.21/28.20 Derived: -banach(real) | hAPP(real,real,of_real(real),hAPP(real,real,exp(real),A)) = hAPP(real,real,exp(real),hAPP(real,real,of_real(real),A)). [resolve(5274,a,5180,a)]. 28.21/28.20 Derived: -banach(complex) | hAPP(real,complex,of_real(complex),hAPP(real,real,exp(real),A)) = hAPP(complex,complex,exp(complex),hAPP(real,complex,of_real(complex),A)). [resolve(5274,a,5224,a)]. 28.21/28.20 5275 -real_normed_field(A) | -banach(A) | zero_zero(A) != hAPP(A,A,exp(A),B) # label(fact_1763_exp__not__eq__zero) # label(axiom). [clausify(4237)]. 28.21/28.20 Derived: -banach(real) | zero_zero(real) != hAPP(real,real,exp(real),A). [resolve(5275,a,5180,a)]. 28.21/28.20 Derived: -banach(complex) | zero_zero(complex) != hAPP(complex,complex,exp(complex),A). [resolve(5275,a,5224,a)]. 28.21/28.20 5276 -real_normed_field(A) | hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),D)) # label(fact_1752_divide_Odiff) # label(axiom). [clausify(4319)]. 28.21/28.20 Derived: hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B)),C) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),C)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),C)). [resolve(5276,a,5180,a)]. 28.21/28.20 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),A),B)),C) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),A),C)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),B),C)). [resolve(5276,a,5224,a)]. 28.50/28.40 5277 -real_normed_field(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),B)) | hBOOL(hAPP(fun(nat,A),bool,summable(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(A,A),fun(fun(nat,A),fun(nat,A)),combb(A,A,nat),uminus_uminus(A)),B))) # label(fact_4025_summable__minus) # label(axiom). [clausify(4342)]. 28.50/28.40 Derived: -hBOOL(hAPP(fun(nat,real),bool,summable(real),A)) | hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),uminus_uminus(real)),A))). [resolve(5277,a,5180,a)]. 28.50/28.40 Derived: -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),A)) | hBOOL(hAPP(fun(nat,complex),bool,summable(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(complex,complex),fun(fun(nat,complex),fun(nat,complex)),combb(complex,complex,nat),uminus_uminus(complex)),A))). [resolve(5277,a,5224,a)]. 28.50/28.40 5278 -real_normed_field(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),B)) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),C)) | hBOOL(hAPP(fun(nat,A),bool,summable(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),minus_minus(A)),B)),C))) # label(fact_4038_summable__diff) # label(axiom). [clausify(4404)]. 28.50/28.40 Derived: -hBOOL(hAPP(fun(nat,real),bool,summable(real),A)) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),B)) | hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),minus_minus(real)),A)),B))). [resolve(5278,a,5180,a)]. 28.50/28.40 Derived: -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),A)) | -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),B)) | hBOOL(hAPP(fun(nat,complex),bool,summable(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),minus_minus(complex)),A)),B))). [resolve(5278,a,5224,a)]. 28.50/28.40 5279 -field_inverse_zero(A) | -real_normed_field(A) | hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(A,real,norm_norm(A),B)),hAPP(A,real,norm_norm(A),C)) = hAPP(A,real,norm_norm(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),C)) # label(fact_1984_norm__divide) # label(axiom). [clausify(4478)]. 28.50/28.40 Derived: -field_inverse_zero(real) | hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,norm_norm(real),A)),hAPP(real,real,norm_norm(real),B)) = hAPP(real,real,norm_norm(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),B)). [resolve(5279,b,5180,a)]. 28.50/28.40 Derived: -field_inverse_zero(complex) | hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(complex,real,norm_norm(complex),A)),hAPP(complex,real,norm_norm(complex),B)) = hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),A),B)). [resolve(5279,b,5224,a)]. 28.50/28.40 5280 -real_normed_field(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),B)) | hBOOL(hAPP(fun(nat,A),bool,summable(A),hAPP(A,fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(A,fun(nat,A)),combc(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),inverse_divide(A)),B)),C))) # label(fact_4035_summable__divide) # label(axiom). [clausify(4526)]. 28.50/28.40 5281 -real_normed_field(A) | -banach(A) | exp(A) = ti(fun(A,A),exp(A)) # label(tsy_c_Transcendental_Oexp_res) # label(axiom). [clausify(4569)]. 28.50/28.40 Derived: -banach(real) | exp(real) = ti(fun(real,real),exp(real)). [resolve(5281,a,5180,a)]. 28.62/28.55 Derived: -banach(complex) | exp(complex) = ti(fun(complex,complex),exp(complex)). [resolve(5281,a,5224,a)]. 28.62/28.55 5282 -banach(A) | -real_normed_field(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),B)),hAPP(A,fun(nat,A),power_power(A),C)))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(A,real,norm_norm(A),D)),hAPP(A,real,norm_norm(A),C))) | hBOOL(hAPP(fun(nat,A),bool,summable(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),B)),hAPP(A,fun(nat,A),power_power(A),D)))) # label(fact_4051_powser__inside) # label(axiom). [clausify(4680)]. 28.62/28.55 Derived: -banach(real) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),A)),hAPP(real,fun(nat,real),power_power(real),B)))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,norm_norm(real),C)),hAPP(real,real,norm_norm(real),B))) | hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),A)),hAPP(real,fun(nat,real),power_power(real),C)))). [resolve(5282,b,5180,a)]. 28.62/28.55 Derived: -banach(complex) | -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),A)),hAPP(complex,fun(nat,complex),power_power(complex),B)))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(complex,real,norm_norm(complex),C)),hAPP(complex,real,norm_norm(complex),B))) | hBOOL(hAPP(fun(nat,complex),bool,summable(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),A)),hAPP(complex,fun(nat,complex),power_power(complex),C)))). [resolve(5282,b,5224,a)]. 28.62/28.55 5283 -real_normed_field(A) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(A,real,norm_norm(A),B)),one_one(real))) | hBOOL(hAPP(fun(nat,A),bool,summable(A),hAPP(A,fun(nat,A),power_power(A),B))) # label(fact_4052_summable__geometric) # label(axiom). [clausify(4721)]. 28.62/28.55 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,norm_norm(real),A)),one_one(real))) | hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(real,fun(nat,real),power_power(real),A))). [resolve(5283,a,5180,a)]. 28.62/28.55 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(complex,real,norm_norm(complex),A)),one_one(real))) | hBOOL(hAPP(fun(nat,complex),bool,summable(complex),hAPP(complex,fun(nat,complex),power_power(complex),A))). [resolve(5283,a,5224,a)]. 28.62/28.55 5284 -real_normed_field(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),B)) | hBOOL(hAPP(filter(nat),bool,hAPP(A,fun(filter(nat),bool),hAPP(fun(nat,A),fun(A,fun(filter(nat),bool)),tendsto(nat,A),B),zero_zero(A)),sequentially)) # label(fact_4514_summable__LIMSEQ__zero) # label(axiom). [clausify(4735)]. 30.53/30.47 Derived: -hBOOL(hAPP(fun(nat,real),bool,summable(real),A)) | hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A),zero_zero(real)),sequentially)). [resolve(5284,a,5180,a)]. 30.53/30.47 Derived: -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),A)) | hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),A),zero_zero(complex)),sequentially)). [resolve(5284,a,5224,a)]. 30.53/30.47 5285 real_normed_algebra(complex) # label(arity_Complex_Ocomplex___RealVector_Oreal__normed__algebra) # label(axiom). [assumption]. 30.53/30.47 5286 -real_normed_algebra(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)) # label(fact_1244_mult__right_Oadd) # label(axiom). [clausify(15)]. 30.53/30.47 5287 -real_normed_algebra(A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),f42(A,B))) # label(fact_2107_mult__right_Ononneg__bounded) # label(axiom). [clausify(166)]. 30.53/30.47 5288 -real_normed_algebra(A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(A,real,norm_norm(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(A,real,norm_norm(A),C)),f42(A,B)))) # label(fact_2107_mult__right_Ononneg__bounded) # label(axiom). [clausify(166)]. 30.53/30.47 5289 -real_normed_algebra(A) | -hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),hAPP(A,fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(A,fun(nat,A)),combc(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),B)),D)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D))) # label(fact_4110_mult__left_Osums) # label(axiom). [clausify(173)]. 30.53/30.47 5290 -real_normed_algebra(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),B)) | hBOOL(hAPP(fun(nat,A),bool,summable(A),hAPP(A,fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(A,fun(nat,A)),combc(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),B)),C))) # label(fact_4029_mult__left_Osummable) # label(axiom). [clausify(193)]. 30.53/30.47 5291 -real_normed_algebra(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),B)) | hAPP(fun(nat,A),A,suminf(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(A,A),fun(fun(nat,A),fun(nat,A)),combb(A,A,nat),hAPP(A,fun(A,A),times_times(A),C)),B)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),hAPP(fun(nat,A),A,suminf(A),B)) # label(fact_4016_suminf__mult) # label(axiom). [clausify(273)]. 30.53/30.47 5292 -real_normed_algebra(A) | -hBOOL(hAPP(filter(nat),bool,hAPP(A,fun(filter(nat),bool),hAPP(fun(nat,A),fun(A,fun(filter(nat),bool)),tendsto(nat,A),B),C),sequentially)) | -hBOOL(hAPP(filter(nat),bool,hAPP(A,fun(filter(nat),bool),hAPP(fun(nat,A),fun(A,fun(filter(nat),bool)),tendsto(nat,A),D),E),sequentially)) | hBOOL(hAPP(filter(nat),bool,hAPP(A,fun(filter(nat),bool),hAPP(fun(nat,A),fun(A,fun(filter(nat),bool)),tendsto(nat,A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),B)),D)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),E)),sequentially)) # label(fact_4484_LIMSEQ__mult) # label(axiom). [clausify(472)]. 30.53/30.47 5293 -real_normed_algebra(A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(A,real,norm_norm(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(A,real,norm_norm(A),C)),f103(A,B)))) # label(fact_2051_mult__right_Opos__bounded) # label(axiom). [clausify(504)]. 30.53/30.47 5294 -real_normed_algebra(A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),f103(A,B))) # label(fact_2051_mult__right_Opos__bounded) # label(axiom). [clausify(504)]. 30.53/30.47 5295 -real_normed_algebra(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D)) # label(fact_1248_mult_Odiff__left) # label(axiom). [clausify(505)]. 30.53/30.47 5296 -real_normed_algebra(A) | hBOOL(hAPP(filter(A),bool,hAPP(A,fun(filter(A),bool),hAPP(fun(A,A),fun(A,fun(filter(A),bool)),tendsto(A,A),hAPP(A,fun(A,A),times_times(A),B)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),hAPP(A,filter(A),at(A),C))) # label(fact_4587_mult__right_Ocont) # label(axiom). [clausify(540)]. 30.53/30.47 5297 -real_normed_algebra(A) | -hBOOL(hAPP(filter(B),bool,hAPP(A,fun(filter(B),bool),hAPP(fun(B,A),fun(A,fun(filter(B),bool)),tendsto(B,A),C),D),E)) | hBOOL(hAPP(filter(B),bool,hAPP(A,fun(filter(B),bool),hAPP(fun(B,A),fun(A,fun(filter(B),bool)),tendsto(B,A),hAPP(fun(B,A),fun(B,A),hAPP(fun(A,A),fun(fun(B,A),fun(B,A)),combb(A,A,B),hAPP(A,fun(A,A),times_times(A),F)),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),F),D)),E)) # label(fact_4479_mult__right_Otendsto) # label(axiom). [clausify(550)]. 30.53/30.47 5298 -real_normed_algebra(A) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),zero_zero(A)),B) # label(fact_1239_mult_Ozero__left) # label(axiom). [clausify(656)]. 30.53/30.47 5299 -real_normed_algebra(A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(A,real,norm_norm(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(A,real,norm_norm(A),B)),hAPP(A,real,norm_norm(A),C))),f130(A)))) # label(fact_2050_mult_Opos__bounded) # label(axiom). [clausify(667)]. 30.53/30.47 5300 -real_normed_algebra(A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),f130(A))) # label(fact_2050_mult_Opos__bounded) # label(axiom). [clausify(667)]. 30.53/30.47 5301 -real_normed_algebra(A) | hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),C)),D) # label(fact_4722_mult_OscaleR__left) # label(axiom). [clausify(709)]. 30.53/30.47 5302 -real_normed_algebra(A) | hBOOL(hAPP(filter(A),bool,hAPP(A,fun(filter(A),bool),hAPP(fun(A,A),fun(A,fun(filter(A),bool)),tendsto(A,A),hAPP(A,fun(A,A),hAPP(fun(A,fun(A,A)),fun(A,fun(A,A)),combc(A,A,A),times_times(A)),B)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),B)),hAPP(A,filter(A),at(A),C))) # label(fact_4588_mult__left_Ocont) # label(axiom). [clausify(804)]. 30.53/30.47 5303 -real_normed_algebra(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D)) # label(fact_1246_mult__left_Oadd) # label(axiom). [clausify(817)]. 30.53/30.47 5304 -real_normed_algebra(A) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),zero_zero(A)),B) # label(fact_1240_mult__left_Ozero) # label(axiom). [clausify(870)]. 30.53/30.47 5305 -real_normed_algebra(A) | -hBOOL(hAPP(fun(nat,A),bool,cauchy(A),B)) | hBOOL(hAPP(fun(nat,A),bool,cauchy(A),hAPP(A,fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(A,fun(nat,A)),combc(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),B)),C))) # label(fact_4128_mult__left_OCauchy) # label(axiom). [clausify(904)]. 30.53/30.47 5306 -topological_space(A) | -power(B) | -real_normed_algebra(B) | -hBOOL(hAPP(A,bool,hAPP(fun(A,B),fun(A,bool),isCont(A,B),C),D)) | hBOOL(hAPP(A,bool,hAPP(fun(A,B),fun(A,bool),isCont(A,B),hAPP(nat,fun(A,B),hAPP(fun(A,fun(nat,B)),fun(nat,fun(A,B)),combc(A,nat,B),hAPP(fun(A,B),fun(A,fun(nat,B)),hAPP(fun(B,fun(nat,B)),fun(fun(A,B),fun(A,fun(nat,B))),combb(B,fun(nat,B),A),power_power(B)),C)),E)),D)) # label(fact_4917_isCont__power) # label(axiom). [clausify(953)]. 30.53/30.47 5307 -real_normed_algebra(A) | -hBOOL(hAPP(filter(nat),bool,hAPP(A,fun(filter(nat),bool),hAPP(fun(nat,A),fun(A,fun(filter(nat),bool)),tendsto(nat,A),B),C),sequentially)) | hBOOL(hAPP(filter(nat),bool,hAPP(A,fun(filter(nat),bool),hAPP(fun(nat,A),fun(A,fun(filter(nat),bool)),tendsto(nat,A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(A,A),fun(fun(nat,A),fun(nat,A)),combb(A,A,nat),hAPP(A,fun(A,A),times_times(A),D)),B)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),C)),sequentially)) # label(fact_4482_mult__right_OLIMSEQ) # label(axiom). [clausify(1143)]. 30.53/30.47 5308 -real_normed_algebra(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),B)) | hAPP(fun(nat,A),A,suminf(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(A,A),fun(fun(nat,A),fun(nat,A)),combb(A,A,nat),hAPP(A,fun(A,A),times_times(A),C)),B)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),hAPP(fun(nat,A),A,suminf(A),B)) # label(fact_4014_mult__right_Osuminf) # label(axiom). [clausify(1174)]. 30.53/30.47 5309 -real_normed_algebra(A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(A,real,norm_norm(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(A,real,norm_norm(A),B)),f223(A,C)))) # label(fact_2110_mult__left_Obounded) # label(axiom). [clausify(1220)]. 30.53/30.47 5310 -real_normed_algebra(A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(A,real,norm_norm(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(A,real,norm_norm(A),B)),f253(A,C)))) # label(fact_2106_mult__left_Ononneg__bounded) # label(axiom). [clausify(1468)]. 30.53/30.47 5311 -real_normed_algebra(A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),f253(A,B))) # label(fact_2106_mult__left_Ononneg__bounded) # label(axiom). [clausify(1468)]. 30.53/30.47 5312 -real_normed_algebra(A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(A,real,norm_norm(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(A,real,norm_norm(A),B)),hAPP(A,real,norm_norm(A),C)))) # label(fact_2004_norm__mult__ineq) # label(axiom). [clausify(1564)]. 30.53/30.47 5313 -topological_space(A) | -real_normed_algebra(B) | -hBOOL(hAPP(A,bool,hAPP(fun(A,B),fun(A,bool),isCont(A,B),C),D)) | -hBOOL(hAPP(A,bool,hAPP(fun(A,B),fun(A,bool),isCont(A,B),E),D)) | hBOOL(hAPP(A,bool,hAPP(fun(A,B),fun(A,bool),isCont(A,B),hAPP(fun(A,B),fun(A,B),hAPP(fun(A,fun(B,B)),fun(fun(A,B),fun(A,B)),combs(A,B,B),hAPP(fun(A,B),fun(A,fun(B,B)),hAPP(fun(B,fun(B,B)),fun(fun(A,B),fun(A,fun(B,B))),combb(B,fun(B,B),A),times_times(B)),C)),E)),D)) # label(fact_4912_isCont__mult) # label(axiom). [clausify(1606)]. 30.53/30.47 5314 -topological_space(A) | -real_normed_algebra(B) | -hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),C),D),hAPP(A,filter(A),at(A),E))) | hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),hAPP(fun(A,B),fun(A,B),hAPP(fun(B,B),fun(fun(A,B),fun(A,B)),combb(B,B,A),hAPP(B,fun(B,B),times_times(B),F)),C)),hAPP(B,B,hAPP(B,fun(B,B),times_times(B),F),D)),hAPP(A,filter(A),at(A),E))) # label(fact_4589_mult__right_OLIM) # label(axiom). [clausify(1698)]. 30.53/30.47 5315 -real_normed_algebra(A) | hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),E)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),C),E))),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),D)),E))),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),C),E))) # label(fact_1238_mult_Oprod__diff__prod) # label(axiom). [clausify(1707)]. 30.53/30.47 5316 -real_normed_algebra(A) | hBOOL(hAPP(A,bool,hAPP(fun(A,A),fun(A,bool),isCont(A,A),hAPP(A,fun(A,A),times_times(A),B)),C)) # label(fact_4914_mult__right_OisCont) # label(axiom). [clausify(1723)]. 30.53/30.47 5317 -power(A) | -real_normed_algebra(A) | -hBOOL(hAPP(filter(nat),bool,hAPP(A,fun(filter(nat),bool),hAPP(fun(nat,A),fun(A,fun(filter(nat),bool)),tendsto(nat,A),B),C),sequentially)) | hBOOL(hAPP(filter(nat),bool,hAPP(A,fun(filter(nat),bool),hAPP(fun(nat,A),fun(A,fun(filter(nat),bool)),tendsto(nat,A),hAPP(nat,fun(nat,A),hAPP(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),combc(nat,nat,A),hAPP(fun(nat,A),fun(nat,fun(nat,A)),hAPP(fun(A,fun(nat,A)),fun(fun(nat,A),fun(nat,fun(nat,A))),combb(A,fun(nat,A),nat),power_power(A)),B)),D)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),D)),sequentially)) # label(fact_4490_LIMSEQ__pow) # label(axiom). [clausify(1886)]. 30.53/30.48 5318 -real_normed_algebra(A) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),hAPP(fun(B,A),fun(B,A),hAPP(fun(A,A),fun(fun(B,A),fun(B,A)),combb(A,A,B),hAPP(A,fun(A,A),times_times(A),C)),D)),E) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),D),E)) # label(fact_4213_mult__right_Osetsum) # label(axiom). [clausify(1904)]. 30.53/30.48 5319 -real_normed_algebra(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D)) # label(fact_1247_mult__left_Odiff) # label(axiom). [clausify(1930)]. 30.53/30.48 5320 -real_normed_algebra(A) | -hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(A,A),fun(fun(nat,A),fun(nat,A)),combb(A,A,nat),hAPP(A,fun(A,A),times_times(A),D)),B)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),C))) # label(fact_4109_mult__right_Osums) # label(axiom). [clausify(1936)]. 30.53/30.48 5321 -real_normed_algebra(A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(A,real,norm_norm(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(A,real,norm_norm(A),B)),f328(A,C)))) # label(fact_2052_mult__left_Opos__bounded) # label(axiom). [clausify(1992)]. 30.53/30.48 5322 -real_normed_algebra(A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),f328(A,B))) # label(fact_2052_mult__left_Opos__bounded) # label(axiom). [clausify(1992)]. 30.53/30.48 5323 -real_normed_algebra(A) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(A,real,norm_norm(A),B)),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(A,real,norm_norm(A),D)),E)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(A,real,norm_norm(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),E))) # label(fact_2005_norm__mult__less) # label(axiom). [clausify(2098)]. 30.53/30.48 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),B),C)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),C)). [resolve(5285,a,5286,a)]. 30.53/30.48 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),f42(complex,A))). [resolve(5285,a,5287,a)]. 30.53/30.48 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),B)),f42(complex,A)))). [resolve(5285,a,5288,a)]. 30.53/30.48 Derived: -hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),A),B)) | hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),hAPP(complex,fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(complex,fun(nat,complex)),combc(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),A)),C)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),B),C))). [resolve(5285,a,5289,a)]. 30.60/30.49 Derived: -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),A)) | hBOOL(hAPP(fun(nat,complex),bool,summable(complex),hAPP(complex,fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(complex,fun(nat,complex)),combc(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),A)),B))). [resolve(5285,a,5290,a)]. 30.60/30.49 Derived: -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),A)) | hAPP(fun(nat,complex),complex,suminf(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(complex,complex),fun(fun(nat,complex),fun(nat,complex)),combb(complex,complex,nat),hAPP(complex,fun(complex,complex),times_times(complex),B)),A)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),B),hAPP(fun(nat,complex),complex,suminf(complex),A)). [resolve(5285,a,5291,a)]. 30.60/30.49 Derived: -hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),A),B),sequentially)) | -hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),C),D),sequentially)) | hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),A)),C)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),B),D)),sequentially)). [resolve(5285,a,5292,a)]. 30.60/30.49 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),B)),f103(complex,A)))). [resolve(5285,a,5293,a)]. 30.60/30.49 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),f103(complex,A))). [resolve(5285,a,5294,a)]. 30.60/30.49 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),A),B)),C) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),C)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),B),C)). [resolve(5285,a,5295,a)]. 30.60/30.49 Derived: hBOOL(hAPP(filter(complex),bool,hAPP(complex,fun(filter(complex),bool),hAPP(fun(complex,complex),fun(complex,fun(filter(complex),bool)),tendsto(complex,complex),hAPP(complex,fun(complex,complex),times_times(complex),A)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)),hAPP(complex,filter(complex),at(complex),B))). [resolve(5285,a,5296,a)]. 30.60/30.49 Derived: -hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),B),C),D)) | hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),hAPP(fun(A,complex),fun(A,complex),hAPP(fun(complex,complex),fun(fun(A,complex),fun(A,complex)),combb(complex,complex,A),hAPP(complex,fun(complex,complex),times_times(complex),E)),B)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),E),C)),D)). [resolve(5285,a,5297,a)]. 30.60/30.49 Derived: zero_zero(complex) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),zero_zero(complex)),A). [resolve(5285,a,5298,a)]. 30.60/30.50 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),A)),hAPP(complex,real,norm_norm(complex),B))),f130(complex)))). [resolve(5285,a,5299,a)]. 30.60/30.50 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),f130(complex))). [resolve(5285,a,5300,a)]. 30.60/30.50 Derived: hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),B),C)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),B)),C). [resolve(5285,a,5301,a)]. 30.60/30.50 Derived: hBOOL(hAPP(filter(complex),bool,hAPP(complex,fun(filter(complex),bool),hAPP(fun(complex,complex),fun(complex,fun(filter(complex),bool)),tendsto(complex,complex),hAPP(complex,fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(complex,fun(complex,complex)),combc(complex,complex,complex),times_times(complex)),A)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),B),A)),hAPP(complex,filter(complex),at(complex),B))). [resolve(5285,a,5302,a)]. 30.60/30.50 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),B)),C) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),C)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),B),C)). [resolve(5285,a,5303,a)]. 30.60/30.50 Derived: -hBOOL(hAPP(fun(nat,complex),bool,cauchy(complex),A)) | hBOOL(hAPP(fun(nat,complex),bool,cauchy(complex),hAPP(complex,fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(complex,fun(nat,complex)),combc(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),A)),B))). [resolve(5285,a,5305,a)]. 30.60/30.50 Derived: -topological_space(A) | -power(complex) | -hBOOL(hAPP(A,bool,hAPP(fun(A,complex),fun(A,bool),isCont(A,complex),B),C)) | hBOOL(hAPP(A,bool,hAPP(fun(A,complex),fun(A,bool),isCont(A,complex),hAPP(nat,fun(A,complex),hAPP(fun(A,fun(nat,complex)),fun(nat,fun(A,complex)),combc(A,nat,complex),hAPP(fun(A,complex),fun(A,fun(nat,complex)),hAPP(fun(complex,fun(nat,complex)),fun(fun(A,complex),fun(A,fun(nat,complex))),combb(complex,fun(nat,complex),A),power_power(complex)),B)),D)),C)). [resolve(5285,a,5306,c)]. 30.60/30.50 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),A)),f223(complex,B)))). [resolve(5285,a,5309,a)]. 30.60/30.50 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),A)),f253(complex,B)))). [resolve(5285,a,5310,a)]. 30.60/30.50 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),f253(complex,A))). [resolve(5285,a,5311,a)]. 30.60/30.50 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),A)),hAPP(complex,real,norm_norm(complex),B)))). [resolve(5285,a,5312,a)]. 30.60/30.50 Derived: -topological_space(A) | -hBOOL(hAPP(A,bool,hAPP(fun(A,complex),fun(A,bool),isCont(A,complex),B),C)) | -hBOOL(hAPP(A,bool,hAPP(fun(A,complex),fun(A,bool),isCont(A,complex),D),C)) | hBOOL(hAPP(A,bool,hAPP(fun(A,complex),fun(A,bool),isCont(A,complex),hAPP(fun(A,complex),fun(A,complex),hAPP(fun(A,fun(complex,complex)),fun(fun(A,complex),fun(A,complex)),combs(A,complex,complex),hAPP(fun(A,complex),fun(A,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(A,complex),fun(A,fun(complex,complex))),combb(complex,fun(complex,complex),A),times_times(complex)),B)),D)),C)). [resolve(5285,a,5313,b)]. 30.60/30.51 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),D)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),A),C)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),B),D))),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),A),C)),D))),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),B),D))). [resolve(5285,a,5315,a)]. 30.60/30.51 Derived: hBOOL(hAPP(complex,bool,hAPP(fun(complex,complex),fun(complex,bool),isCont(complex,complex),hAPP(complex,fun(complex,complex),times_times(complex),A)),B)). [resolve(5285,a,5316,a)]. 30.60/30.51 Derived: -power(complex) | -hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),A),B),sequentially)) | hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),hAPP(nat,fun(nat,complex),hAPP(fun(nat,fun(nat,complex)),fun(nat,fun(nat,complex)),combc(nat,nat,complex),hAPP(fun(nat,complex),fun(nat,fun(nat,complex)),hAPP(fun(complex,fun(nat,complex)),fun(fun(nat,complex),fun(nat,fun(nat,complex))),combb(complex,fun(nat,complex),nat),power_power(complex)),A)),C)),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),B),C)),sequentially)). [resolve(5285,a,5317,b)]. 30.60/30.51 Derived: hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co1399186613setsum(A,complex),hAPP(fun(A,complex),fun(A,complex),hAPP(fun(complex,complex),fun(fun(A,complex),fun(A,complex)),combb(complex,complex,A),hAPP(complex,fun(complex,complex),times_times(complex),B)),C)),D) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),B),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co1399186613setsum(A,complex),C),D)). [resolve(5285,a,5318,a)]. 30.60/30.51 Derived: -hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),A),B)) | hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(complex,complex),fun(fun(nat,complex),fun(nat,complex)),combb(complex,complex,nat),hAPP(complex,fun(complex,complex),times_times(complex),C)),A)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),B))). [resolve(5285,a,5320,a)]. 30.60/30.51 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),A)),f328(complex,B)))). [resolve(5285,a,5321,a)]. 30.60/30.51 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),f328(complex,A))). [resolve(5285,a,5322,a)]. 30.60/30.51 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(complex,real,norm_norm(complex),A)),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(complex,real,norm_norm(complex),C)),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),C))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),D))). [resolve(5285,a,5323,a)]. 30.91/30.81 5324 -real_normed_algebra(A) | hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),D)) # label(fact_4720_mult__right_OscaleR) # label(axiom). [clausify(2316)]. 30.91/30.81 Derived: hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),B),C)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),B),hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),C)). [resolve(5324,a,5285,a)]. 30.91/30.81 5325 -real_normed_algebra(A) | -topological_space(B) | -hBOOL(hAPP(filter(B),bool,hAPP(A,fun(filter(B),bool),hAPP(fun(B,A),fun(A,fun(filter(B),bool)),tendsto(B,A),C),D),hAPP(B,filter(B),at(B),E))) | hBOOL(hAPP(filter(B),bool,hAPP(A,fun(filter(B),bool),hAPP(fun(B,A),fun(A,fun(filter(B),bool)),tendsto(B,A),hAPP(A,fun(B,A),hAPP(fun(B,fun(A,A)),fun(A,fun(B,A)),combc(B,A,A),hAPP(fun(B,A),fun(B,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),combb(A,fun(A,A),B),times_times(A)),C)),F)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),F)),hAPP(B,filter(B),at(B),E))) # label(fact_4590_mult__left_OLIM) # label(axiom). [clausify(2375)]. 30.91/30.81 Derived: -topological_space(A) | -hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),B),C),hAPP(A,filter(A),at(A),D))) | hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),hAPP(complex,fun(A,complex),hAPP(fun(A,fun(complex,complex)),fun(complex,fun(A,complex)),combc(A,complex,complex),hAPP(fun(A,complex),fun(A,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(A,complex),fun(A,fun(complex,complex))),combb(complex,fun(complex,complex),A),times_times(complex)),B)),E)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),E)),hAPP(A,filter(A),at(A),D))). [resolve(5325,a,5285,a)]. 30.91/30.81 5326 -topological_space(A) | -power(B) | -real_normed_algebra(B) | -hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),C),D),hAPP(A,filter(A),at(A),E))) | hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),hAPP(nat,fun(A,B),hAPP(fun(A,fun(nat,B)),fun(nat,fun(A,B)),combc(A,nat,B),hAPP(fun(A,B),fun(A,fun(nat,B)),hAPP(fun(B,fun(nat,B)),fun(fun(A,B),fun(A,fun(nat,B))),combb(B,fun(nat,B),A),power_power(B)),C)),F)),hAPP(nat,B,hAPP(B,fun(nat,B),power_power(B),D),F)),hAPP(A,filter(A),at(A),E))) # label(fact_4585_LIM__power) # label(axiom). [clausify(2794)]. 30.91/30.81 Derived: -topological_space(A) | -power(complex) | -hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),B),C),hAPP(A,filter(A),at(A),D))) | hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),hAPP(nat,fun(A,complex),hAPP(fun(A,fun(nat,complex)),fun(nat,fun(A,complex)),combc(A,nat,complex),hAPP(fun(A,complex),fun(A,fun(nat,complex)),hAPP(fun(complex,fun(nat,complex)),fun(fun(A,complex),fun(A,fun(nat,complex))),combb(complex,fun(nat,complex),A),power_power(complex)),B)),E)),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),C),E)),hAPP(A,filter(A),at(A),D))). [resolve(5326,c,5285,a)]. 30.91/30.81 5327 -real_normed_algebra(A) | -topological_space(B) | -hBOOL(hAPP(filter(B),bool,hAPP(A,fun(filter(B),bool),hAPP(fun(B,A),fun(A,fun(filter(B),bool)),tendsto(B,A),C),zero_zero(A)),hAPP(B,filter(B),at(B),D))) | hBOOL(hAPP(filter(B),bool,hAPP(A,fun(filter(B),bool),hAPP(fun(B,A),fun(A,fun(filter(B),bool)),tendsto(B,A),hAPP(fun(B,A),fun(B,A),hAPP(fun(A,A),fun(fun(B,A),fun(B,A)),combb(A,A,B),hAPP(A,fun(A,A),times_times(A),E)),C)),zero_zero(A)),hAPP(B,filter(B),at(B),D))) # label(fact_4602_LIM__mult__right__zero) # label(axiom). [clausify(2817)]. 31.02/30.91 Derived: -topological_space(A) | -hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),B),zero_zero(complex)),hAPP(A,filter(A),at(A),C))) | hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),hAPP(fun(A,complex),fun(A,complex),hAPP(fun(complex,complex),fun(fun(A,complex),fun(A,complex)),combb(complex,complex,A),hAPP(complex,fun(complex,complex),times_times(complex),D)),B)),zero_zero(complex)),hAPP(A,filter(A),at(A),C))). [resolve(5327,a,5285,a)]. 31.02/30.91 5328 -real_normed_algebra(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)) # label(fact_1243_mult_Oadd__right) # label(axiom). [clausify(2894)]. 31.02/30.91 5329 -real_normed_algebra(A) | -banach(A) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,A),fun(nat,real),hAPP(fun(A,real),fun(fun(nat,A),fun(nat,real)),combb(A,real,nat),norm_norm(A)),B))) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,A),fun(nat,real),hAPP(fun(A,real),fun(fun(nat,A),fun(nat,real)),combb(A,real,nat),norm_norm(A)),C))) | hAPP(fun(nat,A),A,suminf(A),hAPP(fun(nat,fun(nat,bool)),fun(nat,A),hAPP(fun(nat,fun(fun(nat,bool),A)),fun(fun(nat,fun(nat,bool)),fun(nat,A)),combs(nat,fun(nat,bool),A),hAPP(fun(nat,fun(nat,A)),fun(nat,fun(fun(nat,bool),A)),hAPP(fun(fun(nat,A),fun(fun(nat,bool),A)),fun(fun(nat,fun(nat,A)),fun(nat,fun(fun(nat,bool),A))),combb(fun(nat,A),fun(fun(nat,bool),A),nat),big_co1399186613setsum(nat,A)),hAPP(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),hAPP(fun(fun(nat,A),fun(nat,A)),fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A))),combb(fun(nat,A),fun(nat,A),nat),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),B))),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,A)),hAPP(fun(fun(nat,nat),fun(nat,A)),fun(fun(nat,fun(nat,nat)),fun(nat,fun(nat,A))),combb(fun(nat,nat),fun(nat,A),nat),hAPP(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),combb(nat,A,nat),C)),minus_minus(nat))))),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)))) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(fun(nat,A),A,suminf(A),B)),hAPP(fun(nat,A),A,suminf(A),C)) # label(fact_4646_Cauchy__product) # label(axiom). [clausify(2931)]. 31.02/30.91 Derived: -banach(complex) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),norm_norm(complex)),A))) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),norm_norm(complex)),B))) | hAPP(fun(nat,complex),complex,suminf(complex),hAPP(fun(nat,fun(nat,bool)),fun(nat,complex),hAPP(fun(nat,fun(fun(nat,bool),complex)),fun(fun(nat,fun(nat,bool)),fun(nat,complex)),combs(nat,fun(nat,bool),complex),hAPP(fun(nat,fun(nat,complex)),fun(nat,fun(fun(nat,bool),complex)),hAPP(fun(fun(nat,complex),fun(fun(nat,bool),complex)),fun(fun(nat,fun(nat,complex)),fun(nat,fun(fun(nat,bool),complex))),combb(fun(nat,complex),fun(fun(nat,bool),complex),nat),big_co1399186613setsum(nat,complex)),hAPP(fun(nat,fun(nat,complex)),fun(nat,fun(nat,complex)),hAPP(fun(fun(nat,complex),fun(nat,complex)),fun(fun(nat,fun(nat,complex)),fun(nat,fun(nat,complex))),combb(fun(nat,complex),fun(nat,complex),nat),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),A))),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,complex)),hAPP(fun(fun(nat,nat),fun(nat,complex)),fun(fun(nat,fun(nat,nat)),fun(nat,fun(nat,complex))),combb(fun(nat,nat),fun(nat,complex),nat),hAPP(fun(nat,complex),fun(fun(nat,nat),fun(nat,complex)),combb(nat,complex,nat),B)),minus_minus(nat))))),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(fun(nat,complex),complex,suminf(complex),A)),hAPP(fun(nat,complex),complex,suminf(complex),B)). [resolve(5329,a,5285,a)]. 31.20/31.15 5330 -real_normed_algebra(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),B)) | hBOOL(hAPP(fun(nat,A),bool,summable(A),hAPP(A,fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(A,fun(nat,A)),combc(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),B)),C))) # label(fact_4027_summable__mult2) # label(axiom). [clausify(2934)]. 31.20/31.15 5331 -topological_space(A) | -real_normed_algebra(B) | -hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),C),zero_zero(B)),hAPP(A,filter(A),at(A),D))) | hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),hAPP(B,fun(A,B),hAPP(fun(A,fun(B,B)),fun(B,fun(A,B)),combc(A,B,B),hAPP(fun(A,B),fun(A,fun(B,B)),hAPP(fun(B,fun(B,B)),fun(fun(A,B),fun(A,fun(B,B))),combb(B,fun(B,B),A),times_times(B)),C)),E)),zero_zero(B)),hAPP(A,filter(A),at(A),D))) # label(fact_4600_LIM__mult__left__zero) # label(axiom). [clausify(3151)]. 31.20/31.15 Derived: -topological_space(A) | -hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),B),zero_zero(complex)),hAPP(A,filter(A),at(A),C))) | hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),hAPP(complex,fun(A,complex),hAPP(fun(A,fun(complex,complex)),fun(complex,fun(A,complex)),combc(A,complex,complex),hAPP(fun(A,complex),fun(A,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(A,complex),fun(A,fun(complex,complex))),combb(complex,fun(complex,complex),A),times_times(complex)),B)),D)),zero_zero(complex)),hAPP(A,filter(A),at(A),C))). [resolve(5331,b,5285,a)]. 31.20/31.15 5332 -real_normed_algebra(A) | -hBOOL(hAPP(filter(B),bool,hAPP(A,fun(filter(B),bool),hAPP(fun(B,A),fun(A,fun(filter(B),bool)),tendsto(B,A),C),D),E)) | hBOOL(hAPP(filter(B),bool,hAPP(A,fun(filter(B),bool),hAPP(fun(B,A),fun(A,fun(filter(B),bool)),tendsto(B,A),hAPP(A,fun(B,A),hAPP(fun(B,fun(A,A)),fun(A,fun(B,A)),combc(B,A,A),hAPP(fun(B,A),fun(B,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),combb(A,fun(A,A),B),times_times(A)),C)),F)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),F)),E)) # label(fact_4480_mult__left_Otendsto) # label(axiom). [clausify(3248)]. 31.20/31.15 Derived: -hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),B),C),D)) | hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),hAPP(complex,fun(A,complex),hAPP(fun(A,fun(complex,complex)),fun(complex,fun(A,complex)),combc(A,complex,complex),hAPP(fun(A,complex),fun(A,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(A,complex),fun(A,fun(complex,complex))),combb(complex,fun(complex,complex),A),times_times(complex)),B)),E)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),E)),D)). [resolve(5332,a,5285,a)]. 31.20/31.15 5333 -real_normed_algebra(A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(A,real,norm_norm(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(A,real,norm_norm(A),C)),f561(A,B)))) # label(fact_2111_mult__right_Obounded) # label(axiom). [clausify(3311)]. 31.30/31.18 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),B)),f561(complex,A)))). [resolve(5333,a,5285,a)]. 31.30/31.18 5334 -real_normed_algebra(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),B)) | hBOOL(hAPP(fun(nat,A),bool,summable(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(A,A),fun(fun(nat,A),fun(nat,A)),combb(A,A,nat),hAPP(A,fun(A,A),times_times(A),C)),B))) # label(fact_4026_summable__mult) # label(axiom). [clausify(3350)]. 31.30/31.18 Derived: -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),A)) | hBOOL(hAPP(fun(nat,complex),bool,summable(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(complex,complex),fun(fun(nat,complex),fun(nat,complex)),combb(complex,complex,nat),hAPP(complex,fun(complex,complex),times_times(complex),B)),A))). [resolve(5334,a,5285,a)]. 31.30/31.18 5335 -real_normed_algebra(A) | hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),D)) # label(fact_4718_mult_OscaleR__right) # label(axiom). [clausify(3351)]. 31.30/31.18 5336 -real_normed_algebra(A) | -banach(A) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,A),fun(nat,real),hAPP(fun(A,real),fun(fun(nat,A),fun(nat,real)),combb(A,real,nat),norm_norm(A)),B))) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,A),fun(nat,real),hAPP(fun(A,real),fun(fun(nat,A),fun(nat,real)),combb(A,real,nat),norm_norm(A)),C))) | hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),hAPP(fun(nat,fun(nat,bool)),fun(nat,A),hAPP(fun(nat,fun(fun(nat,bool),A)),fun(fun(nat,fun(nat,bool)),fun(nat,A)),combs(nat,fun(nat,bool),A),hAPP(fun(nat,fun(nat,A)),fun(nat,fun(fun(nat,bool),A)),hAPP(fun(fun(nat,A),fun(fun(nat,bool),A)),fun(fun(nat,fun(nat,A)),fun(nat,fun(fun(nat,bool),A))),combb(fun(nat,A),fun(fun(nat,bool),A),nat),big_co1399186613setsum(nat,A)),hAPP(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),hAPP(fun(fun(nat,A),fun(nat,A)),fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A))),combb(fun(nat,A),fun(nat,A),nat),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),B))),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,A)),hAPP(fun(fun(nat,nat),fun(nat,A)),fun(fun(nat,fun(nat,nat)),fun(nat,fun(nat,A))),combb(fun(nat,nat),fun(nat,A),nat),hAPP(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),combb(nat,A,nat),C)),minus_minus(nat))))),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)))),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(fun(nat,A),A,suminf(A),B)),hAPP(fun(nat,A),A,suminf(A),C)))) # label(fact_4648_Cauchy__product__sums) # label(axiom). [clausify(3364)]. 31.30/31.18 Derived: -banach(complex) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),norm_norm(complex)),A))) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),norm_norm(complex)),B))) | hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),hAPP(fun(nat,fun(nat,bool)),fun(nat,complex),hAPP(fun(nat,fun(fun(nat,bool),complex)),fun(fun(nat,fun(nat,bool)),fun(nat,complex)),combs(nat,fun(nat,bool),complex),hAPP(fun(nat,fun(nat,complex)),fun(nat,fun(fun(nat,bool),complex)),hAPP(fun(fun(nat,complex),fun(fun(nat,bool),complex)),fun(fun(nat,fun(nat,complex)),fun(nat,fun(fun(nat,bool),complex))),combb(fun(nat,complex),fun(fun(nat,bool),complex),nat),big_co1399186613setsum(nat,complex)),hAPP(fun(nat,fun(nat,complex)),fun(nat,fun(nat,complex)),hAPP(fun(fun(nat,complex),fun(nat,complex)),fun(fun(nat,fun(nat,complex)),fun(nat,fun(nat,complex))),combb(fun(nat,complex),fun(nat,complex),nat),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),A))),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,complex)),hAPP(fun(fun(nat,nat),fun(nat,complex)),fun(fun(nat,fun(nat,nat)),fun(nat,fun(nat,complex))),combb(fun(nat,nat),fun(nat,complex),nat),hAPP(fun(nat,complex),fun(fun(nat,nat),fun(nat,complex)),combb(nat,complex,nat),B)),minus_minus(nat))))),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)))),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(fun(nat,complex),complex,suminf(complex),A)),hAPP(fun(nat,complex),complex,suminf(complex),B)))). [resolve(5336,a,5285,a)]. 31.71/31.63 5337 -real_normed_algebra(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)) # label(fact_1250_mult_Odiff__right) # label(axiom). [clausify(3459)]. 31.71/31.63 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),B),C)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),C)). [resolve(5337,a,5285,a)]. 31.71/31.63 5338 -real_normed_algebra(A) | -hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(A,A),fun(fun(nat,A),fun(nat,A)),combb(A,A,nat),hAPP(A,fun(A,A),times_times(A),D)),B)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),C))) # label(fact_4107_sums__mult) # label(axiom). [clausify(3523)]. 31.71/31.63 5339 -real_normed_algebra(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),B)) | hBOOL(hAPP(fun(nat,A),bool,summable(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(A,A),fun(fun(nat,A),fun(nat,A)),combb(A,A,nat),hAPP(A,fun(A,A),times_times(A),C)),B))) # label(fact_4028_mult__right_Osummable) # label(axiom). [clausify(3538)]. 31.71/31.63 5340 -real_normed_algebra(A) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),zero_zero(A)) # label(fact_1241_mult_Ozero__right) # label(axiom). [clausify(3619)]. 31.71/31.63 Derived: zero_zero(complex) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),zero_zero(complex)). [resolve(5340,a,5285,a)]. 31.71/31.63 5341 -real_normed_algebra(A) | -hBOOL(hAPP(filter(nat),bool,hAPP(A,fun(filter(nat),bool),hAPP(fun(nat,A),fun(A,fun(filter(nat),bool)),tendsto(nat,A),B),C),sequentially)) | hBOOL(hAPP(filter(nat),bool,hAPP(A,fun(filter(nat),bool),hAPP(fun(nat,A),fun(A,fun(filter(nat),bool)),tendsto(nat,A),hAPP(A,fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(A,fun(nat,A)),combc(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),B)),D)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D)),sequentially)) # label(fact_4483_mult__left_OLIMSEQ) # label(axiom). [clausify(3630)]. 31.71/31.63 5342 -real_normed_algebra(A) | -comm_ring_1(A) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f638(B,A,C,D,E)),E)) | hBOOL(hAPP(filter(nat),bool,hAPP(A,fun(filter(nat),bool),hAPP(fun(nat,A),fun(A,fun(filter(nat),bool)),tendsto(nat,A),hAPP(fun(B,bool),fun(nat,A),hAPP(fun(nat,fun(fun(B,bool),A)),fun(fun(B,bool),fun(nat,A)),combc(nat,fun(B,bool),A),hAPP(fun(nat,fun(B,A)),fun(nat,fun(fun(B,bool),A)),hAPP(fun(fun(B,A),fun(fun(B,bool),A)),fun(fun(nat,fun(B,A)),fun(nat,fun(fun(B,bool),A))),combb(fun(B,A),fun(fun(B,bool),A),nat),big_co302546470etprod(B,A)),hAPP(fun(B,fun(nat,A)),fun(nat,fun(B,A)),combc(B,nat,A),C))),E)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),D),E)),sequentially)) # label(fact_4682_LIMSEQ__setprod) # label(axiom). [clausify(3772)]. 31.82/31.78 Derived: -comm_ring_1(complex) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f638(A,complex,B,C,D)),D)) | hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),hAPP(fun(A,bool),fun(nat,complex),hAPP(fun(nat,fun(fun(A,bool),complex)),fun(fun(A,bool),fun(nat,complex)),combc(nat,fun(A,bool),complex),hAPP(fun(nat,fun(A,complex)),fun(nat,fun(fun(A,bool),complex)),hAPP(fun(fun(A,complex),fun(fun(A,bool),complex)),fun(fun(nat,fun(A,complex)),fun(nat,fun(fun(A,bool),complex))),combb(fun(A,complex),fun(fun(A,bool),complex),nat),big_co302546470etprod(A,complex)),hAPP(fun(A,fun(nat,complex)),fun(nat,fun(A,complex)),combc(A,nat,complex),B))),D)),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),C),D)),sequentially)). [resolve(5342,a,5285,a)]. 31.82/31.78 5343 -real_normed_algebra(A) | -comm_ring_1(A) | -hBOOL(hAPP(filter(nat),bool,hAPP(A,fun(filter(nat),bool),hAPP(fun(nat,A),fun(A,fun(filter(nat),bool)),tendsto(nat,A),hAPP(B,fun(nat,A),C,f638(B,A,C,D,E))),hAPP(B,A,D,f638(B,A,C,D,E))),sequentially)) | hBOOL(hAPP(filter(nat),bool,hAPP(A,fun(filter(nat),bool),hAPP(fun(nat,A),fun(A,fun(filter(nat),bool)),tendsto(nat,A),hAPP(fun(B,bool),fun(nat,A),hAPP(fun(nat,fun(fun(B,bool),A)),fun(fun(B,bool),fun(nat,A)),combc(nat,fun(B,bool),A),hAPP(fun(nat,fun(B,A)),fun(nat,fun(fun(B,bool),A)),hAPP(fun(fun(B,A),fun(fun(B,bool),A)),fun(fun(nat,fun(B,A)),fun(nat,fun(fun(B,bool),A))),combb(fun(B,A),fun(fun(B,bool),A),nat),big_co302546470etprod(B,A)),hAPP(fun(B,fun(nat,A)),fun(nat,fun(B,A)),combc(B,nat,A),C))),E)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),D),E)),sequentially)) # label(fact_4682_LIMSEQ__setprod) # label(axiom). [clausify(3772)]. 31.82/31.78 Derived: -comm_ring_1(complex) | -hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),hAPP(A,fun(nat,complex),B,f638(A,complex,B,C,D))),hAPP(A,complex,C,f638(A,complex,B,C,D))),sequentially)) | hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),hAPP(fun(A,bool),fun(nat,complex),hAPP(fun(nat,fun(fun(A,bool),complex)),fun(fun(A,bool),fun(nat,complex)),combc(nat,fun(A,bool),complex),hAPP(fun(nat,fun(A,complex)),fun(nat,fun(fun(A,bool),complex)),hAPP(fun(fun(A,complex),fun(fun(A,bool),complex)),fun(fun(nat,fun(A,complex)),fun(nat,fun(fun(A,bool),complex))),combb(fun(A,complex),fun(fun(A,bool),complex),nat),big_co302546470etprod(A,complex)),hAPP(fun(A,fun(nat,complex)),fun(nat,fun(A,complex)),combc(A,nat,complex),B))),D)),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),C),D)),sequentially)). [resolve(5343,a,5285,a)]. 31.82/31.78 5344 -real_normed_algebra(A) | -hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),hAPP(A,fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(A,fun(nat,A)),combc(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),B)),D)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D))) # label(fact_4108_sums__mult2) # label(axiom). [clausify(3879)]. 31.82/31.78 5345 -real_normed_algebra(A) | hAPP(A,A,uminus_uminus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(A,A,uminus_uminus(A),C)) # label(fact_1909_mult_Ominus__right) # label(axiom). [clausify(3893)]. 31.82/31.78 Derived: hAPP(complex,complex,uminus_uminus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),hAPP(complex,complex,uminus_uminus(complex),B)). [resolve(5345,a,5285,a)]. 31.82/31.78 5346 -topological_space(A) | -real_normed_algebra(B) | -hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),C),D),hAPP(A,filter(A),at(A),E))) | -hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),F),V6),hAPP(A,filter(A),at(A),E))) | hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),hAPP(fun(A,B),fun(A,B),hAPP(fun(A,fun(B,B)),fun(fun(A,B),fun(A,B)),combs(A,B,B),hAPP(fun(A,B),fun(A,fun(B,B)),hAPP(fun(B,fun(B,B)),fun(fun(A,B),fun(A,fun(B,B))),combb(B,fun(B,B),A),times_times(B)),C)),F)),hAPP(B,B,hAPP(B,fun(B,B),times_times(B),D),V6)),hAPP(A,filter(A),at(A),E))) # label(fact_4591_LIM__mult) # label(axiom). [clausify(3954)]. 31.91/31.88 Derived: -topological_space(A) | -hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),B),C),hAPP(A,filter(A),at(A),D))) | -hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),E),F),hAPP(A,filter(A),at(A),D))) | hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),hAPP(fun(A,complex),fun(A,complex),hAPP(fun(A,fun(complex,complex)),fun(fun(A,complex),fun(A,complex)),combs(A,complex,complex),hAPP(fun(A,complex),fun(A,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(A,complex),fun(A,fun(complex,complex))),combb(complex,fun(complex,complex),A),times_times(complex)),B)),E)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),F)),hAPP(A,filter(A),at(A),D))). [resolve(5346,b,5285,a)]. 31.91/31.88 5347 -metric_space(A) | -real_normed_algebra(B) | -hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),C),zero_zero(B)),hAPP(A,filter(A),at(A),D))) | -hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),E),zero_zero(B)),hAPP(A,filter(A),at(A),D))) | hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),hAPP(fun(A,B),fun(A,B),hAPP(fun(A,fun(B,B)),fun(fun(A,B),fun(A,B)),combs(A,B,B),hAPP(fun(A,B),fun(A,fun(B,B)),hAPP(fun(B,fun(B,B)),fun(fun(A,B),fun(A,fun(B,B))),combb(B,fun(B,B),A),times_times(B)),C)),E)),zero_zero(B)),hAPP(A,filter(A),at(A),D))) # label(fact_4598_LIM__mult__zero) # label(axiom). [clausify(3974)]. 31.91/31.88 Derived: -metric_space(A) | -hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),B),zero_zero(complex)),hAPP(A,filter(A),at(A),C))) | -hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),D),zero_zero(complex)),hAPP(A,filter(A),at(A),C))) | hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),hAPP(fun(A,complex),fun(A,complex),hAPP(fun(A,fun(complex,complex)),fun(fun(A,complex),fun(A,complex)),combs(A,complex,complex),hAPP(fun(A,complex),fun(A,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(A,complex),fun(A,fun(complex,complex))),combb(complex,fun(complex,complex),A),times_times(complex)),B)),D)),zero_zero(complex)),hAPP(A,filter(A),at(A),C))). [resolve(5347,b,5285,a)]. 31.91/31.88 5348 -real_normed_algebra(A) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),zero_zero(A)) # label(fact_1242_mult__right_Ozero) # label(axiom). [clausify(4029)]. 31.91/31.88 5349 -real_normed_algebra(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),B)) | hAPP(fun(nat,A),A,suminf(A),hAPP(A,fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(A,fun(nat,A)),combc(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),B)),C)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(fun(nat,A),A,suminf(A),B)),C) # label(fact_4015_suminf__mult2) # label(axiom). [clausify(4053)]. 31.91/31.88 Derived: -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),A)) | hAPP(fun(nat,complex),complex,suminf(complex),hAPP(complex,fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(complex,fun(nat,complex)),combc(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),A)),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(fun(nat,complex),complex,suminf(complex),A)),B). [resolve(5349,a,5285,a)]. 32.13/32.08 5350 -real_normed_algebra(A) | hAPP(A,A,uminus_uminus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,uminus_uminus(A),B)),C) # label(fact_1912_mult__left_Ominus) # label(axiom). [clausify(4186)]. 32.13/32.08 Derived: hAPP(complex,complex,uminus_uminus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,uminus_uminus(complex),A)),B). [resolve(5350,a,5285,a)]. 32.13/32.08 5351 -topological_space(A) | -real_normed_algebra(B) | -hBOOL(hAPP(A,bool,hAPP(fun(A,B),fun(A,bool),isCont(A,B),C),D)) | -hBOOL(hAPP(A,bool,hAPP(fun(A,B),fun(A,bool),isCont(A,B),E),D)) | hBOOL(hAPP(A,bool,hAPP(fun(A,B),fun(A,bool),isCont(A,B),hAPP(fun(A,B),fun(A,B),hAPP(fun(A,fun(B,B)),fun(fun(A,B),fun(A,B)),combs(A,B,B),hAPP(fun(A,B),fun(A,fun(B,B)),hAPP(fun(B,fun(B,B)),fun(fun(A,B),fun(A,fun(B,B))),combb(B,fun(B,B),A),times_times(B)),C)),E)),D)) # label(fact_4911_mult_OisCont) # label(axiom). [clausify(4200)]. 32.13/32.08 5352 real_normed_algebra(real) # label(arity_RealDef_Oreal___RealVector_Oreal__normed__algebra) # label(axiom). [assumption]. 32.13/32.08 Derived: hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),C)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),C)). [resolve(5352,a,5286,a)]. 32.13/32.08 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),f42(real,A))). [resolve(5352,a,5287,a)]. 32.13/32.08 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),B)),f42(real,A)))). [resolve(5352,a,5288,a)]. 32.13/32.08 Derived: -hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),A)),C)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),C))). [resolve(5352,a,5289,a)]. 32.13/32.08 Derived: -hBOOL(hAPP(fun(nat,real),bool,summable(real),A)) | hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),A)),B))). [resolve(5352,a,5290,a)]. 32.13/32.08 Derived: -hBOOL(hAPP(fun(nat,real),bool,summable(real),A)) | hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),times_times(real),B)),A)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),hAPP(fun(nat,real),real,suminf(real),A)). [resolve(5352,a,5291,a)]. 32.13/32.08 Derived: -hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A),B),sequentially)) | -hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),C),D),sequentially)) | hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),A)),C)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),D)),sequentially)). [resolve(5352,a,5292,a)]. 32.13/32.09 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),B)),f103(real,A)))). [resolve(5352,a,5293,a)]. 32.13/32.09 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),f103(real,A))). [resolve(5352,a,5294,a)]. 32.13/32.09 Derived: hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B)),C) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),C)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),C)). [resolve(5352,a,5295,a)]. 32.13/32.09 Derived: hBOOL(hAPP(filter(real),bool,hAPP(real,fun(filter(real),bool),hAPP(fun(real,real),fun(real,fun(filter(real),bool)),tendsto(real,real),hAPP(real,fun(real,real),times_times(real),A)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),hAPP(real,filter(real),at(real),B))). [resolve(5352,a,5296,a)]. 32.13/32.09 Derived: -hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),B),C),D)) | hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),hAPP(fun(A,real),fun(A,real),hAPP(fun(real,real),fun(fun(A,real),fun(A,real)),combb(real,real,A),hAPP(real,fun(real,real),times_times(real),E)),B)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),E),C)),D)). [resolve(5352,a,5297,a)]. 32.13/32.09 Derived: zero_zero(real) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),zero_zero(real)),A). [resolve(5352,a,5298,a)]. 32.13/32.09 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),A)),hAPP(real,real,norm_norm(real),B))),f130(real)))). [resolve(5352,a,5299,a)]. 32.13/32.09 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),f130(real))). [resolve(5352,a,5300,a)]. 32.13/32.09 Derived: hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),C)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),B)),C). [resolve(5352,a,5301,a)]. 32.13/32.09 Derived: hBOOL(hAPP(filter(real),bool,hAPP(real,fun(filter(real),bool),hAPP(fun(real,real),fun(real,fun(filter(real),bool)),tendsto(real,real),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),times_times(real)),A)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),A)),hAPP(real,filter(real),at(real),B))). [resolve(5352,a,5302,a)]. 32.13/32.09 Derived: hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B)),C) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),C)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),C)). [resolve(5352,a,5303,a)]. 32.13/32.09 Derived: -hBOOL(hAPP(fun(nat,real),bool,cauchy(real),A)) | hBOOL(hAPP(fun(nat,real),bool,cauchy(real),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),A)),B))). [resolve(5352,a,5305,a)]. 32.13/32.09 Derived: -topological_space(A) | -power(real) | -hBOOL(hAPP(A,bool,hAPP(fun(A,real),fun(A,bool),isCont(A,real),B),C)) | hBOOL(hAPP(A,bool,hAPP(fun(A,real),fun(A,bool),isCont(A,real),hAPP(nat,fun(A,real),hAPP(fun(A,fun(nat,real)),fun(nat,fun(A,real)),combc(A,nat,real),hAPP(fun(A,real),fun(A,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(A,real),fun(A,fun(nat,real))),combb(real,fun(nat,real),A),power_power(real)),B)),D)),C)). [resolve(5352,a,5306,c)]. 32.13/32.09 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),A)),f223(real,B)))). [resolve(5352,a,5309,a)]. 32.13/32.09 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),A)),f253(real,B)))). [resolve(5352,a,5310,a)]. 32.13/32.09 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),f253(real,A))). [resolve(5352,a,5311,a)]. 32.13/32.09 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),A)),hAPP(real,real,norm_norm(real),B)))). [resolve(5352,a,5312,a)]. 32.13/32.09 Derived: -topological_space(A) | -hBOOL(hAPP(A,bool,hAPP(fun(A,real),fun(A,bool),isCont(A,real),B),C)) | -hBOOL(hAPP(A,bool,hAPP(fun(A,real),fun(A,bool),isCont(A,real),D),C)) | hBOOL(hAPP(A,bool,hAPP(fun(A,real),fun(A,bool),isCont(A,real),hAPP(fun(A,real),fun(A,real),hAPP(fun(A,fun(real,real)),fun(fun(A,real),fun(A,real)),combs(A,real,real),hAPP(fun(A,real),fun(A,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(A,real),fun(A,fun(real,real))),combb(real,fun(real,real),A),times_times(real)),B)),D)),C)). [resolve(5352,a,5313,b)]. 32.13/32.09 Derived: hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),D)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),C)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),D))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),C)),D))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),D))). [resolve(5352,a,5315,a)]. 32.13/32.09 Derived: hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),hAPP(real,fun(real,real),times_times(real),A)),B)). [resolve(5352,a,5316,a)]. 32.13/32.09 Derived: -power(real) | -hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),A),B),sequentially)) | hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(nat,fun(nat,real),hAPP(fun(nat,fun(nat,real)),fun(nat,fun(nat,real)),combc(nat,nat,real),hAPP(fun(nat,real),fun(nat,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(nat,real),fun(nat,fun(nat,real))),combb(real,fun(nat,real),nat),power_power(real)),A)),C)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),C)),sequentially)). [resolve(5352,a,5317,b)]. 32.13/32.09 Derived: hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),hAPP(fun(A,real),fun(A,real),hAPP(fun(real,real),fun(fun(A,real),fun(A,real)),combb(real,real,A),hAPP(real,fun(real,real),times_times(real),B)),C)),D) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),C),D)). [resolve(5352,a,5318,a)]. 32.21/32.10 Derived: -hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),times_times(real),C)),A)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),B))). [resolve(5352,a,5320,a)]. 32.21/32.10 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),A)),f328(real,B)))). [resolve(5352,a,5321,a)]. 32.21/32.10 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),f328(real,A))). [resolve(5352,a,5322,a)]. 32.21/32.10 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,norm_norm(real),A)),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,norm_norm(real),C)),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,norm_norm(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),C))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),D))). [resolve(5352,a,5323,a)]. 32.21/32.10 Derived: hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),C)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),C)). [resolve(5352,a,5324,a)]. 32.21/32.10 Derived: -topological_space(A) | -hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),B),C),hAPP(A,filter(A),at(A),D))) | hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),hAPP(real,fun(A,real),hAPP(fun(A,fun(real,real)),fun(real,fun(A,real)),combc(A,real,real),hAPP(fun(A,real),fun(A,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(A,real),fun(A,fun(real,real))),combb(real,fun(real,real),A),times_times(real)),B)),E)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),E)),hAPP(A,filter(A),at(A),D))). [resolve(5352,a,5325,a)]. 32.21/32.10 Derived: -topological_space(A) | -power(real) | -hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),B),C),hAPP(A,filter(A),at(A),D))) | hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),hAPP(nat,fun(A,real),hAPP(fun(A,fun(nat,real)),fun(nat,fun(A,real)),combc(A,nat,real),hAPP(fun(A,real),fun(A,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(A,real),fun(A,fun(nat,real))),combb(real,fun(nat,real),A),power_power(real)),B)),E)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),C),E)),hAPP(A,filter(A),at(A),D))). [resolve(5352,a,5326,c)]. 32.21/32.10 Derived: -topological_space(A) | -hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),B),zero_zero(real)),hAPP(A,filter(A),at(A),C))) | hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),hAPP(fun(A,real),fun(A,real),hAPP(fun(real,real),fun(fun(A,real),fun(A,real)),combb(real,real,A),hAPP(real,fun(real,real),times_times(real),D)),B)),zero_zero(real)),hAPP(A,filter(A),at(A),C))). [resolve(5352,a,5327,a)]. 32.21/32.10 Derived: -banach(real) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),norm_norm(real)),A))) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),norm_norm(real)),B))) | hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,fun(nat,bool)),fun(nat,real),hAPP(fun(nat,fun(fun(nat,bool),real)),fun(fun(nat,fun(nat,bool)),fun(nat,real)),combs(nat,fun(nat,bool),real),hAPP(fun(nat,fun(nat,real)),fun(nat,fun(fun(nat,bool),real)),hAPP(fun(fun(nat,real),fun(fun(nat,bool),real)),fun(fun(nat,fun(nat,real)),fun(nat,fun(fun(nat,bool),real))),combb(fun(nat,real),fun(fun(nat,bool),real),nat),big_co1399186613setsum(nat,real)),hAPP(fun(nat,fun(nat,real)),fun(nat,fun(nat,real)),hAPP(fun(fun(nat,real),fun(nat,real)),fun(fun(nat,fun(nat,real)),fun(nat,fun(nat,real))),combb(fun(nat,real),fun(nat,real),nat),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),A))),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,real)),hAPP(fun(fun(nat,nat),fun(nat,real)),fun(fun(nat,fun(nat,nat)),fun(nat,fun(nat,real))),combb(fun(nat,nat),fun(nat,real),nat),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),B)),minus_minus(nat))))),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)))) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(fun(nat,real),real,suminf(real),A)),hAPP(fun(nat,real),real,suminf(real),B)). [resolve(5352,a,5329,a)]. 32.21/32.10 Derived: -topological_space(A) | -hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),B),zero_zero(real)),hAPP(A,filter(A),at(A),C))) | hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),hAPP(real,fun(A,real),hAPP(fun(A,fun(real,real)),fun(real,fun(A,real)),combc(A,real,real),hAPP(fun(A,real),fun(A,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(A,real),fun(A,fun(real,real))),combb(real,fun(real,real),A),times_times(real)),B)),D)),zero_zero(real)),hAPP(A,filter(A),at(A),C))). [resolve(5352,a,5331,b)]. 32.21/32.10 Derived: -hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),B),C),D)) | hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),hAPP(real,fun(A,real),hAPP(fun(A,fun(real,real)),fun(real,fun(A,real)),combc(A,real,real),hAPP(fun(A,real),fun(A,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(A,real),fun(A,fun(real,real))),combb(real,fun(real,real),A),times_times(real)),B)),E)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),E)),D)). [resolve(5352,a,5332,a)]. 32.21/32.10 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),B)),f561(real,A)))). [resolve(5352,a,5333,a)]. 32.21/32.10 Derived: -hBOOL(hAPP(fun(nat,real),bool,summable(real),A)) | hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),times_times(real),B)),A))). [resolve(5352,a,5334,a)]. 32.21/32.10 Derived: -banach(real) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),norm_norm(real)),A))) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),norm_norm(real)),B))) | hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,fun(nat,bool)),fun(nat,real),hAPP(fun(nat,fun(fun(nat,bool),real)),fun(fun(nat,fun(nat,bool)),fun(nat,real)),combs(nat,fun(nat,bool),real),hAPP(fun(nat,fun(nat,real)),fun(nat,fun(fun(nat,bool),real)),hAPP(fun(fun(nat,real),fun(fun(nat,bool),real)),fun(fun(nat,fun(nat,real)),fun(nat,fun(fun(nat,bool),real))),combb(fun(nat,real),fun(fun(nat,bool),real),nat),big_co1399186613setsum(nat,real)),hAPP(fun(nat,fun(nat,real)),fun(nat,fun(nat,real)),hAPP(fun(fun(nat,real),fun(nat,real)),fun(fun(nat,fun(nat,real)),fun(nat,fun(nat,real))),combb(fun(nat,real),fun(nat,real),nat),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),A))),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,real)),hAPP(fun(fun(nat,nat),fun(nat,real)),fun(fun(nat,fun(nat,nat)),fun(nat,fun(nat,real))),combb(fun(nat,nat),fun(nat,real),nat),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),B)),minus_minus(nat))))),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(fun(nat,real),real,suminf(real),A)),hAPP(fun(nat,real),real,suminf(real),B)))). [resolve(5352,a,5336,a)]. 32.21/32.11 Derived: hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),C)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),C)). [resolve(5352,a,5337,a)]. 32.21/32.11 Derived: zero_zero(real) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),zero_zero(real)). [resolve(5352,a,5340,a)]. 32.21/32.11 Derived: -comm_ring_1(real) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f638(A,real,B,C,D)),D)) | hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(A,bool),fun(nat,real),hAPP(fun(nat,fun(fun(A,bool),real)),fun(fun(A,bool),fun(nat,real)),combc(nat,fun(A,bool),real),hAPP(fun(nat,fun(A,real)),fun(nat,fun(fun(A,bool),real)),hAPP(fun(fun(A,real),fun(fun(A,bool),real)),fun(fun(nat,fun(A,real)),fun(nat,fun(fun(A,bool),real))),combb(fun(A,real),fun(fun(A,bool),real),nat),big_co302546470etprod(A,real)),hAPP(fun(A,fun(nat,real)),fun(nat,fun(A,real)),combc(A,nat,real),B))),D)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),C),D)),sequentially)). [resolve(5352,a,5342,a)]. 32.21/32.11 Derived: -comm_ring_1(real) | -hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(A,fun(nat,real),B,f638(A,real,B,C,D))),hAPP(A,real,C,f638(A,real,B,C,D))),sequentially)) | hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(A,bool),fun(nat,real),hAPP(fun(nat,fun(fun(A,bool),real)),fun(fun(A,bool),fun(nat,real)),combc(nat,fun(A,bool),real),hAPP(fun(nat,fun(A,real)),fun(nat,fun(fun(A,bool),real)),hAPP(fun(fun(A,real),fun(fun(A,bool),real)),fun(fun(nat,fun(A,real)),fun(nat,fun(fun(A,bool),real))),combb(fun(A,real),fun(fun(A,bool),real),nat),big_co302546470etprod(A,real)),hAPP(fun(A,fun(nat,real)),fun(nat,fun(A,real)),combc(A,nat,real),B))),D)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),C),D)),sequentially)). [resolve(5352,a,5343,a)]. 32.21/32.11 Derived: hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),hAPP(real,real,uminus_uminus(real),B)). [resolve(5352,a,5345,a)]. 32.21/32.11 Derived: -topological_space(A) | -hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),B),C),hAPP(A,filter(A),at(A),D))) | -hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),E),F),hAPP(A,filter(A),at(A),D))) | hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),hAPP(fun(A,real),fun(A,real),hAPP(fun(A,fun(real,real)),fun(fun(A,real),fun(A,real)),combs(A,real,real),hAPP(fun(A,real),fun(A,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(A,real),fun(A,fun(real,real))),combb(real,fun(real,real),A),times_times(real)),B)),E)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),F)),hAPP(A,filter(A),at(A),D))). [resolve(5352,a,5346,b)]. 32.41/32.29 Derived: -metric_space(A) | -hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),B),zero_zero(real)),hAPP(A,filter(A),at(A),C))) | -hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),D),zero_zero(real)),hAPP(A,filter(A),at(A),C))) | hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),hAPP(fun(A,real),fun(A,real),hAPP(fun(A,fun(real,real)),fun(fun(A,real),fun(A,real)),combs(A,real,real),hAPP(fun(A,real),fun(A,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(A,real),fun(A,fun(real,real))),combb(real,fun(real,real),A),times_times(real)),B)),D)),zero_zero(real)),hAPP(A,filter(A),at(A),C))). [resolve(5352,a,5347,b)]. 32.41/32.29 Derived: -hBOOL(hAPP(fun(nat,real),bool,summable(real),A)) | hAPP(fun(nat,real),real,suminf(real),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),A)),B)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(fun(nat,real),real,suminf(real),A)),B). [resolve(5352,a,5349,a)]. 32.41/32.29 Derived: hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,uminus_uminus(real),A)),B). [resolve(5352,a,5350,a)]. 32.41/32.29 5353 -real_normed_algebra(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D)) # label(fact_1245_mult_Oadd__left) # label(axiom). [clausify(4285)]. 32.41/32.29 5354 -real_normed_algebra(A) | hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),C)),D) # label(fact_4723_mult__left_OscaleR) # label(axiom). [clausify(4293)]. 32.41/32.29 5355 -topological_space(A) | -real_normed_algebra(B) | -hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),C),zero_zero(B)),hAPP(A,filter(A),at(A),D))) | hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),hAPP(fun(A,B),fun(A,B),hAPP(fun(B,B),fun(fun(A,B),fun(A,B)),combb(B,B,A),hAPP(B,fun(B,B),times_times(B),E)),C)),zero_zero(B)),hAPP(A,filter(A),at(A),D))) # label(fact_4601_mult__right_OLIM__zero) # label(axiom). [clausify(4309)]. 32.41/32.29 5356 -topological_space(A) | -real_normed_algebra(B) | -hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),C),zero_zero(B)),hAPP(A,filter(A),at(A),D))) | hBOOL(hAPP(filter(A),bool,hAPP(B,fun(filter(A),bool),hAPP(fun(A,B),fun(B,fun(filter(A),bool)),tendsto(A,B),hAPP(B,fun(A,B),hAPP(fun(A,fun(B,B)),fun(B,fun(A,B)),combc(A,B,B),hAPP(fun(A,B),fun(A,fun(B,B)),hAPP(fun(B,fun(B,B)),fun(fun(A,B),fun(A,fun(B,B))),combb(B,fun(B,B),A),times_times(B)),C)),E)),zero_zero(B)),hAPP(A,filter(A),at(A),D))) # label(fact_4599_mult__left_OLIM__zero) # label(axiom). [clausify(4313)]. 32.41/32.29 5357 -real_normed_algebra(A) | hAPP(A,A,uminus_uminus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(A,A,uminus_uminus(A),C)) # label(fact_1910_mult__right_Ominus) # label(axiom). [clausify(4350)]. 32.41/32.29 5358 -real_normed_algebra(A) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),hAPP(A,fun(B,A),hAPP(fun(B,fun(A,A)),fun(A,fun(B,A)),combc(B,A,A),hAPP(fun(B,A),fun(B,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),combb(A,fun(A,A),B),times_times(A)),C)),D)),E) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),C),E)),D) # label(fact_4212_mult__left_Osetsum) # label(axiom). [clausify(4466)]. 32.59/32.48 Derived: hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co1399186613setsum(A,complex),hAPP(complex,fun(A,complex),hAPP(fun(A,fun(complex,complex)),fun(complex,fun(A,complex)),combc(A,complex,complex),hAPP(fun(A,complex),fun(A,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(A,complex),fun(A,fun(complex,complex))),combb(complex,fun(complex,complex),A),times_times(complex)),B)),C)),D) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co1399186613setsum(A,complex),B),D)),C). [resolve(5358,a,5285,a)]. 32.59/32.48 Derived: hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),hAPP(real,fun(A,real),hAPP(fun(A,fun(real,real)),fun(real,fun(A,real)),combc(A,real,real),hAPP(fun(A,real),fun(A,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(A,real),fun(A,fun(real,real))),combb(real,fun(real,real),A),times_times(real)),B)),C)),D) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),B),D)),C). [resolve(5358,a,5352,a)]. 32.59/32.48 5359 -real_normed_algebra(A) | hAPP(A,A,uminus_uminus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,uminus_uminus(A),B)),C) # label(fact_1911_mult_Ominus__left) # label(axiom). [clausify(4485)]. 32.59/32.48 5360 -real_normed_algebra(A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),f793(A))) # label(fact_2108_mult_Ononneg__bounded) # label(axiom). [clausify(4644)]. 32.59/32.48 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),f793(complex))). [resolve(5360,a,5285,a)]. 32.59/32.48 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),f793(real))). [resolve(5360,a,5352,a)]. 32.59/32.48 5361 -real_normed_algebra(A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(A,real,norm_norm(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(A,real,norm_norm(A),B)),hAPP(A,real,norm_norm(A),C))),f793(A)))) # label(fact_2108_mult_Ononneg__bounded) # label(axiom). [clausify(4644)]. 32.59/32.48 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),A)),hAPP(complex,real,norm_norm(complex),B))),f793(complex)))). [resolve(5361,a,5285,a)]. 32.59/32.48 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),A)),hAPP(real,real,norm_norm(real),B))),f793(real)))). [resolve(5361,a,5352,a)]. 32.59/32.48 5362 -real_normed_algebra(A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(A,real,norm_norm(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(A,real,norm_norm(A),B)),hAPP(A,real,norm_norm(A),C))),f798(A)))) # label(fact_2112_mult_Obounded) # label(axiom). [clausify(4673)]. 32.59/32.48 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),A)),hAPP(complex,real,norm_norm(complex),B))),f798(complex)))). [resolve(5362,a,5285,a)]. 32.75/32.71 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),A)),hAPP(real,real,norm_norm(real),B))),f798(real)))). [resolve(5362,a,5352,a)]. 32.75/32.71 5363 -real_normed_algebra(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)) # label(fact_1249_mult__right_Odiff) # label(axiom). [clausify(4727)]. 32.75/32.71 5364 -real_normed_algebra(A) | hBOOL(hAPP(A,bool,hAPP(fun(A,A),fun(A,bool),isCont(A,A),hAPP(A,fun(A,A),hAPP(fun(A,fun(A,A)),fun(A,fun(A,A)),combc(A,A,A),times_times(A)),B)),C)) # label(fact_4913_mult__left_OisCont) # label(axiom). [clausify(4826)]. 32.75/32.71 Derived: hBOOL(hAPP(complex,bool,hAPP(fun(complex,complex),fun(complex,bool),isCont(complex,complex),hAPP(complex,fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(complex,fun(complex,complex)),combc(complex,complex,complex),times_times(complex)),A)),B)). [resolve(5364,a,5285,a)]. 32.75/32.71 Derived: hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),times_times(real)),A)),B)). [resolve(5364,a,5352,a)]. 32.75/32.71 5365 -real_normed_algebra(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),B)) | hAPP(fun(nat,A),A,suminf(A),hAPP(A,fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(A,fun(nat,A)),combc(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),B)),C)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(fun(nat,A),A,suminf(A),B)),C) # label(fact_4013_mult__left_Osuminf) # label(axiom). [clausify(4886)]. 32.75/32.71 5366 -real_normed_algebra(A) | -hBOOL(hAPP(fun(nat,A),bool,cauchy(A),B)) | hBOOL(hAPP(fun(nat,A),bool,cauchy(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(A,A),fun(fun(nat,A),fun(nat,A)),combb(A,A,nat),hAPP(A,fun(A,A),times_times(A),C)),B))) # label(fact_4127_mult__right_OCauchy) # label(axiom). [clausify(4891)]. 32.75/32.71 Derived: -hBOOL(hAPP(fun(nat,complex),bool,cauchy(complex),A)) | hBOOL(hAPP(fun(nat,complex),bool,cauchy(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(complex,complex),fun(fun(nat,complex),fun(nat,complex)),combb(complex,complex,nat),hAPP(complex,fun(complex,complex),times_times(complex),B)),A))). [resolve(5366,a,5285,a)]. 32.75/32.71 Derived: -hBOOL(hAPP(fun(nat,real),bool,cauchy(real),A)) | hBOOL(hAPP(fun(nat,real),bool,cauchy(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),times_times(real),B)),A))). [resolve(5366,a,5352,a)]. 32.75/32.71 5367 -real_normed_algebra(A) | -hBOOL(hAPP(filter(B),bool,hAPP(A,fun(filter(B),bool),hAPP(fun(B,A),fun(A,fun(filter(B),bool)),tendsto(B,A),C),D),E)) | -hBOOL(hAPP(filter(B),bool,hAPP(A,fun(filter(B),bool),hAPP(fun(B,A),fun(A,fun(filter(B),bool)),tendsto(B,A),F),V6),E)) | hBOOL(hAPP(filter(B),bool,hAPP(A,fun(filter(B),bool),hAPP(fun(B,A),fun(A,fun(filter(B),bool)),tendsto(B,A),hAPP(fun(B,A),fun(B,A),hAPP(fun(B,fun(A,A)),fun(fun(B,A),fun(B,A)),combs(B,A,A),hAPP(fun(B,A),fun(B,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),combb(A,fun(A,A),B),times_times(A)),C)),F)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),V6)),E)) # label(fact_4481_mult_Otendsto) # label(axiom). [clausify(4897)]. 32.75/32.71 Derived: -hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),B),C),D)) | -hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),E),F),D)) | hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),hAPP(fun(A,complex),fun(A,complex),hAPP(fun(A,fun(complex,complex)),fun(fun(A,complex),fun(A,complex)),combs(A,complex,complex),hAPP(fun(A,complex),fun(A,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(A,complex),fun(A,fun(complex,complex))),combb(complex,fun(complex,complex),A),times_times(complex)),B)),E)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),F)),D)). [resolve(5367,a,5285,a)]. 34.24/34.18 Derived: -hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),B),C),D)) | -hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),E),F),D)) | hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),hAPP(fun(A,real),fun(A,real),hAPP(fun(A,fun(real,real)),fun(fun(A,real),fun(A,real)),combs(A,real,real),hAPP(fun(A,real),fun(A,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(A,real),fun(A,fun(real,real))),combb(real,fun(real,real),A),times_times(real)),B)),E)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),F)),D)). [resolve(5367,a,5352,a)]. 34.24/34.18 5368 division_ring(complex) # label(arity_Complex_Ocomplex___Fields_Odivision__ring) # label(axiom). [assumption]. 34.24/34.18 5369 -division_ring(A) | ti(A,B) = zero_zero(A) | ti(A,C) = zero_zero(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,inverse_inverse(A),B)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C))),hAPP(A,A,inverse_inverse(A),C)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,inverse_inverse(A),B)),hAPP(A,A,inverse_inverse(A),C)) # label(fact_2613_division__ring__inverse__add) # label(axiom). [clausify(18)]. 34.24/34.18 5370 -division_ring(A) | hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),D)) # label(fact_1827_times__divide__eq__right) # label(axiom). [clausify(489)]. 34.24/34.18 5371 -division_ring(A) | ti(A,B) = zero_zero(A) | ti(A,C) != hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),D),B) | ti(A,D) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),B) # label(fact_1848_nonzero__eq__divide__eq) # label(axiom). [clausify(932)]. 34.24/34.18 5372 -division_ring(A) | ti(A,B) = zero_zero(A) | ti(A,C) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),D),B) | ti(A,D) != hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),B) # label(fact_1848_nonzero__eq__divide__eq) # label(axiom). [clausify(932)]. 34.24/34.18 5373 -division_ring(A) | zero_zero(A) != hAPP(A,A,inverse_inverse(A),B) | ti(A,B) = zero_zero(A) # label(fact_2577_inverse__zero__imp__zero) # label(axiom). [clausify(934)]. 34.24/34.18 5374 -division_ring(A) | hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),C) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(A,A,inverse_inverse(A),C)) # label(fact_2599_divide__inverse) # label(axiom). [clausify(970)]. 34.24/34.18 5375 -division_ring(A) | inverse_inverse(A) = ti(fun(A,A),inverse_inverse(A)) # label(tsy_c_Fields_Oinverse__class_Oinverse_res) # label(axiom). [clausify(1062)]. 34.24/34.18 5376 -division_ring(A) | ti(A,B) = zero_zero(A) | hAPP(A,A,inverse_inverse(A),B) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),one_one(A)),B) # label(fact_2620_nonzero__inverse__eq__divide) # label(axiom). [clausify(1080)]. 34.24/34.18 5377 -division_ring(A) | hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),D)) # label(fact_1829_add__divide__distrib) # label(axiom). [clausify(1127)]. 34.24/34.18 5378 -division_ring(A) | hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),D)) # label(fact_1831_diff__divide__distrib) # label(axiom). [clausify(1211)]. 34.24/34.19 5379 -division_ring(A) | ti(A,B) = zero_zero(A) | hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),hAPP(A,A,uminus_uminus(A),C)),hAPP(A,A,uminus_uminus(A),B)) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),B) # label(fact_1973_nonzero__minus__divide__divide) # label(axiom). [clausify(1226)]. 34.24/34.19 5380 -division_ring(A) | ti(A,B) = zero_zero(A) | ti(A,B) != ti(A,C) | one_one(A) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),B) # label(fact_1855_right__inverse__eq) # label(axiom). [clausify(1326)]. 34.24/34.19 5381 -division_ring(A) | ti(A,B) = zero_zero(A) | ti(A,B) = ti(A,C) | one_one(A) != hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),B) # label(fact_1855_right__inverse__eq) # label(axiom). [clausify(1326)]. 34.24/34.19 5382 -division_ring(A) | ti(A,B) = zero_zero(A) | ti(A,C) != hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),D),B) | ti(A,D) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),B) # label(fact_1849_nonzero__divide__eq__eq) # label(axiom). [clausify(1853)]. 34.24/34.19 5383 -division_ring(A) | ti(A,B) = zero_zero(A) | ti(A,C) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),D),B) | ti(A,D) != hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),B) # label(fact_1849_nonzero__divide__eq__eq) # label(axiom). [clausify(1853)]. 34.24/34.19 Derived: ti(complex,A) = zero_zero(complex) | ti(complex,B) = zero_zero(complex) | hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,inverse_inverse(complex),A)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),B))),hAPP(complex,complex,inverse_inverse(complex),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,inverse_inverse(complex),A)),hAPP(complex,complex,inverse_inverse(complex),B)). [resolve(5368,a,5369,a)]. 34.24/34.19 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)),C) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),B),C)). [resolve(5368,a,5370,a)]. 34.24/34.19 Derived: ti(complex,A) = zero_zero(complex) | ti(complex,B) != hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),C),A) | ti(complex,C) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),B),A). [resolve(5368,a,5371,a)]. 34.24/34.19 Derived: ti(complex,A) = zero_zero(complex) | ti(complex,B) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),C),A) | ti(complex,C) != hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),B),A). [resolve(5368,a,5372,a)]. 34.24/34.19 Derived: zero_zero(complex) != hAPP(complex,complex,inverse_inverse(complex),A) | ti(complex,A) = zero_zero(complex). [resolve(5368,a,5373,a)]. 34.24/34.19 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),A),B) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),hAPP(complex,complex,inverse_inverse(complex),B)). [resolve(5368,a,5374,a)]. 34.24/34.19 Derived: inverse_inverse(complex) = ti(fun(complex,complex),inverse_inverse(complex)). [resolve(5368,a,5375,a)]. 34.24/34.19 Derived: ti(complex,A) = zero_zero(complex) | hAPP(complex,complex,inverse_inverse(complex),A) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),one_one(complex)),A). [resolve(5368,a,5376,a)]. 34.24/34.19 Derived: ti(complex,A) = zero_zero(complex) | hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),hAPP(complex,complex,uminus_uminus(complex),B)),hAPP(complex,complex,uminus_uminus(complex),A)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),B),A). [resolve(5368,a,5379,a)]. 34.24/34.19 Derived: ti(complex,A) = zero_zero(complex) | ti(complex,A) != ti(complex,B) | one_one(complex) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),B),A). [resolve(5368,a,5380,a)]. 34.61/34.52 Derived: ti(complex,A) = zero_zero(complex) | ti(complex,A) = ti(complex,B) | one_one(complex) != hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),B),A). [resolve(5368,a,5381,a)]. 34.61/34.52 5384 -division_ring(A) | hAPP(A,A,inverse_inverse(A),B) != hAPP(A,A,inverse_inverse(A),C) | ti(A,C) = zero_zero(A) | ti(A,B) = zero_zero(A) | ti(A,B) = ti(A,C) # label(fact_2578_nonzero__inverse__eq__imp__eq) # label(axiom). [clausify(1889)]. 34.61/34.52 Derived: hAPP(complex,complex,inverse_inverse(complex),A) != hAPP(complex,complex,inverse_inverse(complex),B) | ti(complex,B) = zero_zero(complex) | ti(complex,A) = zero_zero(complex) | ti(complex,A) = ti(complex,B). [resolve(5384,a,5368,a)]. 34.61/34.52 5385 -division_ring(A) | ti(A,B) = zero_zero(A) | one_one(A) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,inverse_inverse(A),B)),B) # label(fact_2616_left__inverse) # label(axiom). [clausify(1989)]. 34.61/34.52 Derived: ti(complex,A) = zero_zero(complex) | one_one(complex) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,inverse_inverse(complex),A)),A). [resolve(5385,a,5368,a)]. 34.61/34.52 5386 -division_ring(A) | ti(A,B) = zero_zero(A) | one_one(A) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(A,A,inverse_inverse(A),B)) # label(fact_2618_right__inverse) # label(axiom). [clausify(2025)]. 34.61/34.52 Derived: ti(complex,A) = zero_zero(complex) | one_one(complex) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),hAPP(complex,complex,inverse_inverse(complex),A)). [resolve(5386,a,5368,a)]. 34.61/34.52 5387 -division_ring(A) | ti(A,B) = zero_zero(A) | one_one(A) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),B) # label(fact_1856_divide__self) # label(axiom). [clausify(2046)]. 34.61/34.52 Derived: ti(complex,A) = zero_zero(complex) | one_one(complex) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),A),A). [resolve(5387,a,5368,a)]. 34.61/34.52 5388 -division_ring(A) | ti(A,B) = zero_zero(A) | ti(A,C) = zero_zero(A) | hAPP(A,A,uminus_uminus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,inverse_inverse(A),B)),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C))),hAPP(A,A,inverse_inverse(A),C))) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,inverse_inverse(A),B)),hAPP(A,A,inverse_inverse(A),C)) # label(fact_2631_inverse__diff__inverse) # label(axiom). [clausify(2134)]. 34.61/34.52 Derived: ti(complex,A) = zero_zero(complex) | ti(complex,B) = zero_zero(complex) | hAPP(complex,complex,uminus_uminus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,inverse_inverse(complex),A)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),A),B))),hAPP(complex,complex,inverse_inverse(complex),B))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(complex,complex,inverse_inverse(complex),A)),hAPP(complex,complex,inverse_inverse(complex),B)). [resolve(5388,a,5368,a)]. 34.61/34.52 5389 -division_ring(A) | ti(fun(A,fun(A,A)),inverse_divide(A)) = inverse_divide(A) # label(tsy_c_Fields_Oinverse__class_Odivide_1_res) # label(axiom). [clausify(2520)]. 34.61/34.52 Derived: ti(fun(complex,fun(complex,complex)),inverse_divide(complex)) = inverse_divide(complex). [resolve(5389,a,5368,a)]. 34.61/34.52 5390 -division_ring(A) | ti(A,B) = zero_zero(A) | ti(A,C) = zero_zero(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,inverse_inverse(A),B)),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),C),B))),hAPP(A,A,inverse_inverse(A),C)) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,inverse_inverse(A),B)),hAPP(A,A,inverse_inverse(A),C)) # label(fact_2619_division__ring__inverse__diff) # label(axiom). [clausify(2654)]. 34.61/34.52 Derived: ti(complex,A) = zero_zero(complex) | ti(complex,B) = zero_zero(complex) | hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,inverse_inverse(complex),A)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),B),A))),hAPP(complex,complex,inverse_inverse(complex),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(complex,complex,inverse_inverse(complex),A)),hAPP(complex,complex,inverse_inverse(complex),B)). [resolve(5390,a,5368,a)]. 35.41/35.30 5391 -division_ring(A) | hAPP(A,A,inverse_inverse(A),B) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),one_one(A)),B) # label(fact_2602_inverse__eq__divide) # label(axiom). [clausify(2714)]. 35.41/35.30 Derived: hAPP(complex,complex,inverse_inverse(complex),A) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),one_one(complex)),A). [resolve(5391,a,5368,a)]. 35.41/35.30 5392 -division_ring(A) | ti(A,B) = zero_zero(A) | hAPP(A,A,uminus_uminus(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),B)) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),hAPP(A,A,uminus_uminus(A),B)) # label(fact_1972_nonzero__minus__divide__right) # label(axiom). [clausify(2728)]. 35.41/35.30 Derived: ti(complex,A) = zero_zero(complex) | hAPP(complex,complex,uminus_uminus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),B),A)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),B),hAPP(complex,complex,uminus_uminus(complex),A)). [resolve(5392,a,5368,a)]. 35.41/35.30 5393 -division_ring(A) | ti(A,B) = zero_zero(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(A,A,inverse_inverse(A),B)),C) = hAPP(A,A,inverse_inverse(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C)) # label(fact_2597_nonzero__power__inverse) # label(axiom). [clausify(3144)]. 35.41/35.30 Derived: ti(complex,A) = zero_zero(complex) | hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(complex,complex,inverse_inverse(complex),A)),B) = hAPP(complex,complex,inverse_inverse(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),B)). [resolve(5393,a,5368,a)]. 35.41/35.30 5394 -division_ring(A) | ti(A,B) = zero_zero(A) | hAPP(A,A,inverse_inverse(A),hAPP(A,A,uminus_uminus(A),B)) = hAPP(A,A,uminus_uminus(A),hAPP(A,A,inverse_inverse(A),B)) # label(fact_2598_nonzero__inverse__minus__eq) # label(axiom). [clausify(3236)]. 35.41/35.30 Derived: ti(complex,A) = zero_zero(complex) | hAPP(complex,complex,inverse_inverse(complex),hAPP(complex,complex,uminus_uminus(complex),A)) = hAPP(complex,complex,uminus_uminus(complex),hAPP(complex,complex,inverse_inverse(complex),A)). [resolve(5394,a,5368,a)]. 35.41/35.30 5395 -division_ring(A) | one_one(A) = hAPP(A,A,inverse_inverse(A),one_one(A)) # label(fact_2565_inverse__1) # label(axiom). [clausify(3410)]. 35.41/35.30 Derived: one_one(complex) = hAPP(complex,complex,inverse_inverse(complex),one_one(complex)). [resolve(5395,a,5368,a)]. 35.41/35.30 5396 -division_ring(A) | ti(A,B) = zero_zero(A) | zero_zero(A) != hAPP(A,A,inverse_inverse(A),B) # label(fact_2575_nonzero__imp__inverse__nonzero) # label(axiom). [clausify(3412)]. 35.41/35.30 5397 -division_ring(A) | -number(A) | hAPP(A,A,inverse_inverse(A),hAPP(int,A,number_number_of(A),B)) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),one_one(A)),hAPP(int,A,number_number_of(A),B)) # label(fact_2622_inverse__eq__divide__number__of) # label(axiom). [clausify(3417)]. 35.41/35.30 5398 -division_ring(A) | hAPP(A,A,uminus_uminus(A),hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),hAPP(A,A,uminus_uminus(A),B)),C) # label(fact_1941_minus__divide__left) # label(axiom). [clausify(3732)]. 35.41/35.30 5399 -division_ring(A) | ti(A,B) = zero_zero(A) | ti(A,C) != hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),B) | ti(A,D) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),B) # label(fact_1854_eq__divide__imp) # label(axiom). [clausify(3747)]. 35.41/35.30 5400 division_ring(real) # label(arity_RealDef_Oreal___Fields_Odivision__ring) # label(axiom). [assumption]. 35.41/35.30 Derived: ti(real,A) = zero_zero(real) | ti(real,B) = zero_zero(real) | hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,inverse_inverse(real),A)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B))),hAPP(real,real,inverse_inverse(real),B)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,inverse_inverse(real),A)),hAPP(real,real,inverse_inverse(real),B)). [resolve(5400,a,5369,a)]. 35.41/35.31 Derived: hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),C) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),C)). [resolve(5400,a,5370,a)]. 35.41/35.31 Derived: ti(real,A) = zero_zero(real) | ti(real,B) != hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),C),A) | ti(real,C) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),A). [resolve(5400,a,5371,a)]. 35.41/35.31 Derived: ti(real,A) = zero_zero(real) | ti(real,B) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),C),A) | ti(real,C) != hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),A). [resolve(5400,a,5372,a)]. 35.41/35.31 Derived: zero_zero(real) != hAPP(real,real,inverse_inverse(real),A) | ti(real,A) = zero_zero(real). [resolve(5400,a,5373,a)]. 35.41/35.31 Derived: hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),B) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),hAPP(real,real,inverse_inverse(real),B)). [resolve(5400,a,5374,a)]. 35.41/35.31 Derived: inverse_inverse(real) = ti(fun(real,real),inverse_inverse(real)). [resolve(5400,a,5375,a)]. 35.41/35.31 Derived: ti(real,A) = zero_zero(real) | hAPP(real,real,inverse_inverse(real),A) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),A). [resolve(5400,a,5376,a)]. 35.41/35.31 Derived: ti(real,A) = zero_zero(real) | hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,uminus_uminus(real),B)),hAPP(real,real,uminus_uminus(real),A)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),A). [resolve(5400,a,5379,a)]. 35.41/35.31 Derived: ti(real,A) = zero_zero(real) | ti(real,A) != ti(real,B) | one_one(real) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),A). [resolve(5400,a,5380,a)]. 35.41/35.31 Derived: ti(real,A) = zero_zero(real) | ti(real,A) = ti(real,B) | one_one(real) != hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),A). [resolve(5400,a,5381,a)]. 35.41/35.31 Derived: hAPP(real,real,inverse_inverse(real),A) != hAPP(real,real,inverse_inverse(real),B) | ti(real,B) = zero_zero(real) | ti(real,A) = zero_zero(real) | ti(real,A) = ti(real,B). [resolve(5400,a,5384,a)]. 35.41/35.31 Derived: ti(real,A) = zero_zero(real) | one_one(real) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,inverse_inverse(real),A)),A). [resolve(5400,a,5385,a)]. 35.41/35.31 Derived: ti(real,A) = zero_zero(real) | one_one(real) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),hAPP(real,real,inverse_inverse(real),A)). [resolve(5400,a,5386,a)]. 35.41/35.31 Derived: ti(real,A) = zero_zero(real) | one_one(real) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),A). [resolve(5400,a,5387,a)]. 35.41/35.31 Derived: ti(real,A) = zero_zero(real) | ti(real,B) = zero_zero(real) | hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,inverse_inverse(real),A)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B))),hAPP(real,real,inverse_inverse(real),B))) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,inverse_inverse(real),A)),hAPP(real,real,inverse_inverse(real),B)). [resolve(5400,a,5388,a)]. 35.41/35.31 Derived: ti(fun(real,fun(real,real)),inverse_divide(real)) = inverse_divide(real). [resolve(5400,a,5389,a)]. 35.41/35.31 Derived: ti(real,A) = zero_zero(real) | ti(real,B) = zero_zero(real) | hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,inverse_inverse(real),A)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),A))),hAPP(real,real,inverse_inverse(real),B)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,inverse_inverse(real),A)),hAPP(real,real,inverse_inverse(real),B)). [resolve(5400,a,5390,a)]. 35.63/35.56 Derived: hAPP(real,real,inverse_inverse(real),A) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),A). [resolve(5400,a,5391,a)]. 35.63/35.56 Derived: ti(real,A) = zero_zero(real) | hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),A)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),B),hAPP(real,real,uminus_uminus(real),A)). [resolve(5400,a,5392,a)]. 35.63/35.56 Derived: ti(real,A) = zero_zero(real) | hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,inverse_inverse(real),A)),B) = hAPP(real,real,inverse_inverse(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B)). [resolve(5400,a,5393,a)]. 35.63/35.56 Derived: ti(real,A) = zero_zero(real) | hAPP(real,real,inverse_inverse(real),hAPP(real,real,uminus_uminus(real),A)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,inverse_inverse(real),A)). [resolve(5400,a,5394,a)]. 35.63/35.56 Derived: one_one(real) = hAPP(real,real,inverse_inverse(real),one_one(real)). [resolve(5400,a,5395,a)]. 35.63/35.56 5401 -division_ring(A) | ti(A,B) = zero_zero(A) | ti(A,C) != hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),B) | ti(A,D) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),C),B) # label(fact_1853_divide__eq__imp) # label(axiom). [clausify(3984)]. 35.63/35.56 5402 -division_ring(A) | ti(A,B) = zero_zero(A) | ti(A,C) = zero_zero(A) | hAPP(A,A,inverse_inverse(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,inverse_inverse(A),C)),hAPP(A,A,inverse_inverse(A),B)) # label(fact_2595_nonzero__inverse__mult__distrib) # label(axiom). [clausify(4074)]. 35.63/35.56 Derived: ti(complex,A) = zero_zero(complex) | ti(complex,B) = zero_zero(complex) | hAPP(complex,complex,inverse_inverse(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,inverse_inverse(complex),B)),hAPP(complex,complex,inverse_inverse(complex),A)). [resolve(5402,a,5368,a)]. 35.63/35.56 Derived: ti(real,A) = zero_zero(real) | ti(real,B) = zero_zero(real) | hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,inverse_inverse(real),B)),hAPP(real,real,inverse_inverse(real),A)). [resolve(5402,a,5400,a)]. 35.63/35.56 5403 division_ring(rat) # label(arity_Rat_Orat___Fields_Odivision__ring) # label(axiom). [assumption]. 35.63/35.56 Derived: ti(rat,A) = zero_zero(rat) | ti(rat,B) = zero_zero(rat) | hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,inverse_inverse(rat),A)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B))),hAPP(rat,rat,inverse_inverse(rat),B)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,inverse_inverse(rat),A)),hAPP(rat,rat,inverse_inverse(rat),B)). [resolve(5403,a,5369,a)]. 35.63/35.56 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),C) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),B),C)). [resolve(5403,a,5370,a)]. 35.63/35.56 Derived: ti(rat,A) = zero_zero(rat) | ti(rat,B) != hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),C),A) | ti(rat,C) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),A). [resolve(5403,a,5371,a)]. 35.63/35.56 Derived: ti(rat,A) = zero_zero(rat) | ti(rat,B) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),C),A) | ti(rat,C) != hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),A). [resolve(5403,a,5372,a)]. 35.63/35.56 Derived: zero_zero(rat) != hAPP(rat,rat,inverse_inverse(rat),A) | ti(rat,A) = zero_zero(rat). [resolve(5403,a,5373,a)]. 35.63/35.56 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),B) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),hAPP(rat,rat,inverse_inverse(rat),B)). [resolve(5403,a,5374,a)]. 35.63/35.56 Derived: inverse_inverse(rat) = ti(fun(rat,rat),inverse_inverse(rat)). [resolve(5403,a,5375,a)]. 35.63/35.57 Derived: ti(rat,A) = zero_zero(rat) | hAPP(rat,rat,inverse_inverse(rat),A) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),one_one(rat)),A). [resolve(5403,a,5376,a)]. 35.63/35.57 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B)),C) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),C)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),B),C)). [resolve(5403,a,5377,a)]. 35.63/35.57 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B)),C) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),C)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),B),C)). [resolve(5403,a,5378,a)]. 35.63/35.57 Derived: ti(rat,A) = zero_zero(rat) | hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),hAPP(rat,rat,uminus_uminus(rat),B)),hAPP(rat,rat,uminus_uminus(rat),A)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),B),A). [resolve(5403,a,5379,a)]. 35.63/35.57 Derived: ti(rat,A) = zero_zero(rat) | ti(rat,A) != ti(rat,B) | one_one(rat) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),B),A). [resolve(5403,a,5380,a)]. 35.63/35.57 Derived: ti(rat,A) = zero_zero(rat) | ti(rat,A) = ti(rat,B) | one_one(rat) != hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),B),A). [resolve(5403,a,5381,a)]. 35.63/35.57 Derived: hAPP(rat,rat,inverse_inverse(rat),A) != hAPP(rat,rat,inverse_inverse(rat),B) | ti(rat,B) = zero_zero(rat) | ti(rat,A) = zero_zero(rat) | ti(rat,A) = ti(rat,B). [resolve(5403,a,5384,a)]. 35.63/35.57 Derived: ti(rat,A) = zero_zero(rat) | one_one(rat) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,inverse_inverse(rat),A)),A). [resolve(5403,a,5385,a)]. 35.63/35.57 Derived: ti(rat,A) = zero_zero(rat) | one_one(rat) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),hAPP(rat,rat,inverse_inverse(rat),A)). [resolve(5403,a,5386,a)]. 35.63/35.57 Derived: ti(rat,A) = zero_zero(rat) | one_one(rat) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),A). [resolve(5403,a,5387,a)]. 35.63/35.57 Derived: ti(rat,A) = zero_zero(rat) | ti(rat,B) = zero_zero(rat) | hAPP(rat,rat,uminus_uminus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,inverse_inverse(rat),A)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B))),hAPP(rat,rat,inverse_inverse(rat),B))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(rat,rat,inverse_inverse(rat),A)),hAPP(rat,rat,inverse_inverse(rat),B)). [resolve(5403,a,5388,a)]. 35.63/35.57 Derived: ti(fun(rat,fun(rat,rat)),inverse_divide(rat)) = inverse_divide(rat). [resolve(5403,a,5389,a)]. 35.63/35.57 Derived: ti(rat,A) = zero_zero(rat) | ti(rat,B) = zero_zero(rat) | hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,inverse_inverse(rat),A)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),B),A))),hAPP(rat,rat,inverse_inverse(rat),B)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(rat,rat,inverse_inverse(rat),A)),hAPP(rat,rat,inverse_inverse(rat),B)). [resolve(5403,a,5390,a)]. 35.63/35.57 Derived: hAPP(rat,rat,inverse_inverse(rat),A) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),one_one(rat)),A). [resolve(5403,a,5391,a)]. 35.63/35.57 Derived: ti(rat,A) = zero_zero(rat) | hAPP(rat,rat,uminus_uminus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),B),A)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),B),hAPP(rat,rat,uminus_uminus(rat),A)). [resolve(5403,a,5392,a)]. 35.63/35.57 Derived: ti(rat,A) = zero_zero(rat) | hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),hAPP(rat,rat,inverse_inverse(rat),A)),B) = hAPP(rat,rat,inverse_inverse(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B)). [resolve(5403,a,5393,a)]. 35.63/35.57 Derived: ti(rat,A) = zero_zero(rat) | hAPP(rat,rat,inverse_inverse(rat),hAPP(rat,rat,uminus_uminus(rat),A)) = hAPP(rat,rat,uminus_uminus(rat),hAPP(rat,rat,inverse_inverse(rat),A)). [resolve(5403,a,5394,a)]. 35.63/35.57 Derived: one_one(rat) = hAPP(rat,rat,inverse_inverse(rat),one_one(rat)). [resolve(5403,a,5395,a)]. 37.53/37.44 Derived: hAPP(rat,rat,uminus_uminus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),B)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),hAPP(rat,rat,uminus_uminus(rat),A)),B). [resolve(5403,a,5398,a)]. 37.53/37.44 Derived: ti(rat,A) = zero_zero(rat) | ti(rat,B) = zero_zero(rat) | hAPP(rat,rat,inverse_inverse(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,inverse_inverse(rat),B)),hAPP(rat,rat,inverse_inverse(rat),A)). [resolve(5403,a,5402,a)]. 37.53/37.44 5404 -division_ring(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),one_one(A)) # label(fact_1830_divide__1) # label(axiom). [clausify(4409)]. 37.53/37.44 Derived: ti(complex,A) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),A),one_one(complex)). [resolve(5404,a,5368,a)]. 37.53/37.44 Derived: ti(real,A) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),one_one(real)). [resolve(5404,a,5400,a)]. 37.53/37.44 Derived: ti(rat,A) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),one_one(rat)). [resolve(5404,a,5403,a)]. 37.53/37.44 5405 -division_ring(A) | ti(A,B) = zero_zero(A) | ti(A,B) = hAPP(A,A,inverse_inverse(A),hAPP(A,A,inverse_inverse(A),B)) # label(fact_2576_nonzero__inverse__inverse__eq) # label(axiom). [clausify(4767)]. 37.53/37.44 Derived: ti(complex,A) = zero_zero(complex) | ti(complex,A) = hAPP(complex,complex,inverse_inverse(complex),hAPP(complex,complex,inverse_inverse(complex),A)). [resolve(5405,a,5368,a)]. 37.53/37.44 Derived: ti(real,A) = zero_zero(real) | ti(real,A) = hAPP(real,real,inverse_inverse(real),hAPP(real,real,inverse_inverse(real),A)). [resolve(5405,a,5400,a)]. 37.53/37.44 Derived: ti(rat,A) = zero_zero(rat) | ti(rat,A) = hAPP(rat,rat,inverse_inverse(rat),hAPP(rat,rat,inverse_inverse(rat),A)). [resolve(5405,a,5403,a)]. 37.53/37.44 5406 -division_ring(A) | one_one(A) != hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C) | ti(A,C) = hAPP(A,A,inverse_inverse(A),B) # label(fact_2596_inverse__unique) # label(axiom). [clausify(4795)]. 37.53/37.44 Derived: one_one(complex) != hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B) | ti(complex,B) = hAPP(complex,complex,inverse_inverse(complex),A). [resolve(5406,a,5368,a)]. 37.53/37.44 Derived: one_one(real) != hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B) | ti(real,B) = hAPP(real,real,inverse_inverse(real),A). [resolve(5406,a,5400,a)]. 37.53/37.44 Derived: one_one(rat) != hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B) | ti(rat,B) = hAPP(rat,rat,inverse_inverse(rat),A). [resolve(5406,a,5403,a)]. 37.53/37.44 5407 -division_ring(A) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),zero_zero(A)),B) # label(fact_1826_divide__zero__left) # label(axiom). [clausify(4903)]. 37.53/37.44 Derived: zero_zero(rat) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),zero_zero(rat)),A). [resolve(5407,a,5403,a)]. 37.53/37.44 5408 -ab_group_add(A) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),hAPP(fun(B,A),fun(B,A),hAPP(fun(B,fun(A,A)),fun(fun(B,A),fun(B,A)),combs(B,A,A),hAPP(fun(B,A),fun(B,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),combb(A,fun(A,A),B),minus_minus(A)),C)),D)),E) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),C),E)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),D),E)) # label(fact_4247_setsum__subtractf) # label(axiom). [clausify(468)]. 37.53/37.44 5409 ab_group_add(real) # label(arity_RealDef_Oreal___Groups_Oab__group__add) # label(axiom). [assumption]. 37.53/37.44 Derived: hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),hAPP(fun(A,real),fun(A,real),hAPP(fun(A,fun(real,real)),fun(fun(A,real),fun(A,real)),combs(A,real,real),hAPP(fun(A,real),fun(A,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(A,real),fun(A,fun(real,real))),combb(real,fun(real,real),A),minus_minus(real)),B)),C)),D) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),B),D)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),C),D)). [resolve(5408,a,5409,a)]. 37.60/37.54 5410 -ab_group_add(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),C),D)) | hAPP(fun(nat,bool),A,hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),E),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(fun(nat,bool),A,hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),E),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),B),D))),hAPP(fun(nat,bool),A,hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),E),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),B),C))) # label(fact_4229_setsum__diff__nat__ivl) # label(axiom). [clausify(526)]. 37.60/37.54 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),B),C)) | hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),D),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),B),C)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),D),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),A),C))),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),D),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),A),B))). [resolve(5410,a,5409,a)]. 37.60/37.54 5411 -ab_group_add(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | -hBOOL(hAPP(fun(B,bool),bool,hAPP(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),D),C)) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),E),hAPP(fun(B,bool),fun(B,bool),hAPP(fun(B,bool),fun(fun(B,bool),fun(B,bool)),minus_minus(fun(B,bool)),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),E),C)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),E),D)) # label(fact_4260_setsum__diff) # label(axiom). [clausify(876)]. 37.60/37.54 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),D),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),D),B)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),D),C)). [resolve(5411,a,5409,a)]. 37.60/37.54 5412 ab_group_add(int) # label(arity_Int_Oint___Groups_Oab__group__add) # label(axiom). [assumption]. 37.60/37.54 Derived: hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),hAPP(fun(A,int),fun(A,int),hAPP(fun(A,fun(int,int)),fun(fun(A,int),fun(A,int)),combs(A,int,int),hAPP(fun(A,int),fun(A,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(A,int),fun(A,fun(int,int))),combb(int,fun(int,int),A),minus_minus(int)),B)),C)),D) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),B),D)),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),C),D)). [resolve(5412,a,5408,a)]. 37.60/37.54 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),B),C)) | hAPP(fun(nat,bool),int,hAPP(fun(nat,int),fun(fun(nat,bool),int),big_co1399186613setsum(nat,int),D),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),B),C)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(fun(nat,bool),int,hAPP(fun(nat,int),fun(fun(nat,bool),int),big_co1399186613setsum(nat,int),D),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),A),C))),hAPP(fun(nat,bool),int,hAPP(fun(nat,int),fun(fun(nat,bool),int),big_co1399186613setsum(nat,int),D),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),A),B))). [resolve(5412,a,5410,a)]. 37.83/37.76 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),D),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),D),B)),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),D),C)). [resolve(5412,a,5411,a)]. 37.83/37.76 5413 ab_group_add(complex) # label(arity_Complex_Ocomplex___Groups_Oab__group__add) # label(axiom). [assumption]. 37.83/37.76 Derived: hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co1399186613setsum(A,complex),hAPP(fun(A,complex),fun(A,complex),hAPP(fun(A,fun(complex,complex)),fun(fun(A,complex),fun(A,complex)),combs(A,complex,complex),hAPP(fun(A,complex),fun(A,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(A,complex),fun(A,fun(complex,complex))),combb(complex,fun(complex,complex),A),minus_minus(complex)),B)),C)),D) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co1399186613setsum(A,complex),B),D)),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co1399186613setsum(A,complex),C),D)). [resolve(5413,a,5408,a)]. 37.83/37.76 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),B),C)) | hAPP(fun(nat,bool),complex,hAPP(fun(nat,complex),fun(fun(nat,bool),complex),big_co1399186613setsum(nat,complex),D),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),B),C)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(fun(nat,bool),complex,hAPP(fun(nat,complex),fun(fun(nat,bool),complex),big_co1399186613setsum(nat,complex),D),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),A),C))),hAPP(fun(nat,bool),complex,hAPP(fun(nat,complex),fun(fun(nat,bool),complex),big_co1399186613setsum(nat,complex),D),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),A),B))). [resolve(5413,a,5410,a)]. 37.83/37.76 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co1399186613setsum(A,complex),D),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co1399186613setsum(A,complex),D),B)),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co1399186613setsum(A,complex),D),C)). [resolve(5413,a,5411,a)]. 37.83/37.76 5414 -ab_group_add(A) | hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C) != hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),D),E) | ti(A,C) != ti(A,B) | ti(A,E) = ti(A,D) # label(fact_715_diff__eq__diff__eq) # label(axiom). [clausify(1856)]. 37.83/37.76 Derived: hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B) != hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),C),D) | ti(real,B) != ti(real,A) | ti(real,D) = ti(real,C). [resolve(5414,a,5409,a)]. 37.83/37.76 Derived: hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),C),D) | ti(int,B) != ti(int,A) | ti(int,D) = ti(int,C). [resolve(5414,a,5412,a)]. 38.01/37.90 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),A),B) != hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),C),D) | ti(complex,B) != ti(complex,A) | ti(complex,D) = ti(complex,C). [resolve(5414,a,5413,a)]. 38.01/37.90 5415 -ab_group_add(A) | hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C) != hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),D),E) | ti(A,C) = ti(A,B) | ti(A,E) != ti(A,D) # label(fact_715_diff__eq__diff__eq) # label(axiom). [clausify(1856)]. 38.01/37.90 Derived: hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B) != hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),C),D) | ti(real,B) = ti(real,A) | ti(real,D) != ti(real,C). [resolve(5415,a,5409,a)]. 38.01/37.90 Derived: hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),C),D) | ti(int,B) = ti(int,A) | ti(int,D) != ti(int,C). [resolve(5415,a,5412,a)]. 38.01/37.90 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),A),B) != hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),C),D) | ti(complex,B) = ti(complex,A) | ti(complex,D) != ti(complex,C). [resolve(5415,a,5413,a)]. 38.01/37.90 5416 -ab_group_add(A) | hAPP(A,A,uminus_uminus(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,uminus_uminus(A),B)),hAPP(A,A,uminus_uminus(A),C)) # label(fact_1916_minus__add__distrib) # label(axiom). [clausify(2036)]. 38.01/37.90 Derived: hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,uminus_uminus(real),A)),hAPP(real,real,uminus_uminus(real),B)). [resolve(5416,a,5409,a)]. 38.01/37.90 Derived: hAPP(complex,complex,uminus_uminus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,uminus_uminus(complex),A)),hAPP(complex,complex,uminus_uminus(complex),B)). [resolve(5416,a,5413,a)]. 38.01/37.90 5417 -ab_group_add(A) | hAPP(A,A,uminus_uminus(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),C),B) # label(fact_1918_minus__diff__eq) # label(axiom). [clausify(2085)]. 38.01/37.90 Derived: hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),A). [resolve(5417,a,5409,a)]. 38.01/37.90 Derived: hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),A). [resolve(5417,a,5412,a)]. 38.01/37.90 Derived: hAPP(complex,complex,uminus_uminus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),A),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),B),A). [resolve(5417,a,5413,a)]. 38.01/37.90 5418 ab_group_add(rat) # label(arity_Rat_Orat___Groups_Oab__group__add) # label(axiom). [assumption]. 38.01/37.90 Derived: hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),hAPP(fun(A,rat),fun(A,rat),hAPP(fun(A,fun(rat,rat)),fun(fun(A,rat),fun(A,rat)),combs(A,rat,rat),hAPP(fun(A,rat),fun(A,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(A,rat),fun(A,fun(rat,rat))),combb(rat,fun(rat,rat),A),minus_minus(rat)),B)),C)),D) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),B),D)),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),C),D)). [resolve(5418,a,5408,a)]. 38.01/37.90 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),B),C)) | hAPP(fun(nat,bool),rat,hAPP(fun(nat,rat),fun(fun(nat,bool),rat),big_co1399186613setsum(nat,rat),D),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),B),C)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(fun(nat,bool),rat,hAPP(fun(nat,rat),fun(fun(nat,bool),rat),big_co1399186613setsum(nat,rat),D),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),A),C))),hAPP(fun(nat,bool),rat,hAPP(fun(nat,rat),fun(fun(nat,bool),rat),big_co1399186613setsum(nat,rat),D),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),A),B))). [resolve(5418,a,5410,a)]. 38.20/38.12 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),D),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),D),B)),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),D),C)). [resolve(5418,a,5411,a)]. 38.20/38.12 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B) != hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),C),D) | ti(rat,B) != ti(rat,A) | ti(rat,D) = ti(rat,C). [resolve(5418,a,5414,a)]. 38.20/38.12 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B) != hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),C),D) | ti(rat,B) = ti(rat,A) | ti(rat,D) != ti(rat,C). [resolve(5418,a,5415,a)]. 38.20/38.12 Derived: hAPP(rat,rat,uminus_uminus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,uminus_uminus(rat),A)),hAPP(rat,rat,uminus_uminus(rat),B)). [resolve(5418,a,5416,a)]. 38.20/38.12 Derived: hAPP(rat,rat,uminus_uminus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),B),A). [resolve(5418,a,5417,a)]. 38.20/38.12 5419 -ab_group_add(A) | hAPP(A,A,uminus_uminus(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,uminus_uminus(A),B)),hAPP(A,A,uminus_uminus(A),C)) # label(fact_1917_minus__diff__minus) # label(axiom). [clausify(2448)]. 38.20/38.12 Derived: hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,uminus_uminus(real),A)),hAPP(real,real,uminus_uminus(real),B)). [resolve(5419,a,5409,a)]. 38.20/38.12 Derived: hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,uminus_uminus(int),A)),hAPP(int,int,uminus_uminus(int),B)). [resolve(5419,a,5412,a)]. 38.20/38.12 Derived: hAPP(complex,complex,uminus_uminus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),A),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(complex,complex,uminus_uminus(complex),A)),hAPP(complex,complex,uminus_uminus(complex),B)). [resolve(5419,a,5413,a)]. 38.20/38.12 Derived: hAPP(rat,rat,uminus_uminus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(rat,rat,uminus_uminus(rat),A)),hAPP(rat,rat,uminus_uminus(rat),B)). [resolve(5419,a,5418,a)]. 38.20/38.12 5420 -ab_group_add(A) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,uminus_uminus(A),B)),B) # label(fact_1960_ab__left__minus) # label(axiom). [clausify(2691)]. 38.20/38.12 Derived: zero_zero(real) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,uminus_uminus(real),A)),A). [resolve(5420,a,5409,a)]. 38.20/38.12 Derived: zero_zero(int) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,uminus_uminus(int),A)),A). [resolve(5420,a,5412,a)]. 38.20/38.12 Derived: zero_zero(complex) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,uminus_uminus(complex),A)),A). [resolve(5420,a,5413,a)]. 38.20/38.12 Derived: zero_zero(rat) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,uminus_uminus(rat),A)),A). [resolve(5420,a,5418,a)]. 38.92/38.81 5421 -ab_group_add(A) | hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),hAPP(A,A,uminus_uminus(A),C)) # label(fact_1968_ab__diff__minus) # label(axiom). [clausify(3008)]. 38.92/38.81 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),A),B) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),hAPP(complex,complex,uminus_uminus(complex),B)). [resolve(5421,a,5413,a)]. 38.92/38.81 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),hAPP(rat,rat,uminus_uminus(rat),B)). [resolve(5421,a,5418,a)]. 38.92/38.81 5422 -ab_group_add(A) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),hAPP(fun(B,A),fun(B,A),hAPP(fun(A,A),fun(fun(B,A),fun(B,A)),combb(A,A,B),uminus_uminus(A)),C)),D) = hAPP(A,A,uminus_uminus(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),C),D)) # label(fact_4238_setsum__negf) # label(axiom). [clausify(3345)]. 38.92/38.81 Derived: hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),hAPP(fun(A,real),fun(A,real),hAPP(fun(real,real),fun(fun(A,real),fun(A,real)),combb(real,real,A),uminus_uminus(real)),B)),C) = hAPP(real,real,uminus_uminus(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),B),C)). [resolve(5422,a,5409,a)]. 38.92/38.81 Derived: hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),hAPP(fun(A,int),fun(A,int),hAPP(fun(int,int),fun(fun(A,int),fun(A,int)),combb(int,int,A),uminus_uminus(int)),B)),C) = hAPP(int,int,uminus_uminus(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),B),C)). [resolve(5422,a,5412,a)]. 38.92/38.81 Derived: hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co1399186613setsum(A,complex),hAPP(fun(A,complex),fun(A,complex),hAPP(fun(complex,complex),fun(fun(A,complex),fun(A,complex)),combb(complex,complex,A),uminus_uminus(complex)),B)),C) = hAPP(complex,complex,uminus_uminus(complex),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co1399186613setsum(A,complex),B),C)). [resolve(5422,a,5413,a)]. 38.92/38.81 Derived: hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),hAPP(fun(A,rat),fun(A,rat),hAPP(fun(rat,rat),fun(fun(A,rat),fun(A,rat)),combb(rat,rat,A),uminus_uminus(rat)),B)),C) = hAPP(rat,rat,uminus_uminus(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),B),C)). [resolve(5422,a,5418,a)]. 38.92/38.81 5423 -ab_group_add(A) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) | zero_zero(A) = hAPP(fun(nat,bool),A,hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),minus_minus(A)),D)),hAPP(fun(nat,nat),fun(nat,A),hAPP(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),combb(nat,A,nat),D),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),one_one(nat))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),B),C)) # label(fact_4643_setsum__natinterval__difff) # label(axiom). [clausify(3691)]. 38.92/38.81 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | zero_zero(real) = hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),minus_minus(real)),C)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),C),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),one_one(nat))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),A),B)). [resolve(5423,a,5409,a)]. 38.92/38.81 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | zero_zero(int) = hAPP(fun(nat,bool),int,hAPP(fun(nat,int),fun(fun(nat,bool),int),big_co1399186613setsum(nat,int),hAPP(fun(nat,int),fun(nat,int),hAPP(fun(nat,fun(int,int)),fun(fun(nat,int),fun(nat,int)),combs(nat,int,int),hAPP(fun(nat,int),fun(nat,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(nat,int),fun(nat,fun(int,int))),combb(int,fun(int,int),nat),minus_minus(int)),C)),hAPP(fun(nat,nat),fun(nat,int),hAPP(fun(nat,int),fun(fun(nat,nat),fun(nat,int)),combb(nat,int,nat),C),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),one_one(nat))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),A),B)). [resolve(5423,a,5412,a)]. 38.92/38.81 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | zero_zero(complex) = hAPP(fun(nat,bool),complex,hAPP(fun(nat,complex),fun(fun(nat,bool),complex),big_co1399186613setsum(nat,complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),minus_minus(complex)),C)),hAPP(fun(nat,nat),fun(nat,complex),hAPP(fun(nat,complex),fun(fun(nat,nat),fun(nat,complex)),combb(nat,complex,nat),C),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),one_one(nat))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),A),B)). [resolve(5423,a,5413,a)]. 38.92/38.81 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | zero_zero(rat) = hAPP(fun(nat,bool),rat,hAPP(fun(nat,rat),fun(fun(nat,bool),rat),big_co1399186613setsum(nat,rat),hAPP(fun(nat,rat),fun(nat,rat),hAPP(fun(nat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,rat)),combs(nat,rat,rat),hAPP(fun(nat,rat),fun(nat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,fun(rat,rat))),combb(rat,fun(rat,rat),nat),minus_minus(rat)),C)),hAPP(fun(nat,nat),fun(nat,rat),hAPP(fun(nat,rat),fun(fun(nat,nat),fun(nat,rat)),combb(nat,rat,nat),C),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),one_one(nat))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),A),B)). [resolve(5423,a,5418,a)]. 38.92/38.81 5424 -ab_group_add(A) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) | hAPP(fun(nat,bool),A,hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),minus_minus(A)),D)),hAPP(fun(nat,nat),fun(nat,A),hAPP(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),combb(nat,A,nat),D),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),one_one(nat))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(nat,A,D,B)),hAPP(nat,A,D,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),C),one_one(nat)))) # label(fact_4643_setsum__natinterval__difff) # label(axiom). [clausify(3691)]. 38.92/38.81 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),minus_minus(real)),C)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),C),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),one_one(nat))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),A),B)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(nat,real,C,A)),hAPP(nat,real,C,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),one_one(nat)))). [resolve(5424,a,5409,a)]. 38.92/38.89 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hAPP(fun(nat,bool),int,hAPP(fun(nat,int),fun(fun(nat,bool),int),big_co1399186613setsum(nat,int),hAPP(fun(nat,int),fun(nat,int),hAPP(fun(nat,fun(int,int)),fun(fun(nat,int),fun(nat,int)),combs(nat,int,int),hAPP(fun(nat,int),fun(nat,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(nat,int),fun(nat,fun(int,int))),combb(int,fun(int,int),nat),minus_minus(int)),C)),hAPP(fun(nat,nat),fun(nat,int),hAPP(fun(nat,int),fun(fun(nat,nat),fun(nat,int)),combb(nat,int,nat),C),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),one_one(nat))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),A),B)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,C,A)),hAPP(nat,int,C,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),one_one(nat)))). [resolve(5424,a,5412,a)]. 38.92/38.89 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hAPP(fun(nat,bool),complex,hAPP(fun(nat,complex),fun(fun(nat,bool),complex),big_co1399186613setsum(nat,complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),minus_minus(complex)),C)),hAPP(fun(nat,nat),fun(nat,complex),hAPP(fun(nat,complex),fun(fun(nat,nat),fun(nat,complex)),combb(nat,complex,nat),C),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),one_one(nat))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),A),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(nat,complex,C,A)),hAPP(nat,complex,C,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),one_one(nat)))). [resolve(5424,a,5413,a)]. 38.92/38.89 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hAPP(fun(nat,bool),rat,hAPP(fun(nat,rat),fun(fun(nat,bool),rat),big_co1399186613setsum(nat,rat),hAPP(fun(nat,rat),fun(nat,rat),hAPP(fun(nat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,rat)),combs(nat,rat,rat),hAPP(fun(nat,rat),fun(nat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,fun(rat,rat))),combb(rat,fun(rat,rat),nat),minus_minus(rat)),C)),hAPP(fun(nat,nat),fun(nat,rat),hAPP(fun(nat,rat),fun(fun(nat,nat),fun(nat,rat)),combb(nat,rat,nat),C),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),one_one(nat))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),A),B)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(nat,rat,C,A)),hAPP(nat,rat,C,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),one_one(nat)))). [resolve(5424,a,5418,a)]. 38.92/38.89 5425 -ab_group_add(A) | ti(A,B) != ti(A,C) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),C),B) # label(fact_734_eq__iff__diff__eq__0) # label(axiom). [clausify(3799)]. 38.92/38.89 Derived: ti(real,A) != ti(real,B) | zero_zero(real) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),A). [resolve(5425,a,5409,a)]. 38.92/38.89 Derived: ti(int,A) != ti(int,B) | zero_zero(int) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),A). [resolve(5425,a,5412,a)]. 38.92/38.89 Derived: ti(complex,A) != ti(complex,B) | zero_zero(complex) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),B),A). [resolve(5425,a,5413,a)]. 39.22/39.17 Derived: ti(rat,A) != ti(rat,B) | zero_zero(rat) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),B),A). [resolve(5425,a,5418,a)]. 39.22/39.17 5426 -ab_group_add(A) | ti(A,B) = ti(A,C) | zero_zero(A) != hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),C),B) # label(fact_734_eq__iff__diff__eq__0) # label(axiom). [clausify(3799)]. 39.22/39.17 Derived: ti(real,A) = ti(real,B) | zero_zero(real) != hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),A). [resolve(5426,a,5409,a)]. 39.22/39.17 Derived: ti(int,A) = ti(int,B) | zero_zero(int) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),A). [resolve(5426,a,5412,a)]. 39.22/39.17 Derived: ti(complex,A) = ti(complex,B) | zero_zero(complex) != hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),B),A). [resolve(5426,a,5413,a)]. 39.22/39.17 Derived: ti(rat,A) = ti(rat,B) | zero_zero(rat) != hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),B),A). [resolve(5426,a,5418,a)]. 39.22/39.17 5427 -ab_group_add(A) | hAPP(fun(nat,bool),A,hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),hAPP(fun(nat,nat),fun(nat,A),hAPP(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),combb(nat,A,nat),B),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),C))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),D)) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(fun(nat,bool),A,hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),B),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),D),C)))),hAPP(fun(nat,bool),A,hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),B),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),C))) # label(fact_4203_sumr__offset) # label(axiom). [clausify(4148)]. 39.22/39.17 Derived: hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),A),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),B))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),C)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),A),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),C),B)))),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),A),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),B))). [resolve(5427,a,5409,a)]. 39.22/39.17 Derived: hAPP(fun(nat,bool),int,hAPP(fun(nat,int),fun(fun(nat,bool),int),big_co1399186613setsum(nat,int),hAPP(fun(nat,nat),fun(nat,int),hAPP(fun(nat,int),fun(fun(nat,nat),fun(nat,int)),combb(nat,int,nat),A),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),B))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),C)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(fun(nat,bool),int,hAPP(fun(nat,int),fun(fun(nat,bool),int),big_co1399186613setsum(nat,int),A),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),C),B)))),hAPP(fun(nat,bool),int,hAPP(fun(nat,int),fun(fun(nat,bool),int),big_co1399186613setsum(nat,int),A),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),B))). [resolve(5427,a,5412,a)]. 39.22/39.17 Derived: hAPP(fun(nat,bool),complex,hAPP(fun(nat,complex),fun(fun(nat,bool),complex),big_co1399186613setsum(nat,complex),hAPP(fun(nat,nat),fun(nat,complex),hAPP(fun(nat,complex),fun(fun(nat,nat),fun(nat,complex)),combb(nat,complex,nat),A),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),B))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),C)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(fun(nat,bool),complex,hAPP(fun(nat,complex),fun(fun(nat,bool),complex),big_co1399186613setsum(nat,complex),A),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),C),B)))),hAPP(fun(nat,bool),complex,hAPP(fun(nat,complex),fun(fun(nat,bool),complex),big_co1399186613setsum(nat,complex),A),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),B))). [resolve(5427,a,5413,a)]. 43.10/43.01 Derived: hAPP(fun(nat,bool),rat,hAPP(fun(nat,rat),fun(fun(nat,bool),rat),big_co1399186613setsum(nat,rat),hAPP(fun(nat,nat),fun(nat,rat),hAPP(fun(nat,rat),fun(fun(nat,nat),fun(nat,rat)),combb(nat,rat,nat),A),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),B))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),C)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(fun(nat,bool),rat,hAPP(fun(nat,rat),fun(fun(nat,bool),rat),big_co1399186613setsum(nat,rat),A),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),C),B)))),hAPP(fun(nat,bool),rat,hAPP(fun(nat,rat),fun(fun(nat,bool),rat),big_co1399186613setsum(nat,rat),A),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),B))). [resolve(5427,a,5418,a)]. 43.10/43.01 5428 -ab_group_add(A) | hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),D),E)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),C),E)) # label(fact_741_add__diff__add) # label(axiom). [clausify(4331)]. 43.10/43.01 Derived: hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),C),D)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),C)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),D)). [resolve(5428,a,5409,a)]. 43.10/43.01 Derived: hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),C),D)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),C)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),D)). [resolve(5428,a,5412,a)]. 43.10/43.01 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),B)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),C),D)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),A),C)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),B),D)). [resolve(5428,a,5413,a)]. 43.10/43.01 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),C),D)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),C)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),B),D)). [resolve(5428,a,5418,a)]. 43.10/43.01 5429 archim1727834104eiling(rat) # label(arity_Rat_Orat___Archimedean__Field_Ofloor__ceiling) # label(axiom). [assumption]. 43.10/43.01 5430 -archim1727834104eiling(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(A,int,archimedean_ceiling(A),B)),hAPP(int,int,number_number_of(int),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(int,A,number_number_of(A),C)),one_one(A)))) # label(fact_3336_ceiling__less__number__of) # label(axiom). [clausify(29)]. 43.10/43.01 5431 -archim1727834104eiling(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(A,int,archimedean_ceiling(A),B)),hAPP(int,int,number_number_of(int),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(int,A,number_number_of(A),C)),one_one(A)))) # label(fact_3336_ceiling__less__number__of) # label(axiom). [clausify(29)]. 43.10/43.01 5432 -archim1727834104eiling(A) | one_one(int) = hAPP(A,int,archimedean_ceiling(A),one_one(A)) # label(fact_3307_ceiling__one) # label(axiom). [clausify(170)]. 43.10/43.01 5433 -archim1727834104eiling(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(int,A,ring_1_of_int(A),B)),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(int,A,ring_1_of_int(A),B)),one_one(A)))) | ti(int,B) = hAPP(A,int,archim2085082626_floor(A),C) # label(fact_3630_floor__unique) # label(axiom). [clausify(409)]. 43.10/43.01 5434 -archim1727834104eiling(A) | hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(A,int,archim2085082626_floor(A),B)),one_one(int)) = hAPP(A,int,archim2085082626_floor(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),one_one(A))) # label(fact_2915_floor__diff__one) # label(axiom). [clausify(507)]. 43.10/43.01 5435 -archim1727834104eiling(A) | hAPP(int,int,number_number_of(int),B) = hAPP(A,int,archim2085082626_floor(A),hAPP(int,A,number_number_of(A),B)) # label(fact_2898_floor__number__of) # label(axiom). [clausify(537)]. 43.10/43.01 5436 -archim1727834104eiling(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(int,A,ring_1_of_int(A),B)),one_one(A))),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),hAPP(A,int,archimedean_ceiling(A),C))) # label(fact_3637_le__ceiling__iff) # label(axiom). [clausify(551)]. 43.10/43.01 5437 -archim1727834104eiling(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(int,A,ring_1_of_int(A),B)),one_one(A))),C)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),hAPP(A,int,archimedean_ceiling(A),C))) # label(fact_3637_le__ceiling__iff) # label(axiom). [clausify(551)]. 43.10/43.01 5438 -archim1727834104eiling(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(A,int,archimedean_ceiling(A),B)),zero_zero(int))) # label(fact_3323_ceiling__le__zero) # label(axiom). [clausify(573)]. 43.10/43.01 5439 -archim1727834104eiling(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(A,int,archimedean_ceiling(A),B)),zero_zero(int))) # label(fact_3323_ceiling__le__zero) # label(axiom). [clausify(573)]. 43.10/43.01 5440 -archim1727834104eiling(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(int,A,number_number_of(A),min)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(A,int,archimedean_ceiling(A),B))) # label(fact_3335_zero__le__ceiling) # label(axiom). [clausify(583)]. 43.10/43.01 5441 -archim1727834104eiling(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(int,A,number_number_of(A),min)),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(A,int,archimedean_ceiling(A),B))) # label(fact_3335_zero__le__ceiling) # label(axiom). [clausify(583)]. 43.10/43.01 5442 -archim1727834104eiling(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(A,int,archimedean_ceiling(A),B)),hAPP(A,int,archimedean_ceiling(A),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) # label(fact_3305_ceiling__less__cancel) # label(axiom). [clausify(676)]. 43.10/43.01 5443 -archim1727834104eiling(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),B)),hAPP(A,int,archim2085082626_floor(A),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(int,A,number_number_of(A),B)),C)) # label(fact_2912_number__of__le__floor) # label(axiom). [clausify(734)]. 43.10/43.01 5444 -archim1727834104eiling(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),B)),hAPP(A,int,archim2085082626_floor(A),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(int,A,number_number_of(A),B)),C)) # label(fact_2912_number__of__le__floor) # label(axiom). [clausify(734)]. 43.10/43.01 5445 -archim1727834104eiling(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(int,A,number_number_of(A),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(A,int,archim2085082626_floor(A),B))) # label(fact_2873_one__less__floor) # label(axiom). [clausify(772)]. 43.10/43.01 5446 -archim1727834104eiling(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(int,A,number_number_of(A),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(A,int,archim2085082626_floor(A),B))) # label(fact_2873_one__less__floor) # label(axiom). [clausify(772)]. 43.10/43.01 5447 -archim1727834104eiling(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),hAPP(A,int,archimedean_ceiling(A),B))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) # label(fact_3325_one__le__ceiling) # label(axiom). [clausify(796)]. 43.10/43.01 5448 -archim1727834104eiling(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),hAPP(A,int,archimedean_ceiling(A),B))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) # label(fact_3325_one__le__ceiling) # label(axiom). [clausify(796)]. 43.10/43.01 5449 -archim1727834104eiling(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(A,int,archim2085082626_floor(A),B)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(int,A,ring_1_of_int(A),C)),one_one(A)))) # label(fact_3634_floor__le__iff) # label(axiom). [clausify(798)]. 43.10/43.01 5450 -archim1727834104eiling(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(A,int,archim2085082626_floor(A),B)),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(int,A,ring_1_of_int(A),C)),one_one(A)))) # label(fact_3634_floor__le__iff) # label(axiom). [clausify(798)]. 43.10/43.01 5451 -archim1727834104eiling(A) | hAPP(int,int,uminus_uminus(int),hAPP(A,int,archimedean_ceiling(A),B)) = hAPP(A,int,archim2085082626_floor(A),hAPP(A,A,uminus_uminus(A),B)) # label(fact_3318_floor__minus) # label(axiom). [clausify(821)]. 43.10/43.01 5452 -archim1727834104eiling(A) | hAPP(int,int,number_number_of(int),B) = hAPP(A,int,archimedean_ceiling(A),hAPP(int,A,number_number_of(A),B)) # label(fact_3308_ceiling__number__of) # label(axiom). [clausify(878)]. 43.10/43.01 5453 -archim1727834104eiling(A) | hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(A,int,archimedean_ceiling(A),B)),hAPP(int,int,number_number_of(int),C)) = hAPP(A,int,archimedean_ceiling(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),hAPP(int,A,number_number_of(A),C))) # label(fact_3331_ceiling__add__number__of) # label(axiom). [clausify(936)]. 43.10/43.01 5454 -archim1727834104eiling(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(A,int,archimedean_ceiling(A),B)),one_one(int))) # label(fact_3324_ceiling__less__one) # label(axiom). [clausify(1139)]. 43.10/43.01 5455 -archim1727834104eiling(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(A,int,archimedean_ceiling(A),B)),one_one(int))) # label(fact_3324_ceiling__less__one) # label(axiom). [clausify(1139)]. 43.10/43.01 5456 -archim1727834104eiling(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),hAPP(A,int,archim2085082626_floor(A),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(int,A,ring_1_of_int(A),B)),one_one(A))),C)) # label(fact_3633_less__floor__iff) # label(axiom). [clausify(1244)]. 43.10/43.01 5457 -archim1727834104eiling(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),hAPP(A,int,archim2085082626_floor(A),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(int,A,ring_1_of_int(A),B)),one_one(A))),C)) # label(fact_3633_less__floor__iff) # label(axiom). [clausify(1244)]. 43.10/43.01 5458 -archim1727834104eiling(A) | hAPP(int,int,uminus_uminus(int),hAPP(A,int,archim2085082626_floor(A),hAPP(A,A,uminus_uminus(A),B))) = hAPP(A,int,archimedean_ceiling(A),B) # label(fact_3320_ceiling__def) # label(axiom). [clausify(1261)]. 43.10/43.01 5459 -archim1727834104eiling(A) | hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(A,int,archim2085082626_floor(A),B)),hAPP(int,int,number_number_of(int),C)) = hAPP(A,int,archim2085082626_floor(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),hAPP(int,A,number_number_of(A),C))) # label(fact_2914_floor__add__number__of) # label(axiom). [clausify(1300)]. 43.10/43.01 5460 -archim1727834104eiling(A) | hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(A,int,archim2085082626_floor(A),B)),one_one(int)) = hAPP(A,int,archim2085082626_floor(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),one_one(A))) # label(fact_2913_floor__add__one) # label(axiom). [clausify(1321)]. 43.10/43.01 5461 -archim1727834104eiling(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(A,int,archim2085082626_floor(A),B)),hAPP(int,int,number_number_of(int),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(int,A,number_number_of(A),C))) # label(fact_2911_floor__less__number__of) # label(axiom). [clausify(1330)]. 43.10/43.01 5462 -archim1727834104eiling(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(A,int,archim2085082626_floor(A),B)),hAPP(int,int,number_number_of(int),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(int,A,number_number_of(A),C))) # label(fact_2911_floor__less__number__of) # label(axiom). [clausify(1330)]. 43.10/43.01 5463 -archim1727834104eiling(A) | hAPP(real,int,archimedean_ceiling(real),hAPP(int,real,real_1(int),hAPP(A,int,archim2085082626_floor(A),B))) = hAPP(A,int,archim2085082626_floor(A),B) # label(fact_3382_ceiling__floor) # label(axiom). [clausify(1351)]. 43.10/43.01 5464 -archim1727834104eiling(A) | hAPP(nat,int,semiring_1_of_nat(int),B) = hAPP(A,int,archim2085082626_floor(A),hAPP(nat,A,semiring_1_of_nat(A),B)) # label(fact_2901_floor__of__nat) # label(axiom). [clausify(1390)]. 43.10/43.01 5465 -archim1727834104eiling(A) | hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(A,int,archimedean_ceiling(A),B)),one_one(int)) = hAPP(A,int,archimedean_ceiling(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),one_one(A))) # label(fact_3330_ceiling__add__one) # label(axiom). [clausify(1419)]. 43.10/43.01 5466 -archim1727834104eiling(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(int,A,number_number_of(A),B)),one_one(A))),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),B)),hAPP(A,int,archim2085082626_floor(A),C))) # label(fact_2919_number__of__less__floor) # label(axiom). [clausify(1560)]. 43.10/43.01 5467 -archim1727834104eiling(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(int,A,number_number_of(A),B)),one_one(A))),C)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),B)),hAPP(A,int,archim2085082626_floor(A),C))) # label(fact_2919_number__of__less__floor) # label(axiom). [clausify(1560)]. 43.10/43.01 5468 -archim1727834104eiling(A) | hAPP(real,int,archim2085082626_floor(real),hAPP(int,real,real_1(int),hAPP(A,int,archimedean_ceiling(A),B))) = hAPP(A,int,archimedean_ceiling(A),B) # label(fact_3380_floor__ceiling) # label(axiom). [clausify(1629)]. 43.10/43.01 5469 -archim1727834104eiling(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(int,A,ring_1_of_int(A),hAPP(A,int,archimedean_ceiling(A),B)))) # label(fact_3632_ceiling__correct) # label(axiom). [clausify(1638)]. 43.10/43.01 5470 -archim1727834104eiling(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(int,A,ring_1_of_int(A),hAPP(A,int,archimedean_ceiling(A),B))),one_one(A))),B)) # label(fact_3632_ceiling__correct) # label(axiom). [clausify(1638)]. 43.10/43.01 5471 -archim1727834104eiling(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(A,int,archimedean_ceiling(A),B))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) # label(fact_3322_zero__less__ceiling) # label(axiom). [clausify(1670)]. 43.10/43.01 5472 -archim1727834104eiling(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(A,int,archimedean_ceiling(A),B))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) # label(fact_3322_zero__less__ceiling) # label(axiom). [clausify(1670)]. 43.10/43.01 5473 -archim1727834104eiling(A) | hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(A,int,archimedean_ceiling(A),B)),hAPP(int,int,number_number_of(int),C)) = hAPP(A,int,archimedean_ceiling(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),hAPP(int,A,number_number_of(A),C))) # label(fact_3333_ceiling__diff__number__of) # label(axiom). [clausify(1677)]. 43.10/43.01 5474 -archim1727834104eiling(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(A,int,archim2085082626_floor(A),B))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),B)) # label(fact_2907_zero__less__floor) # label(axiom). [clausify(1694)]. 43.10/43.01 5475 -archim1727834104eiling(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(A,int,archim2085082626_floor(A),B))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),B)) # label(fact_2907_zero__less__floor) # label(axiom). [clausify(1694)]. 43.10/43.01 5476 -archim1727834104eiling(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(A,int,archim2085082626_floor(A),B)),hAPP(A,int,archim2085082626_floor(A),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) # label(fact_2890_floor__less__cancel) # label(axiom). [clausify(1696)]. 43.10/43.01 5477 -archim1727834104eiling(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(A,int,archim2085082626_floor(A),B)),zero_zero(int))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),one_one(A))) # label(fact_2908_floor__le__zero) # label(axiom). [clausify(1871)]. 43.10/43.01 5478 -archim1727834104eiling(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(A,int,archim2085082626_floor(A),B)),zero_zero(int))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),one_one(A))) # label(fact_2908_floor__le__zero) # label(axiom). [clausify(1871)]. 43.10/43.01 5479 -archim1727834104eiling(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(A,int,archimedean_ceiling(A),B))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) # label(fact_3326_one__less__ceiling) # label(axiom). [clausify(2156)]. 43.10/43.01 5480 -archim1727834104eiling(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(A,int,archimedean_ceiling(A),B))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) # label(fact_3326_one__less__ceiling) # label(axiom). [clausify(2156)]. 43.10/43.01 5481 -archim1727834104eiling(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(int,A,ring_1_of_int(A),C)),one_one(A)))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(A,int,archimedean_ceiling(A),B)),C)) # label(fact_3636_ceiling__less__iff) # label(axiom). [clausify(2290)]. 43.10/43.01 5482 -archim1727834104eiling(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(int,A,ring_1_of_int(A),C)),one_one(A)))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(A,int,archimedean_ceiling(A),B)),C)) # label(fact_3636_ceiling__less__iff) # label(axiom). [clausify(2290)]. 43.10/43.01 5483 -archim1727834104eiling(A) | hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(A,int,archim2085082626_floor(A),B)),hAPP(int,int,number_number_of(int),C)) = hAPP(A,int,archim2085082626_floor(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),hAPP(int,A,number_number_of(A),C))) # label(fact_2916_floor__diff__number__of) # label(axiom). [clausify(2315)]. 43.10/43.01 5484 -archim1727834104eiling(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(A,int,archimedean_ceiling(A),B)),hAPP(A,int,archimedean_ceiling(A),C))) # label(fact_3306_ceiling__mono) # label(axiom). [clausify(2436)]. 43.10/43.01 5485 -archim1727834104eiling(A) | hAPP(int,int,uminus_uminus(int),hAPP(A,int,archim2085082626_floor(A),B)) = hAPP(A,int,archimedean_ceiling(A),hAPP(A,A,uminus_uminus(A),B)) # label(fact_3319_ceiling__minus) # label(axiom). [clausify(2462)]. 43.10/43.01 5486 -archim1727834104eiling(A) | ti(int,B) = hAPP(A,int,archimedean_ceiling(A),hAPP(int,A,ring_1_of_int(A),B)) # label(fact_3613_ceiling__of__int) # label(axiom). [clausify(2467)]. 43.10/43.01 5487 -archim1727834104eiling(A) | hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(A,int,archim2085082626_floor(A),B)),C) = hAPP(A,int,archim2085082626_floor(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),hAPP(int,A,ring_1_of_int(A),C))) # label(fact_3623_floor__diff__of__int) # label(axiom). [clausify(2548)]. 43.10/43.01 5488 -archim1727834104eiling(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(int,A,ring_1_of_int(A),B)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),hAPP(A,int,archim2085082626_floor(A),C))) # label(fact_3618_le__floor__iff) # label(axiom). [clausify(2726)]. 43.10/43.01 5489 -archim1727834104eiling(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(int,A,ring_1_of_int(A),B)),C)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),hAPP(A,int,archim2085082626_floor(A),C))) # label(fact_3618_le__floor__iff) # label(axiom). [clausify(2726)]. 43.10/43.01 5490 -archim1727834104eiling(A) | archim2085082626_floor(A) = ti(fun(A,int),archim2085082626_floor(A)) # label(tsy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_res) # label(axiom). [clausify(2808)]. 43.10/43.01 5491 -archim1727834104eiling(A) | one_one(int) = hAPP(A,int,archim2085082626_floor(A),one_one(A)) # label(fact_2896_floor__one) # label(axiom). [clausify(2812)]. 43.10/43.01 5492 -archim1727834104eiling(A) | ti(int,B) = hAPP(A,int,archim2085082626_floor(A),hAPP(int,A,ring_1_of_int(A),B)) # label(fact_3597_floor__of__int) # label(axiom). [clausify(2814)]. 43.10/43.01 5493 -archim1727834104eiling(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),one_one(A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(A,int,archimedean_ceiling(A),B)),one_one(int))) # label(fact_3327_ceiling__le__one) # label(axiom). [clausify(2863)]. 43.10/43.01 5494 -archim1727834104eiling(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),one_one(A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(A,int,archimedean_ceiling(A),B)),one_one(int))) # label(fact_3327_ceiling__le__one) # label(axiom). [clausify(2863)]. 43.10/43.01 5495 -archim1727834104eiling(A) | zero_zero(int) = hAPP(A,int,archim2085082626_floor(A),zero_zero(A)) # label(fact_2889_floor__zero) # label(axiom). [clausify(3042)]. 43.10/43.01 5496 -archim1727834104eiling(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),B)),hAPP(A,int,archimedean_ceiling(A),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(int,A,number_number_of(A),B)),one_one(A))),C)) # label(fact_3236_number__of__le__ceiling) # label(axiom). [clausify(3078)]. 43.10/43.01 5497 -archim1727834104eiling(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),B)),hAPP(A,int,archimedean_ceiling(A),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(int,A,number_number_of(A),B)),one_one(A))),C)) # label(fact_3236_number__of__le__ceiling) # label(axiom). [clausify(3078)]. 43.10/43.01 5498 -archim1727834104eiling(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(A,int,archim2085082626_floor(A),B)),hAPP(A,int,archim2085082626_floor(A),C))) # label(fact_2891_floor__mono) # label(axiom). [clausify(3084)]. 43.10/43.01 5499 -archim1727834104eiling(A) | hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(A,int,archimedean_ceiling(A),B)),C) = hAPP(A,int,archimedean_ceiling(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),hAPP(int,A,ring_1_of_int(A),C))) # label(fact_3622_ceiling__add__of__int) # label(axiom). [clausify(3194)]. 43.10/43.01 5500 -archim1727834104eiling(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(A,int,archimedean_ceiling(A),B)),hAPP(int,int,number_number_of(int),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(int,A,number_number_of(A),C))) # label(fact_3329_ceiling__le__number__of) # label(axiom). [clausify(3213)]. 43.10/43.01 5501 -archim1727834104eiling(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(A,int,archimedean_ceiling(A),B)),hAPP(int,int,number_number_of(int),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(int,A,number_number_of(A),C))) # label(fact_3329_ceiling__le__number__of) # label(axiom). [clausify(3213)]. 43.10/43.01 5502 -archim1727834104eiling(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(int,A,ring_1_of_int(A),hAPP(A,int,archim2085082626_floor(A),B))),B)) # label(fact_3610_of__int__floor__le) # label(axiom). [clausify(3289)]. 43.10/43.01 5503 -archim1727834104eiling(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(A,int,archimedean_ceiling(A),B)),zero_zero(int))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(int,A,number_number_of(A),min))) # label(fact_3334_ceiling__less__zero) # label(axiom). [clausify(3340)]. 43.10/43.01 5504 -archim1727834104eiling(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(A,int,archimedean_ceiling(A),B)),zero_zero(int))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(int,A,number_number_of(A),min))) # label(fact_3334_ceiling__less__zero) # label(axiom). [clausify(3340)]. 43.10/43.01 5505 -archim1727834104eiling(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(int,A,ring_1_of_int(A),C))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(A,int,archimedean_ceiling(A),B)),C)) # label(fact_3621_ceiling__le__iff) # label(axiom). [clausify(3493)]. 43.10/43.01 5506 -archim1727834104eiling(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(int,A,ring_1_of_int(A),C))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(A,int,archimedean_ceiling(A),B)),C)) # label(fact_3621_ceiling__le__iff) # label(axiom). [clausify(3493)]. 43.10/43.01 5507 -archim1727834104eiling(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(int,A,ring_1_of_int(A),hAPP(A,int,archim2085082626_floor(A),B))),B)) # label(fact_3635_floor__correct) # label(axiom). [clausify(3539)]. 43.10/43.01 5508 -archim1727834104eiling(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(int,A,ring_1_of_int(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(A,int,archim2085082626_floor(A),B)),one_one(int))))) # label(fact_3635_floor__correct) # label(axiom). [clausify(3539)]. 43.10/43.01 5509 -archim1727834104eiling(A) | archimedean_ceiling(A) = ti(fun(A,int),archimedean_ceiling(A)) # label(tsy_c_Archimedean__Field_Oceiling_res) # label(axiom). [clausify(3605)]. 43.10/43.01 5510 -archim1727834104eiling(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),hAPP(A,int,archim2085082626_floor(A),B))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),B)) # label(fact_2910_one__le__floor) # label(axiom). [clausify(3789)]. 43.10/43.01 5511 -archim1727834104eiling(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),hAPP(A,int,archim2085082626_floor(A),B))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),B)) # label(fact_2910_one__le__floor) # label(axiom). [clausify(3789)]. 43.10/43.01 5512 -archim1727834104eiling(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(int,A,ring_1_of_int(A),B)),one_one(A))),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),hAPP(int,A,ring_1_of_int(A),B))) | ti(int,B) = hAPP(A,int,archimedean_ceiling(A),C) # label(fact_3631_ceiling__unique) # label(axiom). [clausify(3953)]. 43.10/43.01 5513 -archim1727834104eiling(A) | zero_zero(int) = hAPP(A,int,archimedean_ceiling(A),zero_zero(A)) # label(fact_3304_ceiling__zero) # label(axiom). [clausify(4051)]. 43.10/43.01 5514 -archim1727834104eiling(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(A,int,archim2085082626_floor(A),B))) # label(fact_2906_zero__le__floor) # label(axiom). [clausify(4081)]. 43.10/43.01 5515 -archim1727834104eiling(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(A,int,archim2085082626_floor(A),B))) # label(fact_2906_zero__le__floor) # label(axiom). [clausify(4081)]. 43.10/43.01 5516 -archim1727834104eiling(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(A,int,archim2085082626_floor(A),B)),zero_zero(int))) # label(fact_2905_floor__less__zero) # label(axiom). [clausify(4115)]. 43.10/43.01 5517 -archim1727834104eiling(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(A,int,archim2085082626_floor(A),B)),zero_zero(int))) # label(fact_2905_floor__less__zero) # label(axiom). [clausify(4115)]. 43.10/43.01 5518 -archim1727834104eiling(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(A,int,archim2085082626_floor(A),B)),one_one(int))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),one_one(A))) # label(fact_2909_floor__less__one) # label(axiom). [clausify(4210)]. 43.10/43.01 5519 -archim1727834104eiling(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(A,int,archim2085082626_floor(A),B)),one_one(int))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),one_one(A))) # label(fact_2909_floor__less__one) # label(axiom). [clausify(4210)]. 43.10/43.01 5520 -archim1727834104eiling(A) | hAPP(nat,int,semiring_1_of_nat(int),B) = hAPP(A,int,archimedean_ceiling(A),hAPP(nat,A,semiring_1_of_nat(A),B)) # label(fact_3310_ceiling__of__nat) # label(axiom). [clausify(4267)]. 43.10/43.01 5521 -archim1727834104eiling(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(int,A,ring_1_of_int(A),B)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),hAPP(A,int,archimedean_ceiling(A),C))) # label(fact_3620_less__ceiling__iff) # label(axiom). [clausify(4380)]. 43.10/43.01 5522 -archim1727834104eiling(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(int,A,ring_1_of_int(A),B)),C)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),hAPP(A,int,archimedean_ceiling(A),C))) # label(fact_3620_less__ceiling__iff) # label(axiom). [clausify(4380)]. 43.10/43.01 5523 -archim1727834104eiling(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(int,A,ring_1_of_int(A),hAPP(A,int,archimedean_ceiling(A),B)))) # label(fact_3607_le__of__int__ceiling) # label(axiom). [clausify(4382)]. 43.10/43.01 5524 -archim1727834104eiling(A) | hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(A,int,archim2085082626_floor(A),B)),C) = hAPP(A,int,archim2085082626_floor(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),hAPP(int,A,ring_1_of_int(A),C))) # label(fact_3619_floor__add__of__int) # label(axiom). [clausify(4454)]. 43.10/43.01 5525 -archim1727834104eiling(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(int,A,number_number_of(A),C)),one_one(A)))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(A,int,archim2085082626_floor(A),B)),hAPP(int,int,number_number_of(int),C))) # label(fact_2920_floor__le__number__of) # label(axiom). [clausify(4516)]. 43.10/43.01 5526 -archim1727834104eiling(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(int,A,number_number_of(A),C)),one_one(A)))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(A,int,archim2085082626_floor(A),B)),hAPP(int,int,number_number_of(int),C))) # label(fact_2920_floor__le__number__of) # label(axiom). [clausify(4516)]. 43.10/43.02 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(rat,int,archimedean_ceiling(rat),A)),hAPP(int,int,number_number_of(int),B))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(int,rat,number_number_of(rat),B)),one_one(rat)))). [resolve(5429,a,5430,a)]. 43.10/43.02 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(rat,int,archimedean_ceiling(rat),A)),hAPP(int,int,number_number_of(int),B))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(int,rat,number_number_of(rat),B)),one_one(rat)))). [resolve(5429,a,5431,a)]. 43.10/43.02 Derived: one_one(int) = hAPP(rat,int,archimedean_ceiling(rat),one_one(rat)). [resolve(5429,a,5432,a)]. 43.10/43.02 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,ring_1_of_int(rat),A)),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,ring_1_of_int(rat),A)),one_one(rat)))) | ti(int,A) = hAPP(rat,int,archim2085082626_floor(rat),B). [resolve(5429,a,5433,a)]. 43.10/43.02 Derived: hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(rat,int,archim2085082626_floor(rat),A)),one_one(int)) = hAPP(rat,int,archim2085082626_floor(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),one_one(rat))). [resolve(5429,a,5434,a)]. 43.10/43.02 Derived: hAPP(int,int,number_number_of(int),A) = hAPP(rat,int,archim2085082626_floor(rat),hAPP(int,rat,number_number_of(rat),A)). [resolve(5429,a,5435,a)]. 43.10/43.02 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(int,rat,ring_1_of_int(rat),A)),one_one(rat))),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(rat,int,archimedean_ceiling(rat),B))). [resolve(5429,a,5436,a)]. 43.10/43.02 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(int,rat,ring_1_of_int(rat),A)),one_one(rat))),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(rat,int,archimedean_ceiling(rat),B))). [resolve(5429,a,5437,a)]. 43.10/43.02 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),zero_zero(rat))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(rat,int,archimedean_ceiling(rat),A)),zero_zero(int))). [resolve(5429,a,5438,a)]. 43.10/43.02 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),zero_zero(rat))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(rat,int,archimedean_ceiling(rat),A)),zero_zero(int))). [resolve(5429,a,5439,a)]. 43.10/43.02 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,number_number_of(rat),min)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(rat,int,archimedean_ceiling(rat),A))). [resolve(5429,a,5440,a)]. 43.10/43.02 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,number_number_of(rat),min)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(rat,int,archimedean_ceiling(rat),A))). [resolve(5429,a,5441,a)]. 43.10/43.02 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(rat,int,archimedean_ceiling(rat),A)),hAPP(rat,int,archimedean_ceiling(rat),B))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)). [resolve(5429,a,5442,a)]. 43.10/43.02 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),A)),hAPP(rat,int,archim2085082626_floor(rat),B))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,number_number_of(rat),A)),B)). [resolve(5429,a,5443,a)]. 43.10/43.02 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),A)),hAPP(rat,int,archim2085082626_floor(rat),B))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,number_number_of(rat),A)),B)). [resolve(5429,a,5444,a)]. 43.10/43.02 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(rat,int,archim2085082626_floor(rat),A))). [resolve(5429,a,5445,a)]. 43.10/43.02 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(rat,int,archim2085082626_floor(rat),A))). [resolve(5429,a,5446,a)]. 43.10/43.02 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),hAPP(rat,int,archimedean_ceiling(rat),A))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)). [resolve(5429,a,5447,a)]. 43.10/43.02 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),hAPP(rat,int,archimedean_ceiling(rat),A))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)). [resolve(5429,a,5448,a)]. 43.10/43.02 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(rat,int,archim2085082626_floor(rat),A)),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,ring_1_of_int(rat),B)),one_one(rat)))). [resolve(5429,a,5449,a)]. 43.10/43.02 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(rat,int,archim2085082626_floor(rat),A)),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,ring_1_of_int(rat),B)),one_one(rat)))). [resolve(5429,a,5450,a)]. 43.10/43.02 Derived: hAPP(int,int,uminus_uminus(int),hAPP(rat,int,archimedean_ceiling(rat),A)) = hAPP(rat,int,archim2085082626_floor(rat),hAPP(rat,rat,uminus_uminus(rat),A)). [resolve(5429,a,5451,a)]. 43.10/43.02 Derived: hAPP(int,int,number_number_of(int),A) = hAPP(rat,int,archimedean_ceiling(rat),hAPP(int,rat,number_number_of(rat),A)). [resolve(5429,a,5452,a)]. 43.10/43.02 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(rat,int,archimedean_ceiling(rat),A)),hAPP(int,int,number_number_of(int),B)) = hAPP(rat,int,archimedean_ceiling(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),hAPP(int,rat,number_number_of(rat),B))). [resolve(5429,a,5453,a)]. 43.10/43.02 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),zero_zero(rat))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(rat,int,archimedean_ceiling(rat),A)),one_one(int))). [resolve(5429,a,5454,a)]. 43.10/43.02 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),zero_zero(rat))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(rat,int,archimedean_ceiling(rat),A)),one_one(int))). [resolve(5429,a,5455,a)]. 43.10/43.02 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(rat,int,archim2085082626_floor(rat),B))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,ring_1_of_int(rat),A)),one_one(rat))),B)). [resolve(5429,a,5456,a)]. 43.10/43.02 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(rat,int,archim2085082626_floor(rat),B))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,ring_1_of_int(rat),A)),one_one(rat))),B)). [resolve(5429,a,5457,a)]. 43.10/43.02 Derived: hAPP(int,int,uminus_uminus(int),hAPP(rat,int,archim2085082626_floor(rat),hAPP(rat,rat,uminus_uminus(rat),A))) = hAPP(rat,int,archimedean_ceiling(rat),A). [resolve(5429,a,5458,a)]. 43.10/43.02 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(rat,int,archim2085082626_floor(rat),A)),hAPP(int,int,number_number_of(int),B)) = hAPP(rat,int,archim2085082626_floor(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),hAPP(int,rat,number_number_of(rat),B))). [resolve(5429,a,5459,a)]. 43.10/43.02 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(rat,int,archim2085082626_floor(rat),A)),one_one(int)) = hAPP(rat,int,archim2085082626_floor(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),one_one(rat))). [resolve(5429,a,5460,a)]. 43.10/43.03 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(rat,int,archim2085082626_floor(rat),A)),hAPP(int,int,number_number_of(int),B))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(int,rat,number_number_of(rat),B))). [resolve(5429,a,5461,a)]. 43.10/43.03 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(rat,int,archim2085082626_floor(rat),A)),hAPP(int,int,number_number_of(int),B))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(int,rat,number_number_of(rat),B))). [resolve(5429,a,5462,a)]. 43.10/43.03 Derived: hAPP(real,int,archimedean_ceiling(real),hAPP(int,real,real_1(int),hAPP(rat,int,archim2085082626_floor(rat),A))) = hAPP(rat,int,archim2085082626_floor(rat),A). [resolve(5429,a,5463,a)]. 43.10/43.03 Derived: hAPP(nat,int,semiring_1_of_nat(int),A) = hAPP(rat,int,archim2085082626_floor(rat),hAPP(nat,rat,semiring_1_of_nat(rat),A)). [resolve(5429,a,5464,a)]. 43.10/43.03 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(rat,int,archimedean_ceiling(rat),A)),one_one(int)) = hAPP(rat,int,archimedean_ceiling(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),one_one(rat))). [resolve(5429,a,5465,a)]. 43.10/43.03 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,number_number_of(rat),A)),one_one(rat))),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),A)),hAPP(rat,int,archim2085082626_floor(rat),B))). [resolve(5429,a,5466,a)]. 43.10/43.03 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,number_number_of(rat),A)),one_one(rat))),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),A)),hAPP(rat,int,archim2085082626_floor(rat),B))). [resolve(5429,a,5467,a)]. 43.10/43.03 Derived: hAPP(real,int,archim2085082626_floor(real),hAPP(int,real,real_1(int),hAPP(rat,int,archimedean_ceiling(rat),A))) = hAPP(rat,int,archimedean_ceiling(rat),A). [resolve(5429,a,5468,a)]. 43.10/43.03 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),hAPP(int,rat,ring_1_of_int(rat),hAPP(rat,int,archimedean_ceiling(rat),A)))). [resolve(5429,a,5469,a)]. 43.10/43.03 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(int,rat,ring_1_of_int(rat),hAPP(rat,int,archimedean_ceiling(rat),A))),one_one(rat))),A)). [resolve(5429,a,5470,a)]. 43.10/43.03 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(rat,int,archimedean_ceiling(rat),A))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)). [resolve(5429,a,5471,a)]. 43.10/43.03 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(rat,int,archimedean_ceiling(rat),A))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)). [resolve(5429,a,5472,a)]. 43.10/43.03 Derived: hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(rat,int,archimedean_ceiling(rat),A)),hAPP(int,int,number_number_of(int),B)) = hAPP(rat,int,archimedean_ceiling(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),hAPP(int,rat,number_number_of(rat),B))). [resolve(5429,a,5473,a)]. 43.10/43.03 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(rat,int,archim2085082626_floor(rat),A))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),A)). [resolve(5429,a,5474,a)]. 43.10/43.03 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(rat,int,archim2085082626_floor(rat),A))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),A)). [resolve(5429,a,5475,a)]. 43.10/43.03 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(rat,int,archim2085082626_floor(rat),A)),hAPP(rat,int,archim2085082626_floor(rat),B))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)). [resolve(5429,a,5476,a)]. 43.10/43.03 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(rat,int,archim2085082626_floor(rat),A)),zero_zero(int))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),one_one(rat))). [resolve(5429,a,5477,a)]. 43.10/43.03 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(rat,int,archim2085082626_floor(rat),A)),zero_zero(int))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),one_one(rat))). [resolve(5429,a,5478,a)]. 43.10/43.03 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(rat,int,archimedean_ceiling(rat),A))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),A)). [resolve(5429,a,5479,a)]. 43.10/43.03 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(rat,int,archimedean_ceiling(rat),A))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),A)). [resolve(5429,a,5480,a)]. 43.10/43.03 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(int,rat,ring_1_of_int(rat),B)),one_one(rat)))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(rat,int,archimedean_ceiling(rat),A)),B)). [resolve(5429,a,5481,a)]. 43.10/43.03 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(int,rat,ring_1_of_int(rat),B)),one_one(rat)))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(rat,int,archimedean_ceiling(rat),A)),B)). [resolve(5429,a,5482,a)]. 43.10/43.03 Derived: hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(rat,int,archim2085082626_floor(rat),A)),hAPP(int,int,number_number_of(int),B)) = hAPP(rat,int,archim2085082626_floor(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),hAPP(int,rat,number_number_of(rat),B))). [resolve(5429,a,5483,a)]. 43.10/43.03 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(rat,int,archimedean_ceiling(rat),A)),hAPP(rat,int,archimedean_ceiling(rat),B))). [resolve(5429,a,5484,a)]. 43.10/43.03 Derived: hAPP(int,int,uminus_uminus(int),hAPP(rat,int,archim2085082626_floor(rat),A)) = hAPP(rat,int,archimedean_ceiling(rat),hAPP(rat,rat,uminus_uminus(rat),A)). [resolve(5429,a,5485,a)]. 43.10/43.03 Derived: ti(int,A) = hAPP(rat,int,archimedean_ceiling(rat),hAPP(int,rat,ring_1_of_int(rat),A)). [resolve(5429,a,5486,a)]. 43.10/43.03 Derived: hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(rat,int,archim2085082626_floor(rat),A)),B) = hAPP(rat,int,archim2085082626_floor(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),hAPP(int,rat,ring_1_of_int(rat),B))). [resolve(5429,a,5487,a)]. 43.10/43.03 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,ring_1_of_int(rat),A)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(rat,int,archim2085082626_floor(rat),B))). [resolve(5429,a,5488,a)]. 43.10/43.03 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,ring_1_of_int(rat),A)),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(rat,int,archim2085082626_floor(rat),B))). [resolve(5429,a,5489,a)]. 43.10/43.03 Derived: archim2085082626_floor(rat) = ti(fun(rat,int),archim2085082626_floor(rat)). [resolve(5429,a,5490,a)]. 43.10/43.03 Derived: one_one(int) = hAPP(rat,int,archim2085082626_floor(rat),one_one(rat)). [resolve(5429,a,5491,a)]. 43.10/43.03 Derived: ti(int,A) = hAPP(rat,int,archim2085082626_floor(rat),hAPP(int,rat,ring_1_of_int(rat),A)). [resolve(5429,a,5492,a)]. 43.10/43.03 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),one_one(rat))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(rat,int,archimedean_ceiling(rat),A)),one_one(int))). [resolve(5429,a,5493,a)]. 43.10/43.03 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),one_one(rat))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(rat,int,archimedean_ceiling(rat),A)),one_one(int))). [resolve(5429,a,5494,a)]. 43.10/43.03 Derived: zero_zero(int) = hAPP(rat,int,archim2085082626_floor(rat),zero_zero(rat)). [resolve(5429,a,5495,a)]. 43.10/43.03 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),A)),hAPP(rat,int,archimedean_ceiling(rat),B))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(int,rat,number_number_of(rat),A)),one_one(rat))),B)). [resolve(5429,a,5496,a)]. 43.10/43.04 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),A)),hAPP(rat,int,archimedean_ceiling(rat),B))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(int,rat,number_number_of(rat),A)),one_one(rat))),B)). [resolve(5429,a,5497,a)]. 43.10/43.04 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(rat,int,archim2085082626_floor(rat),A)),hAPP(rat,int,archim2085082626_floor(rat),B))). [resolve(5429,a,5498,a)]. 43.10/43.04 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(rat,int,archimedean_ceiling(rat),A)),B) = hAPP(rat,int,archimedean_ceiling(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),hAPP(int,rat,ring_1_of_int(rat),B))). [resolve(5429,a,5499,a)]. 43.10/43.04 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(rat,int,archimedean_ceiling(rat),A)),hAPP(int,int,number_number_of(int),B))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),hAPP(int,rat,number_number_of(rat),B))). [resolve(5429,a,5500,a)]. 43.10/43.04 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(rat,int,archimedean_ceiling(rat),A)),hAPP(int,int,number_number_of(int),B))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),hAPP(int,rat,number_number_of(rat),B))). [resolve(5429,a,5501,a)]. 43.10/43.04 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,ring_1_of_int(rat),hAPP(rat,int,archim2085082626_floor(rat),A))),A)). [resolve(5429,a,5502,a)]. 43.10/43.04 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(rat,int,archimedean_ceiling(rat),A)),zero_zero(int))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),hAPP(int,rat,number_number_of(rat),min))). [resolve(5429,a,5503,a)]. 43.10/43.04 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(rat,int,archimedean_ceiling(rat),A)),zero_zero(int))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),hAPP(int,rat,number_number_of(rat),min))). [resolve(5429,a,5504,a)]. 43.10/43.04 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),hAPP(int,rat,ring_1_of_int(rat),B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(rat,int,archimedean_ceiling(rat),A)),B)). [resolve(5429,a,5505,a)]. 43.10/43.04 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),hAPP(int,rat,ring_1_of_int(rat),B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(rat,int,archimedean_ceiling(rat),A)),B)). [resolve(5429,a,5506,a)]. 43.10/43.04 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(int,rat,ring_1_of_int(rat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(rat,int,archim2085082626_floor(rat),A)),one_one(int))))). [resolve(5429,a,5508,a)]. 43.10/43.04 Derived: archimedean_ceiling(rat) = ti(fun(rat,int),archimedean_ceiling(rat)). [resolve(5429,a,5509,a)]. 43.10/43.04 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),hAPP(rat,int,archim2085082626_floor(rat),A))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),A)). [resolve(5429,a,5510,a)]. 43.10/43.04 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),hAPP(rat,int,archim2085082626_floor(rat),A))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),A)). [resolve(5429,a,5511,a)]. 43.10/43.04 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(int,rat,ring_1_of_int(rat),A)),one_one(rat))),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),hAPP(int,rat,ring_1_of_int(rat),A))) | ti(int,A) = hAPP(rat,int,archimedean_ceiling(rat),B). [resolve(5429,a,5512,a)]. 43.10/43.06 Derived: zero_zero(int) = hAPP(rat,int,archimedean_ceiling(rat),zero_zero(rat)). [resolve(5429,a,5513,a)]. 43.10/43.06 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(rat,int,archim2085082626_floor(rat),A))). [resolve(5429,a,5514,a)]. 43.10/43.06 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(rat,int,archim2085082626_floor(rat),A))). [resolve(5429,a,5515,a)]. 43.10/43.06 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),zero_zero(rat))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(rat,int,archim2085082626_floor(rat),A)),zero_zero(int))). [resolve(5429,a,5516,a)]. 43.10/43.06 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),zero_zero(rat))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(rat,int,archim2085082626_floor(rat),A)),zero_zero(int))). [resolve(5429,a,5517,a)]. 43.10/43.06 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(rat,int,archim2085082626_floor(rat),A)),one_one(int))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),one_one(rat))). [resolve(5429,a,5518,a)]. 43.10/43.06 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(rat,int,archim2085082626_floor(rat),A)),one_one(int))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),one_one(rat))). [resolve(5429,a,5519,a)]. 43.10/43.06 Derived: hAPP(nat,int,semiring_1_of_nat(int),A) = hAPP(rat,int,archimedean_ceiling(rat),hAPP(nat,rat,semiring_1_of_nat(rat),A)). [resolve(5429,a,5520,a)]. 43.10/43.06 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,ring_1_of_int(rat),A)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(rat,int,archimedean_ceiling(rat),B))). [resolve(5429,a,5521,a)]. 43.10/43.06 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,ring_1_of_int(rat),A)),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(rat,int,archimedean_ceiling(rat),B))). [resolve(5429,a,5522,a)]. 43.10/43.06 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(rat,int,archim2085082626_floor(rat),A)),B) = hAPP(rat,int,archim2085082626_floor(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),hAPP(int,rat,ring_1_of_int(rat),B))). [resolve(5429,a,5524,a)]. 43.10/43.06 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,number_number_of(rat),B)),one_one(rat)))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(rat,int,archim2085082626_floor(rat),A)),hAPP(int,int,number_number_of(int),B))). [resolve(5429,a,5525,a)]. 43.10/43.06 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,number_number_of(rat),B)),one_one(rat)))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(rat,int,archim2085082626_floor(rat),A)),hAPP(int,int,number_number_of(int),B))). [resolve(5429,a,5526,a)]. 43.10/43.06 5527 -archim1727834104eiling(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(int,A,number_number_of(A),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(A,int,archim2085082626_floor(A),B)),one_one(int))) # label(fact_2872_floor__le__one) # label(axiom). [clausify(4556)]. 43.10/43.06 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(rat,int,archim2085082626_floor(rat),A)),one_one(int))). [resolve(5527,a,5429,a)]. 43.10/43.06 5528 -archim1727834104eiling(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(int,A,number_number_of(A),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(A,int,archim2085082626_floor(A),B)),one_one(int))) # label(fact_2872_floor__le__one) # label(axiom). [clausify(4556)]. 43.27/43.16 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(rat,int,archim2085082626_floor(rat),A)),one_one(int))). [resolve(5528,a,5429,a)]. 43.27/43.16 5529 -archim1727834104eiling(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(int,A,ring_1_of_int(A),C))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(A,int,archim2085082626_floor(A),B)),C)) # label(fact_3617_floor__less__iff) # label(axiom). [clausify(4559)]. 43.27/43.16 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(int,rat,ring_1_of_int(rat),B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(rat,int,archim2085082626_floor(rat),A)),B)). [resolve(5529,a,5429,a)]. 43.27/43.16 5530 -archim1727834104eiling(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(int,A,ring_1_of_int(A),C))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(A,int,archim2085082626_floor(A),B)),C)) # label(fact_3617_floor__less__iff) # label(axiom). [clausify(4559)]. 43.27/43.16 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(int,rat,ring_1_of_int(rat),B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(rat,int,archim2085082626_floor(rat),A)),B)). [resolve(5530,a,5429,a)]. 43.27/43.16 5531 -archim1727834104eiling(A) | hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(A,int,archimedean_ceiling(A),B)),one_one(int)) = hAPP(A,int,archimedean_ceiling(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),one_one(A))) # label(fact_3332_ceiling__diff__one) # label(axiom). [clausify(4577)]. 43.27/43.16 Derived: hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(rat,int,archimedean_ceiling(rat),A)),one_one(int)) = hAPP(rat,int,archimedean_ceiling(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),one_one(rat))). [resolve(5531,a,5429,a)]. 43.27/43.16 5532 archim1727834104eiling(real) # label(arity_RealDef_Oreal___Archimedean__Field_Ofloor__ceiling) # label(axiom). [assumption]. 43.27/43.16 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(real,int,archimedean_ceiling(real),A)),hAPP(int,int,number_number_of(int),B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,number_number_of(real),B)),one_one(real)))). [resolve(5532,a,5430,a)]. 43.27/43.16 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(real,int,archimedean_ceiling(real),A)),hAPP(int,int,number_number_of(int),B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,number_number_of(real),B)),one_one(real)))). [resolve(5532,a,5431,a)]. 43.27/43.16 Derived: one_one(int) = hAPP(real,int,archimedean_ceiling(real),one_one(real)). [resolve(5532,a,5432,a)]. 43.27/43.16 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,ring_1_of_int(real),A)),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,ring_1_of_int(real),A)),one_one(real)))) | ti(int,A) = hAPP(real,int,archim2085082626_floor(real),B). [resolve(5532,a,5433,a)]. 43.27/43.16 Derived: hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(real,int,archim2085082626_floor(real),A)),one_one(int)) = hAPP(real,int,archim2085082626_floor(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),one_one(real))). [resolve(5532,a,5434,a)]. 43.27/43.16 Derived: hAPP(int,int,number_number_of(int),A) = hAPP(real,int,archim2085082626_floor(real),hAPP(int,real,number_number_of(real),A)). [resolve(5532,a,5435,a)]. 43.27/43.16 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,ring_1_of_int(real),A)),one_one(real))),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(real,int,archimedean_ceiling(real),B))). [resolve(5532,a,5436,a)]. 43.27/43.16 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,ring_1_of_int(real),A)),one_one(real))),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(real,int,archimedean_ceiling(real),B))). [resolve(5532,a,5437,a)]. 43.27/43.16 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archimedean_ceiling(real),A)),zero_zero(int))). [resolve(5532,a,5438,a)]. 43.27/43.16 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archimedean_ceiling(real),A)),zero_zero(int))). [resolve(5532,a,5439,a)]. 43.27/43.16 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),min)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(real,int,archimedean_ceiling(real),A))). [resolve(5532,a,5440,a)]. 43.27/43.16 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),min)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(real,int,archimedean_ceiling(real),A))). [resolve(5532,a,5441,a)]. 43.27/43.16 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(real,int,archimedean_ceiling(real),A)),hAPP(real,int,archimedean_ceiling(real),B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)). [resolve(5532,a,5442,a)]. 43.27/43.16 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),A)),hAPP(real,int,archim2085082626_floor(real),B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),A)),B)). [resolve(5532,a,5443,a)]. 43.27/43.16 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),A)),hAPP(real,int,archim2085082626_floor(real),B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),A)),B)). [resolve(5532,a,5444,a)]. 43.27/43.16 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(real,int,archim2085082626_floor(real),A))). [resolve(5532,a,5445,a)]. 43.27/43.16 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(real,int,archim2085082626_floor(real),A))). [resolve(5532,a,5446,a)]. 43.27/43.16 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),hAPP(real,int,archimedean_ceiling(real),A))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)). [resolve(5532,a,5447,a)]. 43.27/43.16 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),hAPP(real,int,archimedean_ceiling(real),A))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)). [resolve(5532,a,5448,a)]. 43.27/43.16 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archim2085082626_floor(real),A)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,ring_1_of_int(real),B)),one_one(real)))). [resolve(5532,a,5449,a)]. 43.27/43.16 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archim2085082626_floor(real),A)),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,ring_1_of_int(real),B)),one_one(real)))). [resolve(5532,a,5450,a)]. 43.27/43.16 Derived: hAPP(int,int,uminus_uminus(int),hAPP(real,int,archimedean_ceiling(real),A)) = hAPP(real,int,archim2085082626_floor(real),hAPP(real,real,uminus_uminus(real),A)). [resolve(5532,a,5451,a)]. 43.27/43.17 Derived: hAPP(int,int,number_number_of(int),A) = hAPP(real,int,archimedean_ceiling(real),hAPP(int,real,number_number_of(real),A)). [resolve(5532,a,5452,a)]. 43.27/43.17 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(real,int,archimedean_ceiling(real),A)),hAPP(int,int,number_number_of(int),B)) = hAPP(real,int,archimedean_ceiling(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),hAPP(int,real,number_number_of(real),B))). [resolve(5532,a,5453,a)]. 43.27/43.17 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(real,int,archimedean_ceiling(real),A)),one_one(int))). [resolve(5532,a,5454,a)]. 43.27/43.17 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(real,int,archimedean_ceiling(real),A)),one_one(int))). [resolve(5532,a,5455,a)]. 43.27/43.17 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(real,int,archim2085082626_floor(real),B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,ring_1_of_int(real),A)),one_one(real))),B)). [resolve(5532,a,5456,a)]. 43.27/43.17 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(real,int,archim2085082626_floor(real),B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,ring_1_of_int(real),A)),one_one(real))),B)). [resolve(5532,a,5457,a)]. 43.27/43.17 Derived: hAPP(int,int,uminus_uminus(int),hAPP(real,int,archim2085082626_floor(real),hAPP(real,real,uminus_uminus(real),A))) = hAPP(real,int,archimedean_ceiling(real),A). [resolve(5532,a,5458,a)]. 43.27/43.17 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(real,int,archim2085082626_floor(real),A)),hAPP(int,int,number_number_of(int),B)) = hAPP(real,int,archim2085082626_floor(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),hAPP(int,real,number_number_of(real),B))). [resolve(5532,a,5459,a)]. 43.27/43.17 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(real,int,archim2085082626_floor(real),A)),one_one(int)) = hAPP(real,int,archim2085082626_floor(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),one_one(real))). [resolve(5532,a,5460,a)]. 43.27/43.17 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(real,int,archim2085082626_floor(real),A)),hAPP(int,int,number_number_of(int),B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(int,real,number_number_of(real),B))). [resolve(5532,a,5461,a)]. 43.27/43.17 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(real,int,archim2085082626_floor(real),A)),hAPP(int,int,number_number_of(int),B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(int,real,number_number_of(real),B))). [resolve(5532,a,5462,a)]. 43.27/43.17 Derived: hAPP(real,int,archimedean_ceiling(real),hAPP(int,real,real_1(int),hAPP(real,int,archim2085082626_floor(real),A))) = hAPP(real,int,archim2085082626_floor(real),A). [resolve(5532,a,5463,a)]. 43.27/43.17 Derived: hAPP(nat,int,semiring_1_of_nat(int),A) = hAPP(real,int,archim2085082626_floor(real),hAPP(nat,real,semiring_1_of_nat(real),A)). [resolve(5532,a,5464,a)]. 43.27/43.17 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(real,int,archimedean_ceiling(real),A)),one_one(int)) = hAPP(real,int,archimedean_ceiling(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),one_one(real))). [resolve(5532,a,5465,a)]. 43.27/43.17 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,number_number_of(real),A)),one_one(real))),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),A)),hAPP(real,int,archim2085082626_floor(real),B))). [resolve(5532,a,5466,a)]. 43.27/43.17 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,number_number_of(real),A)),one_one(real))),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),A)),hAPP(real,int,archim2085082626_floor(real),B))). [resolve(5532,a,5467,a)]. 43.27/43.17 Derived: hAPP(real,int,archim2085082626_floor(real),hAPP(int,real,real_1(int),hAPP(real,int,archimedean_ceiling(real),A))) = hAPP(real,int,archimedean_ceiling(real),A). [resolve(5532,a,5468,a)]. 43.27/43.17 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),hAPP(int,real,ring_1_of_int(real),hAPP(real,int,archimedean_ceiling(real),A)))). [resolve(5532,a,5469,a)]. 43.27/43.17 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,ring_1_of_int(real),hAPP(real,int,archimedean_ceiling(real),A))),one_one(real))),A)). [resolve(5532,a,5470,a)]. 43.27/43.17 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(real,int,archimedean_ceiling(real),A))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)). [resolve(5532,a,5471,a)]. 43.27/43.17 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(real,int,archimedean_ceiling(real),A))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)). [resolve(5532,a,5472,a)]. 43.27/43.17 Derived: hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(real,int,archimedean_ceiling(real),A)),hAPP(int,int,number_number_of(int),B)) = hAPP(real,int,archimedean_ceiling(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),hAPP(int,real,number_number_of(real),B))). [resolve(5532,a,5473,a)]. 43.27/43.17 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(real,int,archim2085082626_floor(real),A))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),A)). [resolve(5532,a,5474,a)]. 43.27/43.17 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(real,int,archim2085082626_floor(real),A))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),A)). [resolve(5532,a,5475,a)]. 43.27/43.17 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(real,int,archim2085082626_floor(real),A)),hAPP(real,int,archim2085082626_floor(real),B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)). [resolve(5532,a,5476,a)]. 43.27/43.17 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archim2085082626_floor(real),A)),zero_zero(int))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),one_one(real))). [resolve(5532,a,5477,a)]. 43.27/43.17 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archim2085082626_floor(real),A)),zero_zero(int))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),one_one(real))). [resolve(5532,a,5478,a)]. 43.27/43.17 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(real,int,archimedean_ceiling(real),A))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A)). [resolve(5532,a,5479,a)]. 43.27/43.17 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(real,int,archimedean_ceiling(real),A))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A)). [resolve(5532,a,5480,a)]. 43.27/43.17 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,ring_1_of_int(real),B)),one_one(real)))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(real,int,archimedean_ceiling(real),A)),B)). [resolve(5532,a,5481,a)]. 43.27/43.17 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,ring_1_of_int(real),B)),one_one(real)))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(real,int,archimedean_ceiling(real),A)),B)). [resolve(5532,a,5482,a)]. 43.27/43.18 Derived: hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(real,int,archim2085082626_floor(real),A)),hAPP(int,int,number_number_of(int),B)) = hAPP(real,int,archim2085082626_floor(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),hAPP(int,real,number_number_of(real),B))). [resolve(5532,a,5483,a)]. 43.27/43.18 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archimedean_ceiling(real),A)),hAPP(real,int,archimedean_ceiling(real),B))). [resolve(5532,a,5484,a)]. 43.27/43.18 Derived: hAPP(int,int,uminus_uminus(int),hAPP(real,int,archim2085082626_floor(real),A)) = hAPP(real,int,archimedean_ceiling(real),hAPP(real,real,uminus_uminus(real),A)). [resolve(5532,a,5485,a)]. 43.27/43.18 Derived: ti(int,A) = hAPP(real,int,archimedean_ceiling(real),hAPP(int,real,ring_1_of_int(real),A)). [resolve(5532,a,5486,a)]. 43.27/43.18 Derived: hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(real,int,archim2085082626_floor(real),A)),B) = hAPP(real,int,archim2085082626_floor(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),hAPP(int,real,ring_1_of_int(real),B))). [resolve(5532,a,5487,a)]. 43.27/43.18 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,ring_1_of_int(real),A)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(real,int,archim2085082626_floor(real),B))). [resolve(5532,a,5488,a)]. 43.27/43.18 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,ring_1_of_int(real),A)),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(real,int,archim2085082626_floor(real),B))). [resolve(5532,a,5489,a)]. 43.27/43.18 Derived: archim2085082626_floor(real) = ti(fun(real,int),archim2085082626_floor(real)). [resolve(5532,a,5490,a)]. 43.27/43.18 Derived: one_one(int) = hAPP(real,int,archim2085082626_floor(real),one_one(real)). [resolve(5532,a,5491,a)]. 43.27/43.18 Derived: ti(int,A) = hAPP(real,int,archim2085082626_floor(real),hAPP(int,real,ring_1_of_int(real),A)). [resolve(5532,a,5492,a)]. 43.27/43.18 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),one_one(real))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archimedean_ceiling(real),A)),one_one(int))). [resolve(5532,a,5493,a)]. 43.27/43.18 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),one_one(real))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archimedean_ceiling(real),A)),one_one(int))). [resolve(5532,a,5494,a)]. 43.27/43.18 Derived: zero_zero(int) = hAPP(real,int,archim2085082626_floor(real),zero_zero(real)). [resolve(5532,a,5495,a)]. 43.27/43.18 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),A)),hAPP(real,int,archimedean_ceiling(real),B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,number_number_of(real),A)),one_one(real))),B)). [resolve(5532,a,5496,a)]. 43.27/43.18 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),A)),hAPP(real,int,archimedean_ceiling(real),B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,number_number_of(real),A)),one_one(real))),B)). [resolve(5532,a,5497,a)]. 43.27/43.18 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archim2085082626_floor(real),A)),hAPP(real,int,archim2085082626_floor(real),B))). [resolve(5532,a,5498,a)]. 43.27/43.18 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(real,int,archimedean_ceiling(real),A)),B) = hAPP(real,int,archimedean_ceiling(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),hAPP(int,real,ring_1_of_int(real),B))). [resolve(5532,a,5499,a)]. 43.27/43.18 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archimedean_ceiling(real),A)),hAPP(int,int,number_number_of(int),B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),hAPP(int,real,number_number_of(real),B))). [resolve(5532,a,5500,a)]. 43.27/43.19 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archimedean_ceiling(real),A)),hAPP(int,int,number_number_of(int),B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),hAPP(int,real,number_number_of(real),B))). [resolve(5532,a,5501,a)]. 43.27/43.19 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,ring_1_of_int(real),hAPP(real,int,archim2085082626_floor(real),A))),A)). [resolve(5532,a,5502,a)]. 43.27/43.19 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(real,int,archimedean_ceiling(real),A)),zero_zero(int))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),hAPP(int,real,number_number_of(real),min))). [resolve(5532,a,5503,a)]. 43.27/43.19 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(real,int,archimedean_ceiling(real),A)),zero_zero(int))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),hAPP(int,real,number_number_of(real),min))). [resolve(5532,a,5504,a)]. 43.27/43.19 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),hAPP(int,real,ring_1_of_int(real),B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archimedean_ceiling(real),A)),B)). [resolve(5532,a,5505,a)]. 43.27/43.19 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),hAPP(int,real,ring_1_of_int(real),B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archimedean_ceiling(real),A)),B)). [resolve(5532,a,5506,a)]. 43.27/43.19 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(int,real,ring_1_of_int(real),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(real,int,archim2085082626_floor(real),A)),one_one(int))))). [resolve(5532,a,5508,a)]. 43.27/43.19 Derived: archimedean_ceiling(real) = ti(fun(real,int),archimedean_ceiling(real)). [resolve(5532,a,5509,a)]. 43.27/43.19 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),hAPP(real,int,archim2085082626_floor(real),A))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),A)). [resolve(5532,a,5510,a)]. 43.27/43.19 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),hAPP(real,int,archim2085082626_floor(real),A))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),A)). [resolve(5532,a,5511,a)]. 43.27/43.19 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,ring_1_of_int(real),A)),one_one(real))),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),hAPP(int,real,ring_1_of_int(real),A))) | ti(int,A) = hAPP(real,int,archimedean_ceiling(real),B). [resolve(5532,a,5512,a)]. 43.27/43.19 Derived: zero_zero(int) = hAPP(real,int,archimedean_ceiling(real),zero_zero(real)). [resolve(5532,a,5513,a)]. 43.27/43.19 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(real,int,archim2085082626_floor(real),A))). [resolve(5532,a,5514,a)]. 43.27/43.19 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(real,int,archim2085082626_floor(real),A))). [resolve(5532,a,5515,a)]. 43.27/43.19 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(real,int,archim2085082626_floor(real),A)),zero_zero(int))). [resolve(5532,a,5516,a)]. 43.27/43.19 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(real,int,archim2085082626_floor(real),A)),zero_zero(int))). [resolve(5532,a,5517,a)]. 43.27/43.19 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(real,int,archim2085082626_floor(real),A)),one_one(int))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),one_one(real))). [resolve(5532,a,5518,a)]. 43.27/43.19 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(real,int,archim2085082626_floor(real),A)),one_one(int))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),one_one(real))). [resolve(5532,a,5519,a)]. 43.27/43.19 Derived: hAPP(nat,int,semiring_1_of_nat(int),A) = hAPP(real,int,archimedean_ceiling(real),hAPP(nat,real,semiring_1_of_nat(real),A)). [resolve(5532,a,5520,a)]. 43.27/43.19 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,ring_1_of_int(real),A)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(real,int,archimedean_ceiling(real),B))). [resolve(5532,a,5521,a)]. 43.27/43.19 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,ring_1_of_int(real),A)),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(real,int,archimedean_ceiling(real),B))). [resolve(5532,a,5522,a)]. 43.27/43.19 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(real,int,archim2085082626_floor(real),A)),B) = hAPP(real,int,archim2085082626_floor(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),hAPP(int,real,ring_1_of_int(real),B))). [resolve(5532,a,5524,a)]. 43.27/43.19 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,number_number_of(real),B)),one_one(real)))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archim2085082626_floor(real),A)),hAPP(int,int,number_number_of(int),B))). [resolve(5532,a,5525,a)]. 43.27/43.19 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,number_number_of(real),B)),one_one(real)))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archim2085082626_floor(real),A)),hAPP(int,int,number_number_of(int),B))). [resolve(5532,a,5526,a)]. 43.27/43.19 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archim2085082626_floor(real),A)),one_one(int))). [resolve(5532,a,5527,a)]. 43.27/43.19 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archim2085082626_floor(real),A)),one_one(int))). [resolve(5532,a,5528,a)]. 43.27/43.19 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(int,real,ring_1_of_int(real),B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(real,int,archim2085082626_floor(real),A)),B)). [resolve(5532,a,5529,a)]. 43.27/43.19 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(int,real,ring_1_of_int(real),B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(real,int,archim2085082626_floor(real),A)),B)). [resolve(5532,a,5530,a)]. 43.27/43.19 Derived: hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(real,int,archimedean_ceiling(real),A)),one_one(int)) = hAPP(real,int,archimedean_ceiling(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),one_one(real))). [resolve(5532,a,5531,a)]. 43.27/43.19 5533 -archim1727834104eiling(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(int,A,number_number_of(A),B)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),B)),hAPP(A,int,archimedean_ceiling(A),C))) # label(fact_3328_number__of__less__ceiling) # label(axiom). [clausify(4682)]. 43.27/43.19 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,number_number_of(rat),A)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),A)),hAPP(rat,int,archimedean_ceiling(rat),B))). [resolve(5533,a,5429,a)]. 43.27/43.19 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),A)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),A)),hAPP(real,int,archimedean_ceiling(real),B))). [resolve(5533,a,5532,a)]. 45.13/45.03 5534 -archim1727834104eiling(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(int,A,number_number_of(A),B)),C)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),B)),hAPP(A,int,archimedean_ceiling(A),C))) # label(fact_3328_number__of__less__ceiling) # label(axiom). [clausify(4682)]. 45.13/45.03 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,number_number_of(rat),A)),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),A)),hAPP(rat,int,archimedean_ceiling(rat),B))). [resolve(5534,a,5429,a)]. 45.13/45.03 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),A)),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),A)),hAPP(real,int,archimedean_ceiling(real),B))). [resolve(5534,a,5532,a)]. 45.13/45.03 5535 -archim1727834104eiling(A) | hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(A,int,archimedean_ceiling(A),B)),C) = hAPP(A,int,archimedean_ceiling(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),hAPP(int,A,ring_1_of_int(A),C))) # label(fact_3624_ceiling__diff__of__int) # label(axiom). [clausify(4831)]. 45.13/45.03 Derived: hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(rat,int,archimedean_ceiling(rat),A)),B) = hAPP(rat,int,archimedean_ceiling(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),hAPP(int,rat,ring_1_of_int(rat),B))). [resolve(5535,a,5429,a)]. 45.13/45.03 Derived: hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(real,int,archimedean_ceiling(real),A)),B) = hAPP(real,int,archimedean_ceiling(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),hAPP(int,real,ring_1_of_int(real),B))). [resolve(5535,a,5532,a)]. 45.13/45.03 5536 semiri456707255roduct(real) # label(arity_RealDef_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossp) # label(axiom). [assumption]. 45.13/45.03 5537 -semiri456707255roduct(A) | ti(A,B) = zero_zero(A) | ti(A,C) != ti(A,D) | ti(A,E) = ti(A,F) | hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),E)) != hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),D),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),F)) # label(fact_458_add__scale__eq__noteq) # label(axiom). [clausify(35)]. 45.13/45.03 Derived: ti(real,A) = zero_zero(real) | ti(real,B) != ti(real,C) | ti(real,D) = ti(real,E) | hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),D)) != hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),C),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),E)). [resolve(5536,a,5537,a)]. 45.13/45.03 5538 semiri456707255roduct(rat) # label(arity_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc) # label(axiom). [assumption]. 45.13/45.03 Derived: ti(rat,A) = zero_zero(rat) | ti(rat,B) != ti(rat,C) | ti(rat,D) = ti(rat,E) | hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),D)) != hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),C),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),E)). [resolve(5538,a,5537,a)]. 45.13/45.03 5539 -semiri456707255roduct(A) | hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),E)) != hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),E)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),C)) | ti(A,B) = ti(A,D) | ti(A,C) = ti(A,E) # label(fact_386_crossproduct__eq) # label(axiom). [clausify(1462)]. 45.13/45.03 Derived: hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),D)) != hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),D)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),B)) | ti(real,A) = ti(real,C) | ti(real,B) = ti(real,D). [resolve(5539,a,5536,a)]. 45.22/45.17 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),D)) != hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),D)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),B)) | ti(rat,A) = ti(rat,C) | ti(rat,B) = ti(rat,D). [resolve(5539,a,5538,a)]. 45.22/45.17 5540 -semiri456707255roduct(A) | hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),E)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),E)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),C)) | ti(A,B) != ti(A,D) # label(fact_386_crossproduct__eq) # label(axiom). [clausify(1462)]. 45.22/45.17 Derived: hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),D)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),D)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),B)) | ti(real,A) != ti(real,C). [resolve(5540,a,5536,a)]. 45.22/45.17 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),D)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),D)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),B)) | ti(rat,A) != ti(rat,C). [resolve(5540,a,5538,a)]. 45.22/45.17 5541 -semiri456707255roduct(A) | hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),E)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),E)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),C)) | ti(A,C) != ti(A,E) # label(fact_386_crossproduct__eq) # label(axiom). [clausify(1462)]. 45.22/45.17 Derived: hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),D)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),D)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),B)) | ti(real,B) != ti(real,D). [resolve(5541,a,5536,a)]. 45.22/45.17 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),D)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),D)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),B)) | ti(rat,B) != ti(rat,D). [resolve(5541,a,5538,a)]. 45.22/45.17 5542 semiri456707255roduct(nat) # label(arity_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc) # label(axiom). [assumption]. 45.22/45.17 Derived: ti(nat,A) = zero_zero(nat) | ti(nat,B) != ti(nat,C) | ti(nat,D) = ti(nat,E) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),D)) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),E)). [resolve(5542,a,5537,a)]. 45.22/45.17 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),D)) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),D)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),B)) | ti(nat,A) = ti(nat,C) | ti(nat,B) = ti(nat,D). [resolve(5542,a,5539,a)]. 45.22/45.17 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),D)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),D)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),B)) | ti(nat,A) != ti(nat,C). [resolve(5542,a,5540,a)]. 45.54/45.42 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),D)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),D)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),B)) | ti(nat,B) != ti(nat,D). [resolve(5542,a,5541,a)]. 45.54/45.42 5543 -semiri456707255roduct(A) | hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),E)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),E)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),C)) | ti(A,C) != ti(A,E) # label(fact_383_crossproduct__noteq) # label(axiom). [clausify(2376)]. 45.54/45.42 5544 -semiri456707255roduct(A) | hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),E)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),E)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),C)) | ti(A,D) != ti(A,B) # label(fact_383_crossproduct__noteq) # label(axiom). [clausify(2376)]. 45.54/45.42 Derived: hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),D)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),D)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),B)) | ti(real,C) != ti(real,A). [resolve(5544,a,5536,a)]. 45.54/45.42 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),D)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),D)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),B)) | ti(rat,C) != ti(rat,A). [resolve(5544,a,5538,a)]. 45.54/45.42 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),D)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),D)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),B)) | ti(nat,C) != ti(nat,A). [resolve(5544,a,5542,a)]. 45.54/45.42 5545 -semiri456707255roduct(A) | hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),E)) != hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),E)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),C)) | ti(A,C) = ti(A,E) | ti(A,D) = ti(A,B) # label(fact_383_crossproduct__noteq) # label(axiom). [clausify(2376)]. 45.54/45.42 Derived: hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),D)) != hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),D)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),B)) | ti(real,B) = ti(real,D) | ti(real,C) = ti(real,A). [resolve(5545,a,5536,a)]. 45.54/45.42 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),D)) != hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),D)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),B)) | ti(rat,B) = ti(rat,D) | ti(rat,C) = ti(rat,A). [resolve(5545,a,5538,a)]. 45.54/45.42 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),D)) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),D)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),B)) | ti(nat,B) = ti(nat,D) | ti(nat,C) = ti(nat,A). [resolve(5545,a,5542,a)]. 47.07/46.99 5546 semiri456707255roduct(complex) # label(arity_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__cro) # label(axiom). [assumption]. 47.07/46.99 Derived: ti(complex,A) = zero_zero(complex) | ti(complex,B) != ti(complex,C) | ti(complex,D) = ti(complex,E) | hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),B),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),D)) != hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),C),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),E)). [resolve(5546,a,5537,a)]. 47.07/46.99 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),D)) != hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),D)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),B)) | ti(complex,A) = ti(complex,C) | ti(complex,B) = ti(complex,D). [resolve(5546,a,5539,a)]. 47.07/46.99 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),D)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),D)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),B)) | ti(complex,A) != ti(complex,C). [resolve(5546,a,5540,a)]. 47.07/46.99 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),D)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),D)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),B)) | ti(complex,B) != ti(complex,D). [resolve(5546,a,5541,a)]. 47.07/46.99 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),D)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),D)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),B)) | ti(complex,C) != ti(complex,A). [resolve(5546,a,5544,a)]. 47.07/46.99 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),D)) != hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),D)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),B)) | ti(complex,B) = ti(complex,D) | ti(complex,C) = ti(complex,A). [resolve(5546,a,5545,a)]. 47.07/46.99 5547 semiri456707255roduct(int) # label(arity_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc) # label(axiom). [assumption]. 47.07/46.99 Derived: ti(int,A) = zero_zero(int) | ti(int,B) != ti(int,C) | ti(int,D) = ti(int,E) | hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),D)) != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),C),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),E)). [resolve(5547,a,5537,a)]. 48.82/48.70 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),D)) != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),D)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),B)) | ti(int,A) = ti(int,C) | ti(int,B) = ti(int,D). [resolve(5547,a,5539,a)]. 48.82/48.70 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),D)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),D)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),B)) | ti(int,A) != ti(int,C). [resolve(5547,a,5540,a)]. 48.82/48.70 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),D)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),D)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),B)) | ti(int,B) != ti(int,D). [resolve(5547,a,5541,a)]. 48.82/48.70 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),D)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),D)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),B)) | ti(int,C) != ti(int,A). [resolve(5547,a,5544,a)]. 48.82/48.70 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),D)) != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),D)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),B)) | ti(int,B) = ti(int,D) | ti(int,C) = ti(int,A). [resolve(5547,a,5545,a)]. 48.82/48.70 5548 -semiri456707255roduct(A) | ti(A,B) != hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C) | ti(A,C) = zero_zero(A) # label(fact_317_add__0__iff) # label(axiom). [clausify(4563)]. 48.82/48.70 Derived: ti(real,A) != hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B) | ti(real,B) = zero_zero(real). [resolve(5548,a,5536,a)]. 48.82/48.70 Derived: ti(rat,A) != hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B) | ti(rat,B) = zero_zero(rat). [resolve(5548,a,5538,a)]. 48.82/48.70 Derived: ti(nat,A) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),B) | ti(nat,B) = zero_zero(nat). [resolve(5548,a,5542,a)]. 48.82/48.70 Derived: ti(complex,A) != hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),B) | ti(complex,B) = zero_zero(complex). [resolve(5548,a,5546,a)]. 48.82/48.70 Derived: ti(int,A) != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B) | ti(int,B) = zero_zero(int). [resolve(5548,a,5547,a)]. 48.82/48.70 5549 -semiri456707255roduct(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C) | ti(A,C) != zero_zero(A) # label(fact_317_add__0__iff) # label(axiom). [clausify(4563)]. 48.82/48.70 Derived: ti(real,A) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B) | ti(real,B) != zero_zero(real). [resolve(5549,a,5536,a)]. 48.82/48.70 Derived: ti(rat,A) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B) | ti(rat,B) != zero_zero(rat). [resolve(5549,a,5538,a)]. 48.82/48.70 Derived: ti(nat,A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),B) | ti(nat,B) != zero_zero(nat). [resolve(5549,a,5542,a)]. 48.82/48.70 Derived: ti(complex,A) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),B) | ti(complex,B) != zero_zero(complex). [resolve(5549,a,5546,a)]. 48.82/48.70 Derived: ti(int,A) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B) | ti(int,B) != zero_zero(int). [resolve(5549,a,5547,a)]. 48.82/48.70 5550 comm_monoid_mult(nat) # label(arity_Nat_Onat___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 48.84/48.70 5551 -comm_monoid_mult(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),D),C)) | one_one(A) = hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),hAPP(A,fun(B,A),hAPP(fun(B,fun(A,A)),fun(A,fun(B,A)),combc(B,A,A),hAPP(fun(B,A),fun(B,fun(A,A)),hAPP(fun(B,fun(A,fun(A,A))),fun(fun(B,A),fun(B,fun(A,A))),combs(B,A,fun(A,A)),hAPP(fun(B,bool),fun(B,fun(A,fun(A,A))),hAPP(fun(bool,fun(A,fun(A,A))),fun(fun(B,bool),fun(B,fun(A,fun(A,A)))),combb(bool,fun(A,fun(A,A)),B),if(A)),hAPP(B,fun(B,bool),fequal(B),D))),E)),one_one(A))),C) # label(fact_3983_setprod__delta_H) # label(axiom). [clausify(45)]. 48.84/48.70 5552 -comm_monoid_mult(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | -hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),D),C)) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),hAPP(A,fun(B,A),hAPP(fun(B,fun(A,A)),fun(A,fun(B,A)),combc(B,A,A),hAPP(fun(B,A),fun(B,fun(A,A)),hAPP(fun(B,fun(A,fun(A,A))),fun(fun(B,A),fun(B,fun(A,A))),combs(B,A,fun(A,A)),hAPP(fun(B,bool),fun(B,fun(A,fun(A,A))),hAPP(fun(bool,fun(A,fun(A,A))),fun(fun(B,bool),fun(B,fun(A,fun(A,A)))),combb(bool,fun(A,fun(A,A)),B),if(A)),hAPP(B,fun(B,bool),fequal(B),D))),E)),one_one(A))),C) = hAPP(B,A,E,D) # label(fact_3983_setprod__delta_H) # label(axiom). [clausify(45)]. 48.84/48.70 5553 -comm_monoid_mult(A) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),hAPP(fun(B,A),fun(B,A),hAPP(fun(B,fun(A,A)),fun(fun(B,A),fun(B,A)),combs(B,A,A),hAPP(fun(B,A),fun(B,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),combb(A,fun(A,A),B),times_times(A)),C)),D)),E) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),C),E)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),D),E)) # label(fact_3973_setprod__timesf) # label(axiom). [clausify(135)]. 48.84/48.70 5554 -comm_monoid_add(A) | -comm_monoid_mult(A) | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,B)) | hAPP(fun(nat,bool),A,hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),hAPP(fun(nat,int),fun(nat,A),hAPP(fun(int,A),fun(fun(nat,int),fun(nat,A)),combb(int,A,nat),C),semiring_1_of_nat(int))),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),B)) = hAPP(fun(int,bool),A,hAPP(fun(int,A),fun(fun(int,bool),A),big_co1399186613setsum(int,A),C),B) # label(fact_4293_transfer__int__nat__sum__prod_I1_J) # label(axiom). [clausify(270)]. 48.84/48.70 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | one_one(nat) = hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),hAPP(nat,fun(A,nat),hAPP(fun(A,fun(nat,nat)),fun(nat,fun(A,nat)),combc(A,nat,nat),hAPP(fun(A,nat),fun(A,fun(nat,nat)),hAPP(fun(A,fun(nat,fun(nat,nat))),fun(fun(A,nat),fun(A,fun(nat,nat))),combs(A,nat,fun(nat,nat)),hAPP(fun(A,bool),fun(A,fun(nat,fun(nat,nat))),hAPP(fun(bool,fun(nat,fun(nat,nat))),fun(fun(A,bool),fun(A,fun(nat,fun(nat,nat)))),combb(bool,fun(nat,fun(nat,nat)),A),if(nat)),hAPP(A,fun(A,bool),fequal(A),C))),D)),one_one(nat))),B). [resolve(5550,a,5551,a)]. 48.84/48.70 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),hAPP(nat,fun(A,nat),hAPP(fun(A,fun(nat,nat)),fun(nat,fun(A,nat)),combc(A,nat,nat),hAPP(fun(A,nat),fun(A,fun(nat,nat)),hAPP(fun(A,fun(nat,fun(nat,nat))),fun(fun(A,nat),fun(A,fun(nat,nat))),combs(A,nat,fun(nat,nat)),hAPP(fun(A,bool),fun(A,fun(nat,fun(nat,nat))),hAPP(fun(bool,fun(nat,fun(nat,nat))),fun(fun(A,bool),fun(A,fun(nat,fun(nat,nat)))),combb(bool,fun(nat,fun(nat,nat)),A),if(nat)),hAPP(A,fun(A,bool),fequal(A),C))),D)),one_one(nat))),B) = hAPP(A,nat,D,C). [resolve(5550,a,5552,a)]. 48.84/48.70 Derived: hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),hAPP(fun(A,nat),fun(A,nat),hAPP(fun(A,fun(nat,nat)),fun(fun(A,nat),fun(A,nat)),combs(A,nat,nat),hAPP(fun(A,nat),fun(A,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(A,nat),fun(A,fun(nat,nat))),combb(nat,fun(nat,nat),A),times_times(nat)),B)),C)),D) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),B),D)),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),C),D)). [resolve(5550,a,5553,a)]. 48.84/48.76 Derived: -comm_monoid_add(nat) | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),hAPP(fun(nat,int),fun(nat,nat),hAPP(fun(int,nat),fun(fun(nat,int),fun(nat,nat)),combb(int,nat,nat),B),semiring_1_of_nat(int))),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A)) = hAPP(fun(int,bool),nat,hAPP(fun(int,nat),fun(fun(int,bool),nat),big_co1399186613setsum(int,nat),B),A). [resolve(5550,a,5554,b)]. 48.84/48.76 5555 -comm_monoid_mult(A) | hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | one_one(A) = hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),D),C) # label(fact_3979_setprod_Oinfinite) # label(axiom). [clausify(412)]. 48.84/48.76 Derived: hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | one_one(nat) = hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),C),B). [resolve(5555,a,5550,a)]. 48.84/48.76 5556 -comm_monoid_mult(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),hAPP(fun(B,A),fun(B,A),hAPP(fun(B,fun(A,A)),fun(fun(B,A),fun(B,A)),combs(B,A,A),hAPP(fun(B,A),fun(B,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),combb(A,fun(A,A),B),times_times(A)),D)),E)),C) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),D),C)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),E),C)) # label(fact_3981_setprod_Odistrib) # label(axiom). [clausify(668)]. 48.84/48.76 5557 comm_monoid_mult(complex) # label(arity_Complex_Ocomplex___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 48.84/48.76 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | one_one(complex) = hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),hAPP(complex,fun(A,complex),hAPP(fun(A,fun(complex,complex)),fun(complex,fun(A,complex)),combc(A,complex,complex),hAPP(fun(A,complex),fun(A,fun(complex,complex)),hAPP(fun(A,fun(complex,fun(complex,complex))),fun(fun(A,complex),fun(A,fun(complex,complex))),combs(A,complex,fun(complex,complex)),hAPP(fun(A,bool),fun(A,fun(complex,fun(complex,complex))),hAPP(fun(bool,fun(complex,fun(complex,complex))),fun(fun(A,bool),fun(A,fun(complex,fun(complex,complex)))),combb(bool,fun(complex,fun(complex,complex)),A),if(complex)),hAPP(A,fun(A,bool),fequal(A),C))),D)),one_one(complex))),B). [resolve(5557,a,5551,a)]. 48.84/48.76 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),hAPP(complex,fun(A,complex),hAPP(fun(A,fun(complex,complex)),fun(complex,fun(A,complex)),combc(A,complex,complex),hAPP(fun(A,complex),fun(A,fun(complex,complex)),hAPP(fun(A,fun(complex,fun(complex,complex))),fun(fun(A,complex),fun(A,fun(complex,complex))),combs(A,complex,fun(complex,complex)),hAPP(fun(A,bool),fun(A,fun(complex,fun(complex,complex))),hAPP(fun(bool,fun(complex,fun(complex,complex))),fun(fun(A,bool),fun(A,fun(complex,fun(complex,complex)))),combb(bool,fun(complex,fun(complex,complex)),A),if(complex)),hAPP(A,fun(A,bool),fequal(A),C))),D)),one_one(complex))),B) = hAPP(A,complex,D,C). [resolve(5557,a,5552,a)]. 48.84/48.76 Derived: hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),hAPP(fun(A,complex),fun(A,complex),hAPP(fun(A,fun(complex,complex)),fun(fun(A,complex),fun(A,complex)),combs(A,complex,complex),hAPP(fun(A,complex),fun(A,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(A,complex),fun(A,fun(complex,complex))),combb(complex,fun(complex,complex),A),times_times(complex)),B)),C)),D) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),B),D)),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),C),D)). [resolve(5557,a,5553,a)]. 48.84/48.80 Derived: -comm_monoid_add(complex) | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(fun(nat,bool),complex,hAPP(fun(nat,complex),fun(fun(nat,bool),complex),big_co1399186613setsum(nat,complex),hAPP(fun(nat,int),fun(nat,complex),hAPP(fun(int,complex),fun(fun(nat,int),fun(nat,complex)),combb(int,complex,nat),B),semiring_1_of_nat(int))),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A)) = hAPP(fun(int,bool),complex,hAPP(fun(int,complex),fun(fun(int,bool),complex),big_co1399186613setsum(int,complex),B),A). [resolve(5557,a,5554,b)]. 48.84/48.80 Derived: hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | one_one(complex) = hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),C),B). [resolve(5557,a,5555,a)]. 48.84/48.80 5558 -comm_monoid_mult(A) | one_one(A) = hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),hAPP(A,fun(B,A),combk(A,B),one_one(A))),C) # label(fact_3974_setprod__1) # label(axiom). [clausify(846)]. 48.84/48.80 Derived: one_one(nat) = hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),hAPP(nat,fun(A,nat),combk(nat,A),one_one(nat))),B). [resolve(5558,a,5550,a)]. 48.84/48.80 Derived: one_one(complex) = hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),hAPP(complex,fun(A,complex),combk(complex,A),one_one(complex))),B). [resolve(5558,a,5557,a)]. 48.84/48.80 5559 comm_monoid_mult(rat) # label(arity_Rat_Orat___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 48.84/48.80 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | one_one(rat) = hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),hAPP(rat,fun(A,rat),hAPP(fun(A,fun(rat,rat)),fun(rat,fun(A,rat)),combc(A,rat,rat),hAPP(fun(A,rat),fun(A,fun(rat,rat)),hAPP(fun(A,fun(rat,fun(rat,rat))),fun(fun(A,rat),fun(A,fun(rat,rat))),combs(A,rat,fun(rat,rat)),hAPP(fun(A,bool),fun(A,fun(rat,fun(rat,rat))),hAPP(fun(bool,fun(rat,fun(rat,rat))),fun(fun(A,bool),fun(A,fun(rat,fun(rat,rat)))),combb(bool,fun(rat,fun(rat,rat)),A),if(rat)),hAPP(A,fun(A,bool),fequal(A),C))),D)),one_one(rat))),B). [resolve(5559,a,5551,a)]. 48.84/48.80 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),hAPP(rat,fun(A,rat),hAPP(fun(A,fun(rat,rat)),fun(rat,fun(A,rat)),combc(A,rat,rat),hAPP(fun(A,rat),fun(A,fun(rat,rat)),hAPP(fun(A,fun(rat,fun(rat,rat))),fun(fun(A,rat),fun(A,fun(rat,rat))),combs(A,rat,fun(rat,rat)),hAPP(fun(A,bool),fun(A,fun(rat,fun(rat,rat))),hAPP(fun(bool,fun(rat,fun(rat,rat))),fun(fun(A,bool),fun(A,fun(rat,fun(rat,rat)))),combb(bool,fun(rat,fun(rat,rat)),A),if(rat)),hAPP(A,fun(A,bool),fequal(A),C))),D)),one_one(rat))),B) = hAPP(A,rat,D,C). [resolve(5559,a,5552,a)]. 48.84/48.80 Derived: hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),hAPP(fun(A,rat),fun(A,rat),hAPP(fun(A,fun(rat,rat)),fun(fun(A,rat),fun(A,rat)),combs(A,rat,rat),hAPP(fun(A,rat),fun(A,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(A,rat),fun(A,fun(rat,rat))),combb(rat,fun(rat,rat),A),times_times(rat)),B)),C)),D) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),B),D)),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),C),D)). [resolve(5559,a,5553,a)]. 48.96/48.86 Derived: -comm_monoid_add(rat) | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(fun(nat,bool),rat,hAPP(fun(nat,rat),fun(fun(nat,bool),rat),big_co1399186613setsum(nat,rat),hAPP(fun(nat,int),fun(nat,rat),hAPP(fun(int,rat),fun(fun(nat,int),fun(nat,rat)),combb(int,rat,nat),B),semiring_1_of_nat(int))),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A)) = hAPP(fun(int,bool),rat,hAPP(fun(int,rat),fun(fun(int,bool),rat),big_co1399186613setsum(int,rat),B),A). [resolve(5559,a,5554,b)]. 48.96/48.86 Derived: hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | one_one(rat) = hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),C),B). [resolve(5559,a,5555,a)]. 48.96/48.86 Derived: one_one(rat) = hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),hAPP(rat,fun(A,rat),combk(rat,A),one_one(rat))),B). [resolve(5559,a,5558,a)]. 48.96/48.86 5560 -ring_1(A) | -comm_monoid_mult(A) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),hAPP(fun(B,int),fun(B,A),hAPP(fun(int,A),fun(fun(B,int),fun(B,A)),combb(int,A,B),ring_1_of_int(A)),C)),D) = hAPP(int,A,ring_1_of_int(A),hAPP(fun(B,bool),int,hAPP(fun(B,int),fun(fun(B,bool),int),big_co302546470etprod(B,int),C),D)) # label(fact_3956_of__int__setprod) # label(axiom). [clausify(974)]. 48.96/48.86 Derived: -ring_1(nat) | hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),hAPP(fun(A,int),fun(A,nat),hAPP(fun(int,nat),fun(fun(A,int),fun(A,nat)),combb(int,nat,A),ring_1_of_int(nat)),B)),C) = hAPP(int,nat,ring_1_of_int(nat),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),B),C)). [resolve(5560,b,5550,a)]. 48.96/48.86 Derived: -ring_1(complex) | hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),hAPP(fun(A,int),fun(A,complex),hAPP(fun(int,complex),fun(fun(A,int),fun(A,complex)),combb(int,complex,A),ring_1_of_int(complex)),B)),C) = hAPP(int,complex,ring_1_of_int(complex),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),B),C)). [resolve(5560,b,5557,a)]. 48.96/48.86 Derived: -ring_1(rat) | hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),hAPP(fun(A,int),fun(A,rat),hAPP(fun(int,rat),fun(fun(A,int),fun(A,rat)),combb(int,rat,A),ring_1_of_int(rat)),B)),C) = hAPP(int,rat,ring_1_of_int(rat),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),B),C)). [resolve(5560,b,5559,a)]. 48.96/48.86 5561 -comm_monoid_mult(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f205(B,A,D,C)),C)) | one_one(A) = hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),D),C) # label(fact_4175_setprod_Oneutral) # label(axiom). [clausify(1138)]. 48.96/48.86 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f205(A,nat,C,B)),B)) | one_one(nat) = hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),C),B). [resolve(5561,a,5550,a)]. 48.96/48.86 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f205(A,complex,C,B)),B)) | one_one(complex) = hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),C),B). [resolve(5561,a,5557,a)]. 48.96/48.86 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f205(A,rat,C,B)),B)) | one_one(rat) = hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),C),B). [resolve(5561,a,5559,a)]. 48.96/48.86 5562 -comm_monoid_mult(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | one_one(A) != hAPP(B,A,D,f205(B,A,D,C)) | one_one(A) = hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),D),C) # label(fact_4175_setprod_Oneutral) # label(axiom). [clausify(1138)]. 48.96/48.91 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | one_one(nat) != hAPP(A,nat,C,f205(A,nat,C,B)) | one_one(nat) = hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),C),B). [resolve(5562,a,5550,a)]. 48.96/48.91 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | one_one(complex) != hAPP(A,complex,C,f205(A,complex,C,B)) | one_one(complex) = hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),C),B). [resolve(5562,a,5557,a)]. 48.96/48.91 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | one_one(rat) != hAPP(A,rat,C,f205(A,rat,C,B)) | one_one(rat) = hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),C),B). [resolve(5562,a,5559,a)]. 48.96/48.91 5563 -comm_monoid_mult(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | -hBOOL(hAPP(fun(B,bool),bool,hAPP(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),D),C)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f238(B,A,E,D,C)),hAPP(fun(B,bool),fun(B,bool),hAPP(fun(B,bool),fun(fun(B,bool),fun(B,bool)),minus_minus(fun(B,bool)),C),D))) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),E),C) = hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),E),D) # label(fact_3988_setprod__mono__one__left) # label(axiom). [clausify(1316)]. 48.96/48.91 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f238(A,nat,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),D),B) = hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),D),C). [resolve(5563,a,5550,a)]. 48.96/48.91 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f238(A,complex,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),D),B) = hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),D),C). [resolve(5563,a,5557,a)]. 48.96/48.91 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f238(A,rat,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),D),B) = hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),D),C). [resolve(5563,a,5559,a)]. 48.96/48.91 5564 -comm_monoid_mult(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | -hBOOL(hAPP(fun(B,bool),bool,hAPP(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),D),C)) | one_one(A) != hAPP(B,A,E,f238(B,A,E,D,C)) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),E),C) = hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),E),D) # label(fact_3988_setprod__mono__one__left) # label(axiom). [clausify(1316)]. 48.96/48.91 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | one_one(nat) != hAPP(A,nat,D,f238(A,nat,D,C,B)) | hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),D),B) = hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),D),C). [resolve(5564,a,5550,a)]. 49.05/48.99 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | one_one(complex) != hAPP(A,complex,D,f238(A,complex,D,C,B)) | hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),D),B) = hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),D),C). [resolve(5564,a,5557,a)]. 49.05/48.99 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | one_one(rat) != hAPP(A,rat,D,f238(A,rat,D,C,B)) | hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),D),B) = hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),D),C). [resolve(5564,a,5559,a)]. 49.05/48.99 5565 -comm_monoid_mult(A) | big_co302546470etprod(B,A) = ti(fun(fun(B,A),fun(fun(B,bool),A)),big_co302546470etprod(B,A)) # label(tsy_c_Big__Operators_Ocomm__monoid__mult__class_Osetprod_res) # label(axiom). [clausify(1396)]. 49.05/48.99 Derived: big_co302546470etprod(A,nat) = ti(fun(fun(A,nat),fun(fun(A,bool),nat)),big_co302546470etprod(A,nat)). [resolve(5565,a,5550,a)]. 49.05/48.99 Derived: big_co302546470etprod(A,complex) = ti(fun(fun(A,complex),fun(fun(A,bool),complex)),big_co302546470etprod(A,complex)). [resolve(5565,a,5557,a)]. 49.05/48.99 Derived: big_co302546470etprod(A,rat) = ti(fun(fun(A,rat),fun(fun(A,bool),rat)),big_co302546470etprod(A,rat)). [resolve(5565,a,5559,a)]. 49.05/48.99 5566 -comm_monoid_mult(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),one_one(A)),B) # label(fact_389_mult__1) # label(axiom). [clausify(1487)]. 49.05/48.99 Derived: ti(nat,A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),one_one(nat)),A). [resolve(5566,a,5550,a)]. 49.05/48.99 Derived: ti(complex,A) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),one_one(complex)),A). [resolve(5566,a,5557,a)]. 49.05/48.99 Derived: ti(rat,A) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),one_one(rat)),A). [resolve(5566,a,5559,a)]. 49.05/48.99 5567 -comm_monoid_mult(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,one_one(A)),one_one(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,f266(C,A,D,E,F,B)),f268(C,A,D,E,F,B))) | -hBOOL(hAPP(fun(C,bool),bool,finite_finite_1(C),F)) | hBOOL(hAPP(fun(C,bool),bool,hAPP(C,fun(fun(C,bool),bool),member(C),f269(C,A,D,E,F,B)),F)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,hAPP(fun(C,bool),A,hAPP(fun(C,A),fun(fun(C,bool),A),big_co302546470etprod(C,A),D),F)),hAPP(fun(C,bool),A,hAPP(fun(C,A),fun(fun(C,bool),A),big_co302546470etprod(C,A),E),F))) # label(fact_4804_setprod_Orelated) # label(axiom). [clausify(1567)]. 49.05/48.99 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,one_one(nat)),one_one(nat))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,f266(B,nat,C,D,E,A)),f268(B,nat,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f269(B,nat,C,D,E,A)),E)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,hAPP(fun(B,bool),nat,hAPP(fun(B,nat),fun(fun(B,bool),nat),big_co302546470etprod(B,nat),C),E)),hAPP(fun(B,bool),nat,hAPP(fun(B,nat),fun(fun(B,bool),nat),big_co302546470etprod(B,nat),D),E))). [resolve(5567,a,5550,a)]. 49.05/48.99 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),A,one_one(complex)),one_one(complex))) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),A,f266(B,complex,C,D,E,A)),f268(B,complex,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f269(B,complex,C,D,E,A)),E)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),A,hAPP(fun(B,bool),complex,hAPP(fun(B,complex),fun(fun(B,bool),complex),big_co302546470etprod(B,complex),C),E)),hAPP(fun(B,bool),complex,hAPP(fun(B,complex),fun(fun(B,bool),complex),big_co302546470etprod(B,complex),D),E))). [resolve(5567,a,5557,a)]. 49.05/49.00 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),A,one_one(rat)),one_one(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),A,f266(B,rat,C,D,E,A)),f268(B,rat,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f269(B,rat,C,D,E,A)),E)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),A,hAPP(fun(B,bool),rat,hAPP(fun(B,rat),fun(fun(B,bool),rat),big_co302546470etprod(B,rat),C),E)),hAPP(fun(B,bool),rat,hAPP(fun(B,rat),fun(fun(B,bool),rat),big_co302546470etprod(B,rat),D),E))). [resolve(5567,a,5559,a)]. 49.05/49.00 5568 -comm_monoid_mult(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,one_one(A)),one_one(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,f266(C,A,D,E,F,B)),f268(C,A,D,E,F,B))) | -hBOOL(hAPP(fun(C,bool),bool,finite_finite_1(C),F)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,hAPP(C,A,D,f269(C,A,D,E,F,B))),hAPP(C,A,E,f269(C,A,D,E,F,B)))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,hAPP(fun(C,bool),A,hAPP(fun(C,A),fun(fun(C,bool),A),big_co302546470etprod(C,A),D),F)),hAPP(fun(C,bool),A,hAPP(fun(C,A),fun(fun(C,bool),A),big_co302546470etprod(C,A),E),F))) # label(fact_4804_setprod_Orelated) # label(axiom). [clausify(1567)]. 49.05/49.00 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,one_one(nat)),one_one(nat))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,f266(B,nat,C,D,E,A)),f268(B,nat,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,hAPP(B,nat,C,f269(B,nat,C,D,E,A))),hAPP(B,nat,D,f269(B,nat,C,D,E,A)))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,hAPP(fun(B,bool),nat,hAPP(fun(B,nat),fun(fun(B,bool),nat),big_co302546470etprod(B,nat),C),E)),hAPP(fun(B,bool),nat,hAPP(fun(B,nat),fun(fun(B,bool),nat),big_co302546470etprod(B,nat),D),E))). [resolve(5568,a,5550,a)]. 49.05/49.00 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),A,one_one(complex)),one_one(complex))) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),A,f266(B,complex,C,D,E,A)),f268(B,complex,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),A,hAPP(B,complex,C,f269(B,complex,C,D,E,A))),hAPP(B,complex,D,f269(B,complex,C,D,E,A)))) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),A,hAPP(fun(B,bool),complex,hAPP(fun(B,complex),fun(fun(B,bool),complex),big_co302546470etprod(B,complex),C),E)),hAPP(fun(B,bool),complex,hAPP(fun(B,complex),fun(fun(B,bool),complex),big_co302546470etprod(B,complex),D),E))). [resolve(5568,a,5557,a)]. 49.05/49.00 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),A,one_one(rat)),one_one(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),A,f266(B,rat,C,D,E,A)),f268(B,rat,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),A,hAPP(B,rat,C,f269(B,rat,C,D,E,A))),hAPP(B,rat,D,f269(B,rat,C,D,E,A)))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),A,hAPP(fun(B,bool),rat,hAPP(fun(B,rat),fun(fun(B,bool),rat),big_co302546470etprod(B,rat),C),E)),hAPP(fun(B,bool),rat,hAPP(fun(B,rat),fun(fun(B,bool),rat),big_co302546470etprod(B,rat),D),E))). [resolve(5568,a,5559,a)]. 49.05/49.00 5569 -comm_monoid_mult(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,one_one(A)),one_one(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,f265(C,A,D,E,F,B)),f267(C,A,D,E,F,B))) | -hBOOL(hAPP(fun(C,bool),bool,finite_finite_1(C),F)) | hBOOL(hAPP(fun(C,bool),bool,hAPP(C,fun(fun(C,bool),bool),member(C),f269(C,A,D,E,F,B)),F)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,hAPP(fun(C,bool),A,hAPP(fun(C,A),fun(fun(C,bool),A),big_co302546470etprod(C,A),D),F)),hAPP(fun(C,bool),A,hAPP(fun(C,A),fun(fun(C,bool),A),big_co302546470etprod(C,A),E),F))) # label(fact_4804_setprod_Orelated) # label(axiom). [clausify(1567)]. 49.05/49.00 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,one_one(nat)),one_one(nat))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,f265(B,nat,C,D,E,A)),f267(B,nat,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f269(B,nat,C,D,E,A)),E)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,hAPP(fun(B,bool),nat,hAPP(fun(B,nat),fun(fun(B,bool),nat),big_co302546470etprod(B,nat),C),E)),hAPP(fun(B,bool),nat,hAPP(fun(B,nat),fun(fun(B,bool),nat),big_co302546470etprod(B,nat),D),E))). [resolve(5569,a,5550,a)]. 49.05/49.01 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),A,one_one(complex)),one_one(complex))) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),A,f265(B,complex,C,D,E,A)),f267(B,complex,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f269(B,complex,C,D,E,A)),E)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),A,hAPP(fun(B,bool),complex,hAPP(fun(B,complex),fun(fun(B,bool),complex),big_co302546470etprod(B,complex),C),E)),hAPP(fun(B,bool),complex,hAPP(fun(B,complex),fun(fun(B,bool),complex),big_co302546470etprod(B,complex),D),E))). [resolve(5569,a,5557,a)]. 49.05/49.01 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),A,one_one(rat)),one_one(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),A,f265(B,rat,C,D,E,A)),f267(B,rat,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f269(B,rat,C,D,E,A)),E)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),A,hAPP(fun(B,bool),rat,hAPP(fun(B,rat),fun(fun(B,bool),rat),big_co302546470etprod(B,rat),C),E)),hAPP(fun(B,bool),rat,hAPP(fun(B,rat),fun(fun(B,bool),rat),big_co302546470etprod(B,rat),D),E))). [resolve(5569,a,5559,a)]. 49.05/49.01 5570 -comm_monoid_mult(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,one_one(A)),one_one(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,f265(C,A,D,E,F,B)),f267(C,A,D,E,F,B))) | -hBOOL(hAPP(fun(C,bool),bool,finite_finite_1(C),F)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,hAPP(C,A,D,f269(C,A,D,E,F,B))),hAPP(C,A,E,f269(C,A,D,E,F,B)))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,hAPP(fun(C,bool),A,hAPP(fun(C,A),fun(fun(C,bool),A),big_co302546470etprod(C,A),D),F)),hAPP(fun(C,bool),A,hAPP(fun(C,A),fun(fun(C,bool),A),big_co302546470etprod(C,A),E),F))) # label(fact_4804_setprod_Orelated) # label(axiom). [clausify(1567)]. 49.05/49.01 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,one_one(nat)),one_one(nat))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,f265(B,nat,C,D,E,A)),f267(B,nat,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,hAPP(B,nat,C,f269(B,nat,C,D,E,A))),hAPP(B,nat,D,f269(B,nat,C,D,E,A)))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,hAPP(fun(B,bool),nat,hAPP(fun(B,nat),fun(fun(B,bool),nat),big_co302546470etprod(B,nat),C),E)),hAPP(fun(B,bool),nat,hAPP(fun(B,nat),fun(fun(B,bool),nat),big_co302546470etprod(B,nat),D),E))). [resolve(5570,a,5550,a)]. 49.05/49.01 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),A,one_one(complex)),one_one(complex))) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),A,f265(B,complex,C,D,E,A)),f267(B,complex,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),A,hAPP(B,complex,C,f269(B,complex,C,D,E,A))),hAPP(B,complex,D,f269(B,complex,C,D,E,A)))) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),A,hAPP(fun(B,bool),complex,hAPP(fun(B,complex),fun(fun(B,bool),complex),big_co302546470etprod(B,complex),C),E)),hAPP(fun(B,bool),complex,hAPP(fun(B,complex),fun(fun(B,bool),complex),big_co302546470etprod(B,complex),D),E))). [resolve(5570,a,5557,a)]. 49.05/49.01 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),A,one_one(rat)),one_one(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),A,f265(B,rat,C,D,E,A)),f267(B,rat,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),A,hAPP(B,rat,C,f269(B,rat,C,D,E,A))),hAPP(B,rat,D,f269(B,rat,C,D,E,A)))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),A,hAPP(fun(B,bool),rat,hAPP(fun(B,rat),fun(fun(B,bool),rat),big_co302546470etprod(B,rat),C),E)),hAPP(fun(B,bool),rat,hAPP(fun(B,rat),fun(fun(B,bool),rat),big_co302546470etprod(B,rat),D),E))). [resolve(5570,a,5559,a)]. 49.14/49.01 5571 -comm_monoid_mult(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,one_one(A)),one_one(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,hAPP(A,A,hAPP(A,fun(A,A),times_times(A),f265(C,A,D,E,F,B)),f266(C,A,D,E,F,B))),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),f267(C,A,D,E,F,B)),f268(C,A,D,E,F,B)))) | -hBOOL(hAPP(fun(C,bool),bool,finite_finite_1(C),F)) | hBOOL(hAPP(fun(C,bool),bool,hAPP(C,fun(fun(C,bool),bool),member(C),f269(C,A,D,E,F,B)),F)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,hAPP(fun(C,bool),A,hAPP(fun(C,A),fun(fun(C,bool),A),big_co302546470etprod(C,A),D),F)),hAPP(fun(C,bool),A,hAPP(fun(C,A),fun(fun(C,bool),A),big_co302546470etprod(C,A),E),F))) # label(fact_4804_setprod_Orelated) # label(axiom). [clausify(1567)]. 49.14/49.01 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,one_one(nat)),one_one(nat))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),f265(B,nat,C,D,E,A)),f266(B,nat,C,D,E,A))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),f267(B,nat,C,D,E,A)),f268(B,nat,C,D,E,A)))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f269(B,nat,C,D,E,A)),E)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,hAPP(fun(B,bool),nat,hAPP(fun(B,nat),fun(fun(B,bool),nat),big_co302546470etprod(B,nat),C),E)),hAPP(fun(B,bool),nat,hAPP(fun(B,nat),fun(fun(B,bool),nat),big_co302546470etprod(B,nat),D),E))). [resolve(5571,a,5550,a)]. 49.14/49.01 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),A,one_one(complex)),one_one(complex))) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),A,hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),f265(B,complex,C,D,E,A)),f266(B,complex,C,D,E,A))),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),f267(B,complex,C,D,E,A)),f268(B,complex,C,D,E,A)))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f269(B,complex,C,D,E,A)),E)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),A,hAPP(fun(B,bool),complex,hAPP(fun(B,complex),fun(fun(B,bool),complex),big_co302546470etprod(B,complex),C),E)),hAPP(fun(B,bool),complex,hAPP(fun(B,complex),fun(fun(B,bool),complex),big_co302546470etprod(B,complex),D),E))). [resolve(5571,a,5557,a)]. 49.14/49.01 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),A,one_one(rat)),one_one(rat))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),A,hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),f265(B,rat,C,D,E,A)),f266(B,rat,C,D,E,A))),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),f267(B,rat,C,D,E,A)),f268(B,rat,C,D,E,A)))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f269(B,rat,C,D,E,A)),E)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),A,hAPP(fun(B,bool),rat,hAPP(fun(B,rat),fun(fun(B,bool),rat),big_co302546470etprod(B,rat),C),E)),hAPP(fun(B,bool),rat,hAPP(fun(B,rat),fun(fun(B,bool),rat),big_co302546470etprod(B,rat),D),E))). [resolve(5571,a,5559,a)]. 49.14/49.01 5572 -comm_monoid_mult(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,one_one(A)),one_one(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,hAPP(A,A,hAPP(A,fun(A,A),times_times(A),f265(C,A,D,E,F,B)),f266(C,A,D,E,F,B))),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),f267(C,A,D,E,F,B)),f268(C,A,D,E,F,B)))) | -hBOOL(hAPP(fun(C,bool),bool,finite_finite_1(C),F)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,hAPP(C,A,D,f269(C,A,D,E,F,B))),hAPP(C,A,E,f269(C,A,D,E,F,B)))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),B,hAPP(fun(C,bool),A,hAPP(fun(C,A),fun(fun(C,bool),A),big_co302546470etprod(C,A),D),F)),hAPP(fun(C,bool),A,hAPP(fun(C,A),fun(fun(C,bool),A),big_co302546470etprod(C,A),E),F))) # label(fact_4804_setprod_Orelated) # label(axiom). [clausify(1567)]. 49.14/49.01 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,one_one(nat)),one_one(nat))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),f265(B,nat,C,D,E,A)),f266(B,nat,C,D,E,A))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),f267(B,nat,C,D,E,A)),f268(B,nat,C,D,E,A)))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,hAPP(B,nat,C,f269(B,nat,C,D,E,A))),hAPP(B,nat,D,f269(B,nat,C,D,E,A)))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),A,hAPP(fun(B,bool),nat,hAPP(fun(B,nat),fun(fun(B,bool),nat),big_co302546470etprod(B,nat),C),E)),hAPP(fun(B,bool),nat,hAPP(fun(B,nat),fun(fun(B,bool),nat),big_co302546470etprod(B,nat),D),E))). [resolve(5572,a,5550,a)]. 49.14/49.03 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),A,one_one(complex)),one_one(complex))) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),A,hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),f265(B,complex,C,D,E,A)),f266(B,complex,C,D,E,A))),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),f267(B,complex,C,D,E,A)),f268(B,complex,C,D,E,A)))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),A,hAPP(B,complex,C,f269(B,complex,C,D,E,A))),hAPP(B,complex,D,f269(B,complex,C,D,E,A)))) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),A,hAPP(fun(B,bool),complex,hAPP(fun(B,complex),fun(fun(B,bool),complex),big_co302546470etprod(B,complex),C),E)),hAPP(fun(B,bool),complex,hAPP(fun(B,complex),fun(fun(B,bool),complex),big_co302546470etprod(B,complex),D),E))). [resolve(5572,a,5557,a)]. 49.14/49.03 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),A,one_one(rat)),one_one(rat))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),A,hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),f265(B,rat,C,D,E,A)),f266(B,rat,C,D,E,A))),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),f267(B,rat,C,D,E,A)),f268(B,rat,C,D,E,A)))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),A,hAPP(B,rat,C,f269(B,rat,C,D,E,A))),hAPP(B,rat,D,f269(B,rat,C,D,E,A)))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),A,hAPP(fun(B,bool),rat,hAPP(fun(B,rat),fun(fun(B,bool),rat),big_co302546470etprod(B,rat),C),E)),hAPP(fun(B,bool),rat,hAPP(fun(B,rat),fun(fun(B,bool),rat),big_co302546470etprod(B,rat),D),E))). [resolve(5572,a,5559,a)]. 49.14/49.03 5573 -comm_monoid_mult(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),D)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),D)) # label(fact_394_power__mult__distrib) # label(axiom). [clausify(1580)]. 49.14/49.03 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),C) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),C)). [resolve(5573,a,5550,a)]. 49.14/49.03 Derived: hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)),C) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),C)),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),B),C)). [resolve(5573,a,5557,a)]. 49.14/49.03 Derived: hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),C) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),C)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),B),C)). [resolve(5573,a,5559,a)]. 49.14/49.03 5574 -comm_monoid_mult(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),one_one(A)) # label(fact_392_mult_Ocomm__neutral) # label(axiom). [clausify(1601)]. 49.14/49.03 Derived: ti(nat,A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),one_one(nat)). [resolve(5574,a,5550,a)]. 49.14/49.03 Derived: ti(complex,A) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),one_one(complex)). [resolve(5574,a,5557,a)]. 49.14/49.03 Derived: ti(rat,A) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),one_one(rat)). [resolve(5574,a,5559,a)]. 49.14/49.05 5575 comm_monoid_mult(real) # label(arity_RealDef_Oreal___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 49.14/49.05 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | one_one(real) = hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),hAPP(real,fun(A,real),hAPP(fun(A,fun(real,real)),fun(real,fun(A,real)),combc(A,real,real),hAPP(fun(A,real),fun(A,fun(real,real)),hAPP(fun(A,fun(real,fun(real,real))),fun(fun(A,real),fun(A,fun(real,real))),combs(A,real,fun(real,real)),hAPP(fun(A,bool),fun(A,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(A,bool),fun(A,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),A),if(real)),hAPP(A,fun(A,bool),fequal(A),C))),D)),one_one(real))),B). [resolve(5575,a,5551,a)]. 49.14/49.05 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),hAPP(real,fun(A,real),hAPP(fun(A,fun(real,real)),fun(real,fun(A,real)),combc(A,real,real),hAPP(fun(A,real),fun(A,fun(real,real)),hAPP(fun(A,fun(real,fun(real,real))),fun(fun(A,real),fun(A,fun(real,real))),combs(A,real,fun(real,real)),hAPP(fun(A,bool),fun(A,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(A,bool),fun(A,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),A),if(real)),hAPP(A,fun(A,bool),fequal(A),C))),D)),one_one(real))),B) = hAPP(A,real,D,C). [resolve(5575,a,5552,a)]. 49.14/49.05 Derived: hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),hAPP(fun(A,real),fun(A,real),hAPP(fun(A,fun(real,real)),fun(fun(A,real),fun(A,real)),combs(A,real,real),hAPP(fun(A,real),fun(A,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(A,real),fun(A,fun(real,real))),combb(real,fun(real,real),A),times_times(real)),B)),C)),D) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),B),D)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),C),D)). [resolve(5575,a,5553,a)]. 49.14/49.05 Derived: -comm_monoid_add(real) | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,int),fun(nat,real),hAPP(fun(int,real),fun(fun(nat,int),fun(nat,real)),combb(int,real,nat),B),semiring_1_of_nat(int))),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A)) = hAPP(fun(int,bool),real,hAPP(fun(int,real),fun(fun(int,bool),real),big_co1399186613setsum(int,real),B),A). [resolve(5575,a,5554,b)]. 49.14/49.05 Derived: hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | one_one(real) = hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),C),B). [resolve(5575,a,5555,a)]. 49.14/49.05 Derived: one_one(real) = hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),hAPP(real,fun(A,real),combk(real,A),one_one(real))),B). [resolve(5575,a,5558,a)]. 49.14/49.05 Derived: -ring_1(real) | hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),hAPP(fun(A,int),fun(A,real),hAPP(fun(int,real),fun(fun(A,int),fun(A,real)),combb(int,real,A),ring_1_of_int(real)),B)),C) = hAPP(int,real,ring_1_of_int(real),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),B),C)). [resolve(5575,a,5560,b)]. 49.14/49.05 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f205(A,real,C,B)),B)) | one_one(real) = hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),C),B). [resolve(5575,a,5561,a)]. 49.14/49.05 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | one_one(real) != hAPP(A,real,C,f205(A,real,C,B)) | one_one(real) = hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),C),B). [resolve(5575,a,5562,a)]. 49.14/49.06 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f238(A,real,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),D),B) = hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),D),C). [resolve(5575,a,5563,a)]. 49.14/49.06 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | one_one(real) != hAPP(A,real,D,f238(A,real,D,C,B)) | hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),D),B) = hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),D),C). [resolve(5575,a,5564,a)]. 49.14/49.06 Derived: big_co302546470etprod(A,real) = ti(fun(fun(A,real),fun(fun(A,bool),real)),big_co302546470etprod(A,real)). [resolve(5575,a,5565,a)]. 49.14/49.06 Derived: ti(real,A) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),one_one(real)),A). [resolve(5575,a,5566,a)]. 49.14/49.06 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),A,one_one(real)),one_one(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),A,f266(B,real,C,D,E,A)),f268(B,real,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f269(B,real,C,D,E,A)),E)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),A,hAPP(fun(B,bool),real,hAPP(fun(B,real),fun(fun(B,bool),real),big_co302546470etprod(B,real),C),E)),hAPP(fun(B,bool),real,hAPP(fun(B,real),fun(fun(B,bool),real),big_co302546470etprod(B,real),D),E))). [resolve(5575,a,5567,a)]. 49.14/49.06 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),A,one_one(real)),one_one(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),A,f266(B,real,C,D,E,A)),f268(B,real,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),A,hAPP(B,real,C,f269(B,real,C,D,E,A))),hAPP(B,real,D,f269(B,real,C,D,E,A)))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),A,hAPP(fun(B,bool),real,hAPP(fun(B,real),fun(fun(B,bool),real),big_co302546470etprod(B,real),C),E)),hAPP(fun(B,bool),real,hAPP(fun(B,real),fun(fun(B,bool),real),big_co302546470etprod(B,real),D),E))). [resolve(5575,a,5568,a)]. 49.14/49.06 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),A,one_one(real)),one_one(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),A,f265(B,real,C,D,E,A)),f267(B,real,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f269(B,real,C,D,E,A)),E)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),A,hAPP(fun(B,bool),real,hAPP(fun(B,real),fun(fun(B,bool),real),big_co302546470etprod(B,real),C),E)),hAPP(fun(B,bool),real,hAPP(fun(B,real),fun(fun(B,bool),real),big_co302546470etprod(B,real),D),E))). [resolve(5575,a,5569,a)]. 49.14/49.06 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),A,one_one(real)),one_one(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),A,f265(B,real,C,D,E,A)),f267(B,real,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),A,hAPP(B,real,C,f269(B,real,C,D,E,A))),hAPP(B,real,D,f269(B,real,C,D,E,A)))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),A,hAPP(fun(B,bool),real,hAPP(fun(B,real),fun(fun(B,bool),real),big_co302546470etprod(B,real),C),E)),hAPP(fun(B,bool),real,hAPP(fun(B,real),fun(fun(B,bool),real),big_co302546470etprod(B,real),D),E))). [resolve(5575,a,5570,a)]. 49.14/49.06 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),A,one_one(real)),one_one(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),A,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),f265(B,real,C,D,E,A)),f266(B,real,C,D,E,A))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),f267(B,real,C,D,E,A)),f268(B,real,C,D,E,A)))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f269(B,real,C,D,E,A)),E)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),A,hAPP(fun(B,bool),real,hAPP(fun(B,real),fun(fun(B,bool),real),big_co302546470etprod(B,real),C),E)),hAPP(fun(B,bool),real,hAPP(fun(B,real),fun(fun(B,bool),real),big_co302546470etprod(B,real),D),E))). [resolve(5575,a,5571,a)]. 49.22/49.13 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),A,one_one(real)),one_one(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),A,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),f265(B,real,C,D,E,A)),f266(B,real,C,D,E,A))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),f267(B,real,C,D,E,A)),f268(B,real,C,D,E,A)))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),A,hAPP(B,real,C,f269(B,real,C,D,E,A))),hAPP(B,real,D,f269(B,real,C,D,E,A)))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),A,hAPP(fun(B,bool),real,hAPP(fun(B,real),fun(fun(B,bool),real),big_co302546470etprod(B,real),C),E)),hAPP(fun(B,bool),real,hAPP(fun(B,real),fun(fun(B,bool),real),big_co302546470etprod(B,real),D),E))). [resolve(5575,a,5572,a)]. 49.22/49.13 Derived: hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),C) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),C)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),C)). [resolve(5575,a,5573,a)]. 49.22/49.13 Derived: ti(real,A) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),one_one(real)). [resolve(5575,a,5574,a)]. 49.22/49.13 5576 -comm_monoid_mult(A) | -dvd(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | -hBOOL(hAPP(fun(B,bool),bool,hAPP(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),D),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),E),D)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),E),C))) # label(fact_3985_setprod__dvd__setprod__subset) # label(axiom). [clausify(1831)]. 49.22/49.13 Derived: -dvd(nat) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),D),C)),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),D),B))). [resolve(5576,a,5550,a)]. 49.22/49.13 Derived: -dvd(complex) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),D),C)),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),D),B))). [resolve(5576,a,5557,a)]. 49.22/49.13 Derived: -dvd(rat) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),D),C)),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),D),B))). [resolve(5576,a,5559,a)]. 49.22/49.13 Derived: -dvd(real) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),D),C)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),D),B))). [resolve(5576,a,5575,a)]. 49.35/49.24 5577 -comm_monoid_mult(A) | -comm_monoid_add(A) | hAPP(fun(nat,bool),A,hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co302546470etprod(nat,A),B),C) = hAPP(fun(int,bool),A,hAPP(fun(int,A),fun(fun(int,bool),A),big_co302546470etprod(int,A),hAPP(fun(int,nat),fun(int,A),hAPP(fun(nat,A),fun(fun(int,nat),fun(int,A)),combb(nat,A,int),B),nat_1)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),C)) # label(fact_3971_transfer__nat__int__sum__prod_I2_J) # label(axiom). [clausify(2139)]. 49.35/49.24 Derived: -comm_monoid_add(nat) | hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co302546470etprod(nat,nat),A),B) = hAPP(fun(int,bool),nat,hAPP(fun(int,nat),fun(fun(int,bool),nat),big_co302546470etprod(int,nat),hAPP(fun(int,nat),fun(int,nat),hAPP(fun(nat,nat),fun(fun(int,nat),fun(int,nat)),combb(nat,nat,int),A),nat_1)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B)). [resolve(5577,a,5550,a)]. 49.35/49.24 Derived: -comm_monoid_add(complex) | hAPP(fun(nat,bool),complex,hAPP(fun(nat,complex),fun(fun(nat,bool),complex),big_co302546470etprod(nat,complex),A),B) = hAPP(fun(int,bool),complex,hAPP(fun(int,complex),fun(fun(int,bool),complex),big_co302546470etprod(int,complex),hAPP(fun(int,nat),fun(int,complex),hAPP(fun(nat,complex),fun(fun(int,nat),fun(int,complex)),combb(nat,complex,int),A),nat_1)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B)). [resolve(5577,a,5557,a)]. 49.35/49.24 Derived: -comm_monoid_add(rat) | hAPP(fun(nat,bool),rat,hAPP(fun(nat,rat),fun(fun(nat,bool),rat),big_co302546470etprod(nat,rat),A),B) = hAPP(fun(int,bool),rat,hAPP(fun(int,rat),fun(fun(int,bool),rat),big_co302546470etprod(int,rat),hAPP(fun(int,nat),fun(int,rat),hAPP(fun(nat,rat),fun(fun(int,nat),fun(int,rat)),combb(nat,rat,int),A),nat_1)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B)). [resolve(5577,a,5559,a)]. 49.35/49.24 Derived: -comm_monoid_add(real) | hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co302546470etprod(nat,real),A),B) = hAPP(fun(int,bool),real,hAPP(fun(int,real),fun(fun(int,bool),real),big_co302546470etprod(int,real),hAPP(fun(int,nat),fun(int,real),hAPP(fun(nat,real),fun(fun(int,nat),fun(int,real)),combb(nat,real,int),A),nat_1)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B)). [resolve(5577,a,5575,a)]. 49.35/49.24 5578 comm_monoid_mult(code_code_numeral) # label(arity_Code__Numeral_Ocode__numeral___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 49.35/49.24 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | one_one(code_code_numeral) = hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),hAPP(code_code_numeral,fun(A,code_code_numeral),hAPP(fun(A,fun(code_code_numeral,code_code_numeral)),fun(code_code_numeral,fun(A,code_code_numeral)),combc(A,code_code_numeral,code_code_numeral),hAPP(fun(A,code_code_numeral),fun(A,fun(code_code_numeral,code_code_numeral)),hAPP(fun(A,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral))),fun(fun(A,code_code_numeral),fun(A,fun(code_code_numeral,code_code_numeral))),combs(A,code_code_numeral,fun(code_code_numeral,code_code_numeral)),hAPP(fun(A,bool),fun(A,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral))),hAPP(fun(bool,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral))),fun(fun(A,bool),fun(A,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)))),combb(bool,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)),A),if(code_code_numeral)),hAPP(A,fun(A,bool),fequal(A),C))),D)),one_one(code_code_numeral))),B). [resolve(5578,a,5551,a)]. 49.35/49.25 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),hAPP(code_code_numeral,fun(A,code_code_numeral),hAPP(fun(A,fun(code_code_numeral,code_code_numeral)),fun(code_code_numeral,fun(A,code_code_numeral)),combc(A,code_code_numeral,code_code_numeral),hAPP(fun(A,code_code_numeral),fun(A,fun(code_code_numeral,code_code_numeral)),hAPP(fun(A,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral))),fun(fun(A,code_code_numeral),fun(A,fun(code_code_numeral,code_code_numeral))),combs(A,code_code_numeral,fun(code_code_numeral,code_code_numeral)),hAPP(fun(A,bool),fun(A,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral))),hAPP(fun(bool,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral))),fun(fun(A,bool),fun(A,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)))),combb(bool,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)),A),if(code_code_numeral)),hAPP(A,fun(A,bool),fequal(A),C))),D)),one_one(code_code_numeral))),B) = hAPP(A,code_code_numeral,D,C). [resolve(5578,a,5552,a)]. 49.35/49.25 Derived: hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),hAPP(fun(A,code_code_numeral),fun(A,code_code_numeral),hAPP(fun(A,fun(code_code_numeral,code_code_numeral)),fun(fun(A,code_code_numeral),fun(A,code_code_numeral)),combs(A,code_code_numeral,code_code_numeral),hAPP(fun(A,code_code_numeral),fun(A,fun(code_code_numeral,code_code_numeral)),hAPP(fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)),fun(fun(A,code_code_numeral),fun(A,fun(code_code_numeral,code_code_numeral))),combb(code_code_numeral,fun(code_code_numeral,code_code_numeral),A),times_times(code_code_numeral)),B)),C)),D) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),B),D)),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),C),D)). [resolve(5578,a,5553,a)]. 49.35/49.25 Derived: -comm_monoid_add(code_code_numeral) | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(fun(nat,bool),code_code_numeral,hAPP(fun(nat,code_code_numeral),fun(fun(nat,bool),code_code_numeral),big_co1399186613setsum(nat,code_code_numeral),hAPP(fun(nat,int),fun(nat,code_code_numeral),hAPP(fun(int,code_code_numeral),fun(fun(nat,int),fun(nat,code_code_numeral)),combb(int,code_code_numeral,nat),B),semiring_1_of_nat(int))),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A)) = hAPP(fun(int,bool),code_code_numeral,hAPP(fun(int,code_code_numeral),fun(fun(int,bool),code_code_numeral),big_co1399186613setsum(int,code_code_numeral),B),A). [resolve(5578,a,5554,b)]. 49.35/49.25 Derived: hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | one_one(code_code_numeral) = hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),C),B). [resolve(5578,a,5555,a)]. 49.35/49.25 Derived: one_one(code_code_numeral) = hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),hAPP(code_code_numeral,fun(A,code_code_numeral),combk(code_code_numeral,A),one_one(code_code_numeral))),B). [resolve(5578,a,5558,a)]. 49.35/49.25 Derived: -ring_1(code_code_numeral) | hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),hAPP(fun(A,int),fun(A,code_code_numeral),hAPP(fun(int,code_code_numeral),fun(fun(A,int),fun(A,code_code_numeral)),combb(int,code_code_numeral,A),ring_1_of_int(code_code_numeral)),B)),C) = hAPP(int,code_code_numeral,ring_1_of_int(code_code_numeral),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),B),C)). [resolve(5578,a,5560,b)]. 49.35/49.26 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f205(A,code_code_numeral,C,B)),B)) | one_one(code_code_numeral) = hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),C),B). [resolve(5578,a,5561,a)]. 49.35/49.26 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | one_one(code_code_numeral) != hAPP(A,code_code_numeral,C,f205(A,code_code_numeral,C,B)) | one_one(code_code_numeral) = hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),C),B). [resolve(5578,a,5562,a)]. 49.35/49.26 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f238(A,code_code_numeral,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),D),B) = hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),D),C). [resolve(5578,a,5563,a)]. 49.35/49.26 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | one_one(code_code_numeral) != hAPP(A,code_code_numeral,D,f238(A,code_code_numeral,D,C,B)) | hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),D),B) = hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),D),C). [resolve(5578,a,5564,a)]. 49.35/49.26 Derived: big_co302546470etprod(A,code_code_numeral) = ti(fun(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral)),big_co302546470etprod(A,code_code_numeral)). [resolve(5578,a,5565,a)]. 49.35/49.26 Derived: ti(code_code_numeral,A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),one_one(code_code_numeral)),A). [resolve(5578,a,5566,a)]. 49.35/49.26 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),A,one_one(code_code_numeral)),one_one(code_code_numeral))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),A,f266(B,code_code_numeral,C,D,E,A)),f268(B,code_code_numeral,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f269(B,code_code_numeral,C,D,E,A)),E)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),A,hAPP(fun(B,bool),code_code_numeral,hAPP(fun(B,code_code_numeral),fun(fun(B,bool),code_code_numeral),big_co302546470etprod(B,code_code_numeral),C),E)),hAPP(fun(B,bool),code_code_numeral,hAPP(fun(B,code_code_numeral),fun(fun(B,bool),code_code_numeral),big_co302546470etprod(B,code_code_numeral),D),E))). [resolve(5578,a,5567,a)]. 49.35/49.26 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),A,one_one(code_code_numeral)),one_one(code_code_numeral))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),A,f266(B,code_code_numeral,C,D,E,A)),f268(B,code_code_numeral,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),A,hAPP(B,code_code_numeral,C,f269(B,code_code_numeral,C,D,E,A))),hAPP(B,code_code_numeral,D,f269(B,code_code_numeral,C,D,E,A)))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),A,hAPP(fun(B,bool),code_code_numeral,hAPP(fun(B,code_code_numeral),fun(fun(B,bool),code_code_numeral),big_co302546470etprod(B,code_code_numeral),C),E)),hAPP(fun(B,bool),code_code_numeral,hAPP(fun(B,code_code_numeral),fun(fun(B,bool),code_code_numeral),big_co302546470etprod(B,code_code_numeral),D),E))). [resolve(5578,a,5568,a)]. 49.35/49.26 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),A,one_one(code_code_numeral)),one_one(code_code_numeral))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),A,f265(B,code_code_numeral,C,D,E,A)),f267(B,code_code_numeral,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f269(B,code_code_numeral,C,D,E,A)),E)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),A,hAPP(fun(B,bool),code_code_numeral,hAPP(fun(B,code_code_numeral),fun(fun(B,bool),code_code_numeral),big_co302546470etprod(B,code_code_numeral),C),E)),hAPP(fun(B,bool),code_code_numeral,hAPP(fun(B,code_code_numeral),fun(fun(B,bool),code_code_numeral),big_co302546470etprod(B,code_code_numeral),D),E))). [resolve(5578,a,5569,a)]. 49.35/49.26 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),A,one_one(code_code_numeral)),one_one(code_code_numeral))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),A,f265(B,code_code_numeral,C,D,E,A)),f267(B,code_code_numeral,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),A,hAPP(B,code_code_numeral,C,f269(B,code_code_numeral,C,D,E,A))),hAPP(B,code_code_numeral,D,f269(B,code_code_numeral,C,D,E,A)))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),A,hAPP(fun(B,bool),code_code_numeral,hAPP(fun(B,code_code_numeral),fun(fun(B,bool),code_code_numeral),big_co302546470etprod(B,code_code_numeral),C),E)),hAPP(fun(B,bool),code_code_numeral,hAPP(fun(B,code_code_numeral),fun(fun(B,bool),code_code_numeral),big_co302546470etprod(B,code_code_numeral),D),E))). [resolve(5578,a,5570,a)]. 49.35/49.26 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),A,one_one(code_code_numeral)),one_one(code_code_numeral))) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),A,hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),f265(B,code_code_numeral,C,D,E,A)),f266(B,code_code_numeral,C,D,E,A))),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),f267(B,code_code_numeral,C,D,E,A)),f268(B,code_code_numeral,C,D,E,A)))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f269(B,code_code_numeral,C,D,E,A)),E)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),A,hAPP(fun(B,bool),code_code_numeral,hAPP(fun(B,code_code_numeral),fun(fun(B,bool),code_code_numeral),big_co302546470etprod(B,code_code_numeral),C),E)),hAPP(fun(B,bool),code_code_numeral,hAPP(fun(B,code_code_numeral),fun(fun(B,bool),code_code_numeral),big_co302546470etprod(B,code_code_numeral),D),E))). [resolve(5578,a,5571,a)]. 49.35/49.26 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),A,one_one(code_code_numeral)),one_one(code_code_numeral))) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),A,hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),f265(B,code_code_numeral,C,D,E,A)),f266(B,code_code_numeral,C,D,E,A))),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),f267(B,code_code_numeral,C,D,E,A)),f268(B,code_code_numeral,C,D,E,A)))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),A,hAPP(B,code_code_numeral,C,f269(B,code_code_numeral,C,D,E,A))),hAPP(B,code_code_numeral,D,f269(B,code_code_numeral,C,D,E,A)))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),A,hAPP(fun(B,bool),code_code_numeral,hAPP(fun(B,code_code_numeral),fun(fun(B,bool),code_code_numeral),big_co302546470etprod(B,code_code_numeral),C),E)),hAPP(fun(B,bool),code_code_numeral,hAPP(fun(B,code_code_numeral),fun(fun(B,bool),code_code_numeral),big_co302546470etprod(B,code_code_numeral),D),E))). [resolve(5578,a,5572,a)]. 49.52/49.45 Derived: hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B)),C) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),C)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),B),C)). [resolve(5578,a,5573,a)]. 49.52/49.45 Derived: ti(code_code_numeral,A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),one_one(code_code_numeral)). [resolve(5578,a,5574,a)]. 49.52/49.45 Derived: -dvd(code_code_numeral) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),D),C)),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),D),B))). [resolve(5578,a,5576,a)]. 49.52/49.45 Derived: -comm_monoid_add(code_code_numeral) | hAPP(fun(nat,bool),code_code_numeral,hAPP(fun(nat,code_code_numeral),fun(fun(nat,bool),code_code_numeral),big_co302546470etprod(nat,code_code_numeral),A),B) = hAPP(fun(int,bool),code_code_numeral,hAPP(fun(int,code_code_numeral),fun(fun(int,bool),code_code_numeral),big_co302546470etprod(int,code_code_numeral),hAPP(fun(int,nat),fun(int,code_code_numeral),hAPP(fun(nat,code_code_numeral),fun(fun(int,nat),fun(int,code_code_numeral)),combb(nat,code_code_numeral,int),A),nat_1)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B)). [resolve(5578,a,5577,a)]. 49.52/49.45 5579 -comm_monoid_mult(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),D),C)) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),hAPP(A,fun(B,A),hAPP(fun(B,fun(A,A)),fun(A,fun(B,A)),combc(B,A,A),hAPP(fun(B,A),fun(B,fun(A,A)),hAPP(fun(B,fun(A,fun(A,A))),fun(fun(B,A),fun(B,fun(A,A))),combs(B,A,fun(A,A)),hAPP(fun(B,bool),fun(B,fun(A,fun(A,A))),hAPP(fun(bool,fun(A,fun(A,A))),fun(fun(B,bool),fun(B,fun(A,fun(A,A)))),combb(bool,fun(A,fun(A,A)),B),if(A)),hAPP(B,fun(B,bool),hAPP(fun(B,fun(B,bool)),fun(B,fun(B,bool)),combc(B,B,bool),fequal(B)),D))),E)),F)),C) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),F),hAPP(fun(B,bool),nat,finite_card(B),C)) # label(fact_3963_setprod__gen__delta) # label(axiom). [clausify(2580)]. 49.52/49.45 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),hAPP(nat,fun(A,nat),hAPP(fun(A,fun(nat,nat)),fun(nat,fun(A,nat)),combc(A,nat,nat),hAPP(fun(A,nat),fun(A,fun(nat,nat)),hAPP(fun(A,fun(nat,fun(nat,nat))),fun(fun(A,nat),fun(A,fun(nat,nat))),combs(A,nat,fun(nat,nat)),hAPP(fun(A,bool),fun(A,fun(nat,fun(nat,nat))),hAPP(fun(bool,fun(nat,fun(nat,nat))),fun(fun(A,bool),fun(A,fun(nat,fun(nat,nat)))),combb(bool,fun(nat,fun(nat,nat)),A),if(nat)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),E)),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),E),hAPP(fun(A,bool),nat,finite_card(A),B)). [resolve(5579,a,5550,a)]. 49.52/49.45 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),hAPP(complex,fun(A,complex),hAPP(fun(A,fun(complex,complex)),fun(complex,fun(A,complex)),combc(A,complex,complex),hAPP(fun(A,complex),fun(A,fun(complex,complex)),hAPP(fun(A,fun(complex,fun(complex,complex))),fun(fun(A,complex),fun(A,fun(complex,complex))),combs(A,complex,fun(complex,complex)),hAPP(fun(A,bool),fun(A,fun(complex,fun(complex,complex))),hAPP(fun(bool,fun(complex,fun(complex,complex))),fun(fun(A,bool),fun(A,fun(complex,fun(complex,complex)))),combb(bool,fun(complex,fun(complex,complex)),A),if(complex)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),E)),B) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),E),hAPP(fun(A,bool),nat,finite_card(A),B)). [resolve(5579,a,5557,a)]. 49.52/49.45 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),hAPP(rat,fun(A,rat),hAPP(fun(A,fun(rat,rat)),fun(rat,fun(A,rat)),combc(A,rat,rat),hAPP(fun(A,rat),fun(A,fun(rat,rat)),hAPP(fun(A,fun(rat,fun(rat,rat))),fun(fun(A,rat),fun(A,fun(rat,rat))),combs(A,rat,fun(rat,rat)),hAPP(fun(A,bool),fun(A,fun(rat,fun(rat,rat))),hAPP(fun(bool,fun(rat,fun(rat,rat))),fun(fun(A,bool),fun(A,fun(rat,fun(rat,rat)))),combb(bool,fun(rat,fun(rat,rat)),A),if(rat)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),E)),B) = hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),E),hAPP(fun(A,bool),nat,finite_card(A),B)). [resolve(5579,a,5559,a)]. 49.52/49.45 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),hAPP(real,fun(A,real),hAPP(fun(A,fun(real,real)),fun(real,fun(A,real)),combc(A,real,real),hAPP(fun(A,real),fun(A,fun(real,real)),hAPP(fun(A,fun(real,fun(real,real))),fun(fun(A,real),fun(A,fun(real,real))),combs(A,real,fun(real,real)),hAPP(fun(A,bool),fun(A,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(A,bool),fun(A,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),A),if(real)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),E)),B) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),E),hAPP(fun(A,bool),nat,finite_card(A),B)). [resolve(5579,a,5575,a)]. 49.52/49.45 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),hAPP(code_code_numeral,fun(A,code_code_numeral),hAPP(fun(A,fun(code_code_numeral,code_code_numeral)),fun(code_code_numeral,fun(A,code_code_numeral)),combc(A,code_code_numeral,code_code_numeral),hAPP(fun(A,code_code_numeral),fun(A,fun(code_code_numeral,code_code_numeral)),hAPP(fun(A,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral))),fun(fun(A,code_code_numeral),fun(A,fun(code_code_numeral,code_code_numeral))),combs(A,code_code_numeral,fun(code_code_numeral,code_code_numeral)),hAPP(fun(A,bool),fun(A,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral))),hAPP(fun(bool,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral))),fun(fun(A,bool),fun(A,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)))),combb(bool,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)),A),if(code_code_numeral)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),E)),B) = hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),E),hAPP(fun(A,bool),nat,finite_card(A),B)). [resolve(5579,a,5578,a)]. 49.52/49.45 5580 -comm_monoid_mult(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | -hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),D),C)) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),hAPP(A,fun(B,A),hAPP(fun(B,fun(A,A)),fun(A,fun(B,A)),combc(B,A,A),hAPP(fun(B,A),fun(B,fun(A,A)),hAPP(fun(B,fun(A,fun(A,A))),fun(fun(B,A),fun(B,fun(A,A))),combs(B,A,fun(A,A)),hAPP(fun(B,bool),fun(B,fun(A,fun(A,A))),hAPP(fun(bool,fun(A,fun(A,A))),fun(fun(B,bool),fun(B,fun(A,fun(A,A)))),combb(bool,fun(A,fun(A,A)),B),if(A)),hAPP(B,fun(B,bool),hAPP(fun(B,fun(B,bool)),fun(B,fun(B,bool)),combc(B,B,bool),fequal(B)),D))),E)),F)),C) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(B,A,E,D)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),F),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(B,bool),nat,finite_card(B),C)),one_one(nat)))) # label(fact_3963_setprod__gen__delta) # label(axiom). [clausify(2580)]. 49.52/49.45 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),hAPP(nat,fun(A,nat),hAPP(fun(A,fun(nat,nat)),fun(nat,fun(A,nat)),combc(A,nat,nat),hAPP(fun(A,nat),fun(A,fun(nat,nat)),hAPP(fun(A,fun(nat,fun(nat,nat))),fun(fun(A,nat),fun(A,fun(nat,nat))),combs(A,nat,fun(nat,nat)),hAPP(fun(A,bool),fun(A,fun(nat,fun(nat,nat))),hAPP(fun(bool,fun(nat,fun(nat,nat))),fun(fun(A,bool),fun(A,fun(nat,fun(nat,nat)))),combb(bool,fun(nat,fun(nat,nat)),A),if(nat)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),E)),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(A,nat,D,C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),E),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(A,bool),nat,finite_card(A),B)),one_one(nat)))). [resolve(5580,a,5550,a)]. 49.52/49.45 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),hAPP(complex,fun(A,complex),hAPP(fun(A,fun(complex,complex)),fun(complex,fun(A,complex)),combc(A,complex,complex),hAPP(fun(A,complex),fun(A,fun(complex,complex)),hAPP(fun(A,fun(complex,fun(complex,complex))),fun(fun(A,complex),fun(A,fun(complex,complex))),combs(A,complex,fun(complex,complex)),hAPP(fun(A,bool),fun(A,fun(complex,fun(complex,complex))),hAPP(fun(bool,fun(complex,fun(complex,complex))),fun(fun(A,bool),fun(A,fun(complex,fun(complex,complex)))),combb(bool,fun(complex,fun(complex,complex)),A),if(complex)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),E)),B) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(A,complex,D,C)),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),E),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(A,bool),nat,finite_card(A),B)),one_one(nat)))). [resolve(5580,a,5557,a)]. 49.52/49.45 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),hAPP(rat,fun(A,rat),hAPP(fun(A,fun(rat,rat)),fun(rat,fun(A,rat)),combc(A,rat,rat),hAPP(fun(A,rat),fun(A,fun(rat,rat)),hAPP(fun(A,fun(rat,fun(rat,rat))),fun(fun(A,rat),fun(A,fun(rat,rat))),combs(A,rat,fun(rat,rat)),hAPP(fun(A,bool),fun(A,fun(rat,fun(rat,rat))),hAPP(fun(bool,fun(rat,fun(rat,rat))),fun(fun(A,bool),fun(A,fun(rat,fun(rat,rat)))),combb(bool,fun(rat,fun(rat,rat)),A),if(rat)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),E)),B) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(A,rat,D,C)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),E),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(A,bool),nat,finite_card(A),B)),one_one(nat)))). [resolve(5580,a,5559,a)]. 49.74/49.62 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),hAPP(real,fun(A,real),hAPP(fun(A,fun(real,real)),fun(real,fun(A,real)),combc(A,real,real),hAPP(fun(A,real),fun(A,fun(real,real)),hAPP(fun(A,fun(real,fun(real,real))),fun(fun(A,real),fun(A,fun(real,real))),combs(A,real,fun(real,real)),hAPP(fun(A,bool),fun(A,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(A,bool),fun(A,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),A),if(real)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),E)),B) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(A,real,D,C)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),E),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(A,bool),nat,finite_card(A),B)),one_one(nat)))). [resolve(5580,a,5575,a)]. 49.74/49.62 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),hAPP(code_code_numeral,fun(A,code_code_numeral),hAPP(fun(A,fun(code_code_numeral,code_code_numeral)),fun(code_code_numeral,fun(A,code_code_numeral)),combc(A,code_code_numeral,code_code_numeral),hAPP(fun(A,code_code_numeral),fun(A,fun(code_code_numeral,code_code_numeral)),hAPP(fun(A,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral))),fun(fun(A,code_code_numeral),fun(A,fun(code_code_numeral,code_code_numeral))),combs(A,code_code_numeral,fun(code_code_numeral,code_code_numeral)),hAPP(fun(A,bool),fun(A,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral))),hAPP(fun(bool,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral))),fun(fun(A,bool),fun(A,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)))),combb(bool,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)),A),if(code_code_numeral)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),E)),B) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(A,code_code_numeral,D,C)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),E),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(A,bool),nat,finite_card(A),B)),one_one(nat)))). [resolve(5580,a,5578,a)]. 49.74/49.62 5581 -comm_monoid_mult(A) | hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | one_one(A) = hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),D),C) # label(fact_3978_setprod__infinite) # label(axiom). [clausify(2608)]. 49.74/49.62 5582 comm_monoid_mult(int) # label(arity_Int_Oint___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 49.74/49.62 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | one_one(int) = hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),hAPP(int,fun(A,int),hAPP(fun(A,fun(int,int)),fun(int,fun(A,int)),combc(A,int,int),hAPP(fun(A,int),fun(A,fun(int,int)),hAPP(fun(A,fun(int,fun(int,int))),fun(fun(A,int),fun(A,fun(int,int))),combs(A,int,fun(int,int)),hAPP(fun(A,bool),fun(A,fun(int,fun(int,int))),hAPP(fun(bool,fun(int,fun(int,int))),fun(fun(A,bool),fun(A,fun(int,fun(int,int)))),combb(bool,fun(int,fun(int,int)),A),if(int)),hAPP(A,fun(A,bool),fequal(A),C))),D)),one_one(int))),B). [resolve(5582,a,5551,a)]. 49.74/49.63 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),hAPP(int,fun(A,int),hAPP(fun(A,fun(int,int)),fun(int,fun(A,int)),combc(A,int,int),hAPP(fun(A,int),fun(A,fun(int,int)),hAPP(fun(A,fun(int,fun(int,int))),fun(fun(A,int),fun(A,fun(int,int))),combs(A,int,fun(int,int)),hAPP(fun(A,bool),fun(A,fun(int,fun(int,int))),hAPP(fun(bool,fun(int,fun(int,int))),fun(fun(A,bool),fun(A,fun(int,fun(int,int)))),combb(bool,fun(int,fun(int,int)),A),if(int)),hAPP(A,fun(A,bool),fequal(A),C))),D)),one_one(int))),B) = hAPP(A,int,D,C). [resolve(5582,a,5552,a)]. 49.74/49.63 Derived: hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),hAPP(fun(A,int),fun(A,int),hAPP(fun(A,fun(int,int)),fun(fun(A,int),fun(A,int)),combs(A,int,int),hAPP(fun(A,int),fun(A,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(A,int),fun(A,fun(int,int))),combb(int,fun(int,int),A),times_times(int)),B)),C)),D) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),B),D)),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),C),D)). [resolve(5582,a,5553,a)]. 49.74/49.63 Derived: -comm_monoid_add(int) | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(fun(nat,bool),int,hAPP(fun(nat,int),fun(fun(nat,bool),int),big_co1399186613setsum(nat,int),hAPP(fun(nat,int),fun(nat,int),hAPP(fun(int,int),fun(fun(nat,int),fun(nat,int)),combb(int,int,nat),B),semiring_1_of_nat(int))),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A)) = hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co1399186613setsum(int,int),B),A). [resolve(5582,a,5554,b)]. 49.74/49.63 Derived: hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | one_one(int) = hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),C),B). [resolve(5582,a,5555,a)]. 49.74/49.63 Derived: one_one(int) = hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),hAPP(int,fun(A,int),combk(int,A),one_one(int))),B). [resolve(5582,a,5558,a)]. 49.74/49.63 Derived: -ring_1(int) | hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),hAPP(fun(A,int),fun(A,int),hAPP(fun(int,int),fun(fun(A,int),fun(A,int)),combb(int,int,A),ring_1_of_int(int)),B)),C) = hAPP(int,int,ring_1_of_int(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),B),C)). [resolve(5582,a,5560,b)]. 49.74/49.63 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f205(A,int,C,B)),B)) | one_one(int) = hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),C),B). [resolve(5582,a,5561,a)]. 49.74/49.63 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | one_one(int) != hAPP(A,int,C,f205(A,int,C,B)) | one_one(int) = hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),C),B). [resolve(5582,a,5562,a)]. 49.74/49.63 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f238(A,int,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),D),B) = hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),D),C). [resolve(5582,a,5563,a)]. 49.74/49.64 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | one_one(int) != hAPP(A,int,D,f238(A,int,D,C,B)) | hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),D),B) = hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),D),C). [resolve(5582,a,5564,a)]. 49.74/49.64 Derived: big_co302546470etprod(A,int) = ti(fun(fun(A,int),fun(fun(A,bool),int)),big_co302546470etprod(A,int)). [resolve(5582,a,5565,a)]. 49.74/49.64 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),A,one_one(int)),one_one(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),A,f266(B,int,C,D,E,A)),f268(B,int,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f269(B,int,C,D,E,A)),E)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),A,hAPP(fun(B,bool),int,hAPP(fun(B,int),fun(fun(B,bool),int),big_co302546470etprod(B,int),C),E)),hAPP(fun(B,bool),int,hAPP(fun(B,int),fun(fun(B,bool),int),big_co302546470etprod(B,int),D),E))). [resolve(5582,a,5567,a)]. 49.74/49.64 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),A,one_one(int)),one_one(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),A,f266(B,int,C,D,E,A)),f268(B,int,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),A,hAPP(B,int,C,f269(B,int,C,D,E,A))),hAPP(B,int,D,f269(B,int,C,D,E,A)))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),A,hAPP(fun(B,bool),int,hAPP(fun(B,int),fun(fun(B,bool),int),big_co302546470etprod(B,int),C),E)),hAPP(fun(B,bool),int,hAPP(fun(B,int),fun(fun(B,bool),int),big_co302546470etprod(B,int),D),E))). [resolve(5582,a,5568,a)]. 49.74/49.64 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),A,one_one(int)),one_one(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),A,f265(B,int,C,D,E,A)),f267(B,int,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f269(B,int,C,D,E,A)),E)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),A,hAPP(fun(B,bool),int,hAPP(fun(B,int),fun(fun(B,bool),int),big_co302546470etprod(B,int),C),E)),hAPP(fun(B,bool),int,hAPP(fun(B,int),fun(fun(B,bool),int),big_co302546470etprod(B,int),D),E))). [resolve(5582,a,5569,a)]. 49.74/49.64 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),A,one_one(int)),one_one(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),A,f265(B,int,C,D,E,A)),f267(B,int,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),A,hAPP(B,int,C,f269(B,int,C,D,E,A))),hAPP(B,int,D,f269(B,int,C,D,E,A)))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),A,hAPP(fun(B,bool),int,hAPP(fun(B,int),fun(fun(B,bool),int),big_co302546470etprod(B,int),C),E)),hAPP(fun(B,bool),int,hAPP(fun(B,int),fun(fun(B,bool),int),big_co302546470etprod(B,int),D),E))). [resolve(5582,a,5570,a)]. 49.74/49.64 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),A,one_one(int)),one_one(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),A,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),f265(B,int,C,D,E,A)),f266(B,int,C,D,E,A))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),f267(B,int,C,D,E,A)),f268(B,int,C,D,E,A)))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f269(B,int,C,D,E,A)),E)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),A,hAPP(fun(B,bool),int,hAPP(fun(B,int),fun(fun(B,bool),int),big_co302546470etprod(B,int),C),E)),hAPP(fun(B,bool),int,hAPP(fun(B,int),fun(fun(B,bool),int),big_co302546470etprod(B,int),D),E))). [resolve(5582,a,5571,a)]. 49.74/49.64 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),A,one_one(int)),one_one(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),A,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),f265(B,int,C,D,E,A)),f266(B,int,C,D,E,A))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),f267(B,int,C,D,E,A)),f268(B,int,C,D,E,A)))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),A,hAPP(B,int,C,f269(B,int,C,D,E,A))),hAPP(B,int,D,f269(B,int,C,D,E,A)))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),A,hAPP(fun(B,bool),int,hAPP(fun(B,int),fun(fun(B,bool),int),big_co302546470etprod(B,int),C),E)),hAPP(fun(B,bool),int,hAPP(fun(B,int),fun(fun(B,bool),int),big_co302546470etprod(B,int),D),E))). [resolve(5582,a,5572,a)]. 49.74/49.64 Derived: hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),C) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),C)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),C)). [resolve(5582,a,5573,a)]. 49.74/49.64 Derived: -dvd(int) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),D),C)),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),D),B))). [resolve(5582,a,5576,a)]. 49.74/49.64 Derived: -comm_monoid_add(int) | hAPP(fun(nat,bool),int,hAPP(fun(nat,int),fun(fun(nat,bool),int),big_co302546470etprod(nat,int),A),B) = hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),hAPP(fun(int,nat),fun(int,int),hAPP(fun(nat,int),fun(fun(int,nat),fun(int,int)),combb(nat,int,int),A),nat_1)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B)). [resolve(5582,a,5577,a)]. 49.74/49.64 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),hAPP(int,fun(A,int),hAPP(fun(A,fun(int,int)),fun(int,fun(A,int)),combc(A,int,int),hAPP(fun(A,int),fun(A,fun(int,int)),hAPP(fun(A,fun(int,fun(int,int))),fun(fun(A,int),fun(A,fun(int,int))),combs(A,int,fun(int,int)),hAPP(fun(A,bool),fun(A,fun(int,fun(int,int))),hAPP(fun(bool,fun(int,fun(int,int))),fun(fun(A,bool),fun(A,fun(int,fun(int,int)))),combb(bool,fun(int,fun(int,int)),A),if(int)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),E)),B) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),E),hAPP(fun(A,bool),nat,finite_card(A),B)). [resolve(5582,a,5579,a)]. 49.74/49.64 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),hAPP(int,fun(A,int),hAPP(fun(A,fun(int,int)),fun(int,fun(A,int)),combc(A,int,int),hAPP(fun(A,int),fun(A,fun(int,int)),hAPP(fun(A,fun(int,fun(int,int))),fun(fun(A,int),fun(A,fun(int,int))),combs(A,int,fun(int,int)),hAPP(fun(A,bool),fun(A,fun(int,fun(int,int))),hAPP(fun(bool,fun(int,fun(int,int))),fun(fun(A,bool),fun(A,fun(int,fun(int,int)))),combb(bool,fun(int,fun(int,int)),A),if(int)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),E)),B) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(A,int,D,C)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),E),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(A,bool),nat,finite_card(A),B)),one_one(nat)))). [resolve(5582,a,5580,a)]. 49.74/49.64 5583 -comm_monoid_mult(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | -hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),D),C)) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),hAPP(A,fun(B,A),hAPP(fun(B,fun(A,A)),fun(A,fun(B,A)),combc(B,A,A),hAPP(fun(B,A),fun(B,fun(A,A)),hAPP(fun(B,fun(A,fun(A,A))),fun(fun(B,A),fun(B,fun(A,A))),combs(B,A,fun(A,A)),hAPP(fun(B,bool),fun(B,fun(A,fun(A,A))),hAPP(fun(bool,fun(A,fun(A,A))),fun(fun(B,bool),fun(B,fun(A,fun(A,A)))),combb(bool,fun(A,fun(A,A)),B),if(A)),hAPP(B,fun(B,bool),hAPP(fun(B,fun(B,bool)),fun(B,fun(B,bool)),combc(B,B,bool),fequal(B)),D))),E)),one_one(A))),C) = hAPP(B,A,E,D) # label(fact_3982_setprod__delta) # label(axiom). [clausify(2885)]. 49.74/49.65 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),hAPP(nat,fun(A,nat),hAPP(fun(A,fun(nat,nat)),fun(nat,fun(A,nat)),combc(A,nat,nat),hAPP(fun(A,nat),fun(A,fun(nat,nat)),hAPP(fun(A,fun(nat,fun(nat,nat))),fun(fun(A,nat),fun(A,fun(nat,nat))),combs(A,nat,fun(nat,nat)),hAPP(fun(A,bool),fun(A,fun(nat,fun(nat,nat))),hAPP(fun(bool,fun(nat,fun(nat,nat))),fun(fun(A,bool),fun(A,fun(nat,fun(nat,nat)))),combb(bool,fun(nat,fun(nat,nat)),A),if(nat)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),one_one(nat))),B) = hAPP(A,nat,D,C). [resolve(5583,a,5550,a)]. 49.74/49.65 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),hAPP(complex,fun(A,complex),hAPP(fun(A,fun(complex,complex)),fun(complex,fun(A,complex)),combc(A,complex,complex),hAPP(fun(A,complex),fun(A,fun(complex,complex)),hAPP(fun(A,fun(complex,fun(complex,complex))),fun(fun(A,complex),fun(A,fun(complex,complex))),combs(A,complex,fun(complex,complex)),hAPP(fun(A,bool),fun(A,fun(complex,fun(complex,complex))),hAPP(fun(bool,fun(complex,fun(complex,complex))),fun(fun(A,bool),fun(A,fun(complex,fun(complex,complex)))),combb(bool,fun(complex,fun(complex,complex)),A),if(complex)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),one_one(complex))),B) = hAPP(A,complex,D,C). [resolve(5583,a,5557,a)]. 49.74/49.65 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),hAPP(rat,fun(A,rat),hAPP(fun(A,fun(rat,rat)),fun(rat,fun(A,rat)),combc(A,rat,rat),hAPP(fun(A,rat),fun(A,fun(rat,rat)),hAPP(fun(A,fun(rat,fun(rat,rat))),fun(fun(A,rat),fun(A,fun(rat,rat))),combs(A,rat,fun(rat,rat)),hAPP(fun(A,bool),fun(A,fun(rat,fun(rat,rat))),hAPP(fun(bool,fun(rat,fun(rat,rat))),fun(fun(A,bool),fun(A,fun(rat,fun(rat,rat)))),combb(bool,fun(rat,fun(rat,rat)),A),if(rat)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),one_one(rat))),B) = hAPP(A,rat,D,C). [resolve(5583,a,5559,a)]. 49.74/49.65 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),hAPP(real,fun(A,real),hAPP(fun(A,fun(real,real)),fun(real,fun(A,real)),combc(A,real,real),hAPP(fun(A,real),fun(A,fun(real,real)),hAPP(fun(A,fun(real,fun(real,real))),fun(fun(A,real),fun(A,fun(real,real))),combs(A,real,fun(real,real)),hAPP(fun(A,bool),fun(A,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(A,bool),fun(A,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),A),if(real)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),one_one(real))),B) = hAPP(A,real,D,C). [resolve(5583,a,5575,a)]. 49.74/49.65 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),hAPP(code_code_numeral,fun(A,code_code_numeral),hAPP(fun(A,fun(code_code_numeral,code_code_numeral)),fun(code_code_numeral,fun(A,code_code_numeral)),combc(A,code_code_numeral,code_code_numeral),hAPP(fun(A,code_code_numeral),fun(A,fun(code_code_numeral,code_code_numeral)),hAPP(fun(A,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral))),fun(fun(A,code_code_numeral),fun(A,fun(code_code_numeral,code_code_numeral))),combs(A,code_code_numeral,fun(code_code_numeral,code_code_numeral)),hAPP(fun(A,bool),fun(A,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral))),hAPP(fun(bool,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral))),fun(fun(A,bool),fun(A,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)))),combb(bool,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)),A),if(code_code_numeral)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),one_one(code_code_numeral))),B) = hAPP(A,code_code_numeral,D,C). [resolve(5583,a,5578,a)]. 49.74/49.65 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),hAPP(int,fun(A,int),hAPP(fun(A,fun(int,int)),fun(int,fun(A,int)),combc(A,int,int),hAPP(fun(A,int),fun(A,fun(int,int)),hAPP(fun(A,fun(int,fun(int,int))),fun(fun(A,int),fun(A,fun(int,int))),combs(A,int,fun(int,int)),hAPP(fun(A,bool),fun(A,fun(int,fun(int,int))),hAPP(fun(bool,fun(int,fun(int,int))),fun(fun(A,bool),fun(A,fun(int,fun(int,int)))),combb(bool,fun(int,fun(int,int)),A),if(int)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),one_one(int))),B) = hAPP(A,int,D,C). [resolve(5583,a,5582,a)]. 49.74/49.65 5584 -comm_monoid_mult(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),D),C)) | one_one(A) = hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),hAPP(A,fun(B,A),hAPP(fun(B,fun(A,A)),fun(A,fun(B,A)),combc(B,A,A),hAPP(fun(B,A),fun(B,fun(A,A)),hAPP(fun(B,fun(A,fun(A,A))),fun(fun(B,A),fun(B,fun(A,A))),combs(B,A,fun(A,A)),hAPP(fun(B,bool),fun(B,fun(A,fun(A,A))),hAPP(fun(bool,fun(A,fun(A,A))),fun(fun(B,bool),fun(B,fun(A,fun(A,A)))),combb(bool,fun(A,fun(A,A)),B),if(A)),hAPP(B,fun(B,bool),hAPP(fun(B,fun(B,bool)),fun(B,fun(B,bool)),combc(B,B,bool),fequal(B)),D))),E)),one_one(A))),C) # label(fact_3982_setprod__delta) # label(axiom). [clausify(2885)]. 49.74/49.65 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | one_one(nat) = hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),hAPP(nat,fun(A,nat),hAPP(fun(A,fun(nat,nat)),fun(nat,fun(A,nat)),combc(A,nat,nat),hAPP(fun(A,nat),fun(A,fun(nat,nat)),hAPP(fun(A,fun(nat,fun(nat,nat))),fun(fun(A,nat),fun(A,fun(nat,nat))),combs(A,nat,fun(nat,nat)),hAPP(fun(A,bool),fun(A,fun(nat,fun(nat,nat))),hAPP(fun(bool,fun(nat,fun(nat,nat))),fun(fun(A,bool),fun(A,fun(nat,fun(nat,nat)))),combb(bool,fun(nat,fun(nat,nat)),A),if(nat)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),one_one(nat))),B). [resolve(5584,a,5550,a)]. 49.74/49.65 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | one_one(complex) = hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),hAPP(complex,fun(A,complex),hAPP(fun(A,fun(complex,complex)),fun(complex,fun(A,complex)),combc(A,complex,complex),hAPP(fun(A,complex),fun(A,fun(complex,complex)),hAPP(fun(A,fun(complex,fun(complex,complex))),fun(fun(A,complex),fun(A,fun(complex,complex))),combs(A,complex,fun(complex,complex)),hAPP(fun(A,bool),fun(A,fun(complex,fun(complex,complex))),hAPP(fun(bool,fun(complex,fun(complex,complex))),fun(fun(A,bool),fun(A,fun(complex,fun(complex,complex)))),combb(bool,fun(complex,fun(complex,complex)),A),if(complex)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),one_one(complex))),B). [resolve(5584,a,5557,a)]. 49.74/49.65 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | one_one(rat) = hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),hAPP(rat,fun(A,rat),hAPP(fun(A,fun(rat,rat)),fun(rat,fun(A,rat)),combc(A,rat,rat),hAPP(fun(A,rat),fun(A,fun(rat,rat)),hAPP(fun(A,fun(rat,fun(rat,rat))),fun(fun(A,rat),fun(A,fun(rat,rat))),combs(A,rat,fun(rat,rat)),hAPP(fun(A,bool),fun(A,fun(rat,fun(rat,rat))),hAPP(fun(bool,fun(rat,fun(rat,rat))),fun(fun(A,bool),fun(A,fun(rat,fun(rat,rat)))),combb(bool,fun(rat,fun(rat,rat)),A),if(rat)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),one_one(rat))),B). [resolve(5584,a,5559,a)]. 49.91/49.77 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | one_one(real) = hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),hAPP(real,fun(A,real),hAPP(fun(A,fun(real,real)),fun(real,fun(A,real)),combc(A,real,real),hAPP(fun(A,real),fun(A,fun(real,real)),hAPP(fun(A,fun(real,fun(real,real))),fun(fun(A,real),fun(A,fun(real,real))),combs(A,real,fun(real,real)),hAPP(fun(A,bool),fun(A,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(A,bool),fun(A,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),A),if(real)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),one_one(real))),B). [resolve(5584,a,5575,a)]. 49.91/49.77 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | one_one(code_code_numeral) = hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),hAPP(code_code_numeral,fun(A,code_code_numeral),hAPP(fun(A,fun(code_code_numeral,code_code_numeral)),fun(code_code_numeral,fun(A,code_code_numeral)),combc(A,code_code_numeral,code_code_numeral),hAPP(fun(A,code_code_numeral),fun(A,fun(code_code_numeral,code_code_numeral)),hAPP(fun(A,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral))),fun(fun(A,code_code_numeral),fun(A,fun(code_code_numeral,code_code_numeral))),combs(A,code_code_numeral,fun(code_code_numeral,code_code_numeral)),hAPP(fun(A,bool),fun(A,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral))),hAPP(fun(bool,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral))),fun(fun(A,bool),fun(A,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)))),combb(bool,fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)),A),if(code_code_numeral)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),one_one(code_code_numeral))),B). [resolve(5584,a,5578,a)]. 49.91/49.77 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | one_one(int) = hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),hAPP(int,fun(A,int),hAPP(fun(A,fun(int,int)),fun(int,fun(A,int)),combc(A,int,int),hAPP(fun(A,int),fun(A,fun(int,int)),hAPP(fun(A,fun(int,fun(int,int))),fun(fun(A,int),fun(A,fun(int,int))),combs(A,int,fun(int,int)),hAPP(fun(A,bool),fun(A,fun(int,fun(int,int))),hAPP(fun(bool,fun(int,fun(int,int))),fun(fun(A,bool),fun(A,fun(int,fun(int,int)))),combb(bool,fun(int,fun(int,int)),A),if(int)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),one_one(int))),B). [resolve(5584,a,5582,a)]. 49.91/49.77 5585 -comm_monoid_mult(A) | -comm_monoid_add(A) | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,B)) | hAPP(fun(nat,bool),A,hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co302546470etprod(nat,A),hAPP(fun(nat,int),fun(nat,A),hAPP(fun(int,A),fun(fun(nat,int),fun(nat,A)),combb(int,A,nat),C),semiring_1_of_nat(int))),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),B)) = hAPP(fun(int,bool),A,hAPP(fun(int,A),fun(fun(int,bool),A),big_co302546470etprod(int,A),C),B) # label(fact_3972_transfer__int__nat__sum__prod_I2_J) # label(axiom). [clausify(3120)]. 49.91/49.77 Derived: -comm_monoid_add(nat) | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co302546470etprod(nat,nat),hAPP(fun(nat,int),fun(nat,nat),hAPP(fun(int,nat),fun(fun(nat,int),fun(nat,nat)),combb(int,nat,nat),B),semiring_1_of_nat(int))),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A)) = hAPP(fun(int,bool),nat,hAPP(fun(int,nat),fun(fun(int,bool),nat),big_co302546470etprod(int,nat),B),A). [resolve(5585,a,5550,a)]. 49.91/49.77 Derived: -comm_monoid_add(complex) | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(fun(nat,bool),complex,hAPP(fun(nat,complex),fun(fun(nat,bool),complex),big_co302546470etprod(nat,complex),hAPP(fun(nat,int),fun(nat,complex),hAPP(fun(int,complex),fun(fun(nat,int),fun(nat,complex)),combb(int,complex,nat),B),semiring_1_of_nat(int))),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A)) = hAPP(fun(int,bool),complex,hAPP(fun(int,complex),fun(fun(int,bool),complex),big_co302546470etprod(int,complex),B),A). [resolve(5585,a,5557,a)]. 49.91/49.77 Derived: -comm_monoid_add(rat) | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(fun(nat,bool),rat,hAPP(fun(nat,rat),fun(fun(nat,bool),rat),big_co302546470etprod(nat,rat),hAPP(fun(nat,int),fun(nat,rat),hAPP(fun(int,rat),fun(fun(nat,int),fun(nat,rat)),combb(int,rat,nat),B),semiring_1_of_nat(int))),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A)) = hAPP(fun(int,bool),rat,hAPP(fun(int,rat),fun(fun(int,bool),rat),big_co302546470etprod(int,rat),B),A). [resolve(5585,a,5559,a)]. 49.91/49.77 Derived: -comm_monoid_add(real) | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co302546470etprod(nat,real),hAPP(fun(nat,int),fun(nat,real),hAPP(fun(int,real),fun(fun(nat,int),fun(nat,real)),combb(int,real,nat),B),semiring_1_of_nat(int))),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A)) = hAPP(fun(int,bool),real,hAPP(fun(int,real),fun(fun(int,bool),real),big_co302546470etprod(int,real),B),A). [resolve(5585,a,5575,a)]. 49.91/49.77 Derived: -comm_monoid_add(code_code_numeral) | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(fun(nat,bool),code_code_numeral,hAPP(fun(nat,code_code_numeral),fun(fun(nat,bool),code_code_numeral),big_co302546470etprod(nat,code_code_numeral),hAPP(fun(nat,int),fun(nat,code_code_numeral),hAPP(fun(int,code_code_numeral),fun(fun(nat,int),fun(nat,code_code_numeral)),combb(int,code_code_numeral,nat),B),semiring_1_of_nat(int))),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A)) = hAPP(fun(int,bool),code_code_numeral,hAPP(fun(int,code_code_numeral),fun(fun(int,bool),code_code_numeral),big_co302546470etprod(int,code_code_numeral),B),A). [resolve(5585,a,5578,a)]. 49.91/49.77 Derived: -comm_monoid_add(int) | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(fun(nat,bool),int,hAPP(fun(nat,int),fun(fun(nat,bool),int),big_co302546470etprod(nat,int),hAPP(fun(nat,int),fun(nat,int),hAPP(fun(int,int),fun(fun(nat,int),fun(nat,int)),combb(int,int,nat),B),semiring_1_of_nat(int))),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A)) = hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),B),A). [resolve(5585,a,5582,a)]. 49.91/49.77 5586 -comm_monoid_add(A) | -comm_monoid_mult(A) | hAPP(fun(nat,bool),A,hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),B),C) = hAPP(fun(int,bool),A,hAPP(fun(int,A),fun(fun(int,bool),A),big_co1399186613setsum(int,A),hAPP(fun(int,nat),fun(int,A),hAPP(fun(nat,A),fun(fun(int,nat),fun(int,A)),combb(nat,A,int),B),nat_1)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),C)) # label(fact_4291_transfer__nat__int__sum__prod_I1_J) # label(axiom). [clausify(3121)]. 49.93/49.81 Derived: -comm_monoid_add(nat) | hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),A),B) = hAPP(fun(int,bool),nat,hAPP(fun(int,nat),fun(fun(int,bool),nat),big_co1399186613setsum(int,nat),hAPP(fun(int,nat),fun(int,nat),hAPP(fun(nat,nat),fun(fun(int,nat),fun(int,nat)),combb(nat,nat,int),A),nat_1)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B)). [resolve(5586,b,5550,a)]. 49.93/49.81 Derived: -comm_monoid_add(complex) | hAPP(fun(nat,bool),complex,hAPP(fun(nat,complex),fun(fun(nat,bool),complex),big_co1399186613setsum(nat,complex),A),B) = hAPP(fun(int,bool),complex,hAPP(fun(int,complex),fun(fun(int,bool),complex),big_co1399186613setsum(int,complex),hAPP(fun(int,nat),fun(int,complex),hAPP(fun(nat,complex),fun(fun(int,nat),fun(int,complex)),combb(nat,complex,int),A),nat_1)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B)). [resolve(5586,b,5557,a)]. 49.93/49.81 Derived: -comm_monoid_add(rat) | hAPP(fun(nat,bool),rat,hAPP(fun(nat,rat),fun(fun(nat,bool),rat),big_co1399186613setsum(nat,rat),A),B) = hAPP(fun(int,bool),rat,hAPP(fun(int,rat),fun(fun(int,bool),rat),big_co1399186613setsum(int,rat),hAPP(fun(int,nat),fun(int,rat),hAPP(fun(nat,rat),fun(fun(int,nat),fun(int,rat)),combb(nat,rat,int),A),nat_1)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B)). [resolve(5586,b,5559,a)]. 49.93/49.81 Derived: -comm_monoid_add(real) | hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),A),B) = hAPP(fun(int,bool),real,hAPP(fun(int,real),fun(fun(int,bool),real),big_co1399186613setsum(int,real),hAPP(fun(int,nat),fun(int,real),hAPP(fun(nat,real),fun(fun(int,nat),fun(int,real)),combb(nat,real,int),A),nat_1)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B)). [resolve(5586,b,5575,a)]. 49.93/49.81 Derived: -comm_monoid_add(code_code_numeral) | hAPP(fun(nat,bool),code_code_numeral,hAPP(fun(nat,code_code_numeral),fun(fun(nat,bool),code_code_numeral),big_co1399186613setsum(nat,code_code_numeral),A),B) = hAPP(fun(int,bool),code_code_numeral,hAPP(fun(int,code_code_numeral),fun(fun(int,bool),code_code_numeral),big_co1399186613setsum(int,code_code_numeral),hAPP(fun(int,nat),fun(int,code_code_numeral),hAPP(fun(nat,code_code_numeral),fun(fun(int,nat),fun(int,code_code_numeral)),combb(nat,code_code_numeral,int),A),nat_1)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B)). [resolve(5586,b,5578,a)]. 49.93/49.81 Derived: -comm_monoid_add(int) | hAPP(fun(nat,bool),int,hAPP(fun(nat,int),fun(fun(nat,bool),int),big_co1399186613setsum(nat,int),A),B) = hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co1399186613setsum(int,int),hAPP(fun(int,nat),fun(int,int),hAPP(fun(nat,int),fun(fun(int,nat),fun(int,int)),combb(nat,int,int),A),nat_1)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B)). [resolve(5586,b,5582,a)]. 49.93/49.81 5587 comm_monoid_mult(quickcheck_code_int) # label(arity_Quickcheck__Narrowing_Ocode__int___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 49.93/49.81 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | one_one(quickcheck_code_int) = hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),hAPP(quickcheck_code_int,fun(A,quickcheck_code_int),hAPP(fun(A,fun(quickcheck_code_int,quickcheck_code_int)),fun(quickcheck_code_int,fun(A,quickcheck_code_int)),combc(A,quickcheck_code_int,quickcheck_code_int),hAPP(fun(A,quickcheck_code_int),fun(A,fun(quickcheck_code_int,quickcheck_code_int)),hAPP(fun(A,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int))),fun(fun(A,quickcheck_code_int),fun(A,fun(quickcheck_code_int,quickcheck_code_int))),combs(A,quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),hAPP(fun(A,bool),fun(A,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int))),hAPP(fun(bool,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int))),fun(fun(A,bool),fun(A,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)))),combb(bool,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),A),if(quickcheck_code_int)),hAPP(A,fun(A,bool),fequal(A),C))),D)),one_one(quickcheck_code_int))),B). [resolve(5587,a,5551,a)]. 49.93/49.81 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),hAPP(quickcheck_code_int,fun(A,quickcheck_code_int),hAPP(fun(A,fun(quickcheck_code_int,quickcheck_code_int)),fun(quickcheck_code_int,fun(A,quickcheck_code_int)),combc(A,quickcheck_code_int,quickcheck_code_int),hAPP(fun(A,quickcheck_code_int),fun(A,fun(quickcheck_code_int,quickcheck_code_int)),hAPP(fun(A,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int))),fun(fun(A,quickcheck_code_int),fun(A,fun(quickcheck_code_int,quickcheck_code_int))),combs(A,quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),hAPP(fun(A,bool),fun(A,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int))),hAPP(fun(bool,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int))),fun(fun(A,bool),fun(A,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)))),combb(bool,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),A),if(quickcheck_code_int)),hAPP(A,fun(A,bool),fequal(A),C))),D)),one_one(quickcheck_code_int))),B) = hAPP(A,quickcheck_code_int,D,C). [resolve(5587,a,5552,a)]. 49.93/49.81 Derived: hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),hAPP(fun(A,quickcheck_code_int),fun(A,quickcheck_code_int),hAPP(fun(A,fun(quickcheck_code_int,quickcheck_code_int)),fun(fun(A,quickcheck_code_int),fun(A,quickcheck_code_int)),combs(A,quickcheck_code_int,quickcheck_code_int),hAPP(fun(A,quickcheck_code_int),fun(A,fun(quickcheck_code_int,quickcheck_code_int)),hAPP(fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),fun(fun(A,quickcheck_code_int),fun(A,fun(quickcheck_code_int,quickcheck_code_int))),combb(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),A),times_times(quickcheck_code_int)),B)),C)),D) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),B),D)),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),C),D)). [resolve(5587,a,5553,a)]. 49.93/49.81 Derived: -comm_monoid_add(quickcheck_code_int) | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(fun(nat,bool),quickcheck_code_int,hAPP(fun(nat,quickcheck_code_int),fun(fun(nat,bool),quickcheck_code_int),big_co1399186613setsum(nat,quickcheck_code_int),hAPP(fun(nat,int),fun(nat,quickcheck_code_int),hAPP(fun(int,quickcheck_code_int),fun(fun(nat,int),fun(nat,quickcheck_code_int)),combb(int,quickcheck_code_int,nat),B),semiring_1_of_nat(int))),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A)) = hAPP(fun(int,bool),quickcheck_code_int,hAPP(fun(int,quickcheck_code_int),fun(fun(int,bool),quickcheck_code_int),big_co1399186613setsum(int,quickcheck_code_int),B),A). [resolve(5587,a,5554,b)]. 49.93/49.81 Derived: hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | one_one(quickcheck_code_int) = hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),C),B). [resolve(5587,a,5555,a)]. 49.96/49.82 Derived: one_one(quickcheck_code_int) = hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),hAPP(quickcheck_code_int,fun(A,quickcheck_code_int),combk(quickcheck_code_int,A),one_one(quickcheck_code_int))),B). [resolve(5587,a,5558,a)]. 49.96/49.82 Derived: -ring_1(quickcheck_code_int) | hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),hAPP(fun(A,int),fun(A,quickcheck_code_int),hAPP(fun(int,quickcheck_code_int),fun(fun(A,int),fun(A,quickcheck_code_int)),combb(int,quickcheck_code_int,A),ring_1_of_int(quickcheck_code_int)),B)),C) = hAPP(int,quickcheck_code_int,ring_1_of_int(quickcheck_code_int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),B),C)). [resolve(5587,a,5560,b)]. 49.96/49.82 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f205(A,quickcheck_code_int,C,B)),B)) | one_one(quickcheck_code_int) = hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),C),B). [resolve(5587,a,5561,a)]. 49.96/49.82 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | one_one(quickcheck_code_int) != hAPP(A,quickcheck_code_int,C,f205(A,quickcheck_code_int,C,B)) | one_one(quickcheck_code_int) = hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),C),B). [resolve(5587,a,5562,a)]. 49.96/49.82 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f238(A,quickcheck_code_int,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),D),B) = hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),D),C). [resolve(5587,a,5563,a)]. 49.96/49.82 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | one_one(quickcheck_code_int) != hAPP(A,quickcheck_code_int,D,f238(A,quickcheck_code_int,D,C,B)) | hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),D),B) = hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),D),C). [resolve(5587,a,5564,a)]. 49.96/49.82 Derived: big_co302546470etprod(A,quickcheck_code_int) = ti(fun(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int)),big_co302546470etprod(A,quickcheck_code_int)). [resolve(5587,a,5565,a)]. 49.96/49.82 Derived: ti(quickcheck_code_int,A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),one_one(quickcheck_code_int)),A). [resolve(5587,a,5566,a)]. 49.96/49.82 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),A,one_one(quickcheck_code_int)),one_one(quickcheck_code_int))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),A,f266(B,quickcheck_code_int,C,D,E,A)),f268(B,quickcheck_code_int,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f269(B,quickcheck_code_int,C,D,E,A)),E)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),A,hAPP(fun(B,bool),quickcheck_code_int,hAPP(fun(B,quickcheck_code_int),fun(fun(B,bool),quickcheck_code_int),big_co302546470etprod(B,quickcheck_code_int),C),E)),hAPP(fun(B,bool),quickcheck_code_int,hAPP(fun(B,quickcheck_code_int),fun(fun(B,bool),quickcheck_code_int),big_co302546470etprod(B,quickcheck_code_int),D),E))). [resolve(5587,a,5567,a)]. 49.96/49.82 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),A,one_one(quickcheck_code_int)),one_one(quickcheck_code_int))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),A,f266(B,quickcheck_code_int,C,D,E,A)),f268(B,quickcheck_code_int,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),A,hAPP(B,quickcheck_code_int,C,f269(B,quickcheck_code_int,C,D,E,A))),hAPP(B,quickcheck_code_int,D,f269(B,quickcheck_code_int,C,D,E,A)))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),A,hAPP(fun(B,bool),quickcheck_code_int,hAPP(fun(B,quickcheck_code_int),fun(fun(B,bool),quickcheck_code_int),big_co302546470etprod(B,quickcheck_code_int),C),E)),hAPP(fun(B,bool),quickcheck_code_int,hAPP(fun(B,quickcheck_code_int),fun(fun(B,bool),quickcheck_code_int),big_co302546470etprod(B,quickcheck_code_int),D),E))). [resolve(5587,a,5568,a)]. 49.96/49.82 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),A,one_one(quickcheck_code_int)),one_one(quickcheck_code_int))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),A,f265(B,quickcheck_code_int,C,D,E,A)),f267(B,quickcheck_code_int,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f269(B,quickcheck_code_int,C,D,E,A)),E)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),A,hAPP(fun(B,bool),quickcheck_code_int,hAPP(fun(B,quickcheck_code_int),fun(fun(B,bool),quickcheck_code_int),big_co302546470etprod(B,quickcheck_code_int),C),E)),hAPP(fun(B,bool),quickcheck_code_int,hAPP(fun(B,quickcheck_code_int),fun(fun(B,bool),quickcheck_code_int),big_co302546470etprod(B,quickcheck_code_int),D),E))). [resolve(5587,a,5569,a)]. 49.96/49.82 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),A,one_one(quickcheck_code_int)),one_one(quickcheck_code_int))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),A,f265(B,quickcheck_code_int,C,D,E,A)),f267(B,quickcheck_code_int,C,D,E,A))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),A,hAPP(B,quickcheck_code_int,C,f269(B,quickcheck_code_int,C,D,E,A))),hAPP(B,quickcheck_code_int,D,f269(B,quickcheck_code_int,C,D,E,A)))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),A,hAPP(fun(B,bool),quickcheck_code_int,hAPP(fun(B,quickcheck_code_int),fun(fun(B,bool),quickcheck_code_int),big_co302546470etprod(B,quickcheck_code_int),C),E)),hAPP(fun(B,bool),quickcheck_code_int,hAPP(fun(B,quickcheck_code_int),fun(fun(B,bool),quickcheck_code_int),big_co302546470etprod(B,quickcheck_code_int),D),E))). [resolve(5587,a,5570,a)]. 49.96/49.82 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),A,one_one(quickcheck_code_int)),one_one(quickcheck_code_int))) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),A,hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),f265(B,quickcheck_code_int,C,D,E,A)),f266(B,quickcheck_code_int,C,D,E,A))),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),f267(B,quickcheck_code_int,C,D,E,A)),f268(B,quickcheck_code_int,C,D,E,A)))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f269(B,quickcheck_code_int,C,D,E,A)),E)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),A,hAPP(fun(B,bool),quickcheck_code_int,hAPP(fun(B,quickcheck_code_int),fun(fun(B,bool),quickcheck_code_int),big_co302546470etprod(B,quickcheck_code_int),C),E)),hAPP(fun(B,bool),quickcheck_code_int,hAPP(fun(B,quickcheck_code_int),fun(fun(B,bool),quickcheck_code_int),big_co302546470etprod(B,quickcheck_code_int),D),E))). [resolve(5587,a,5571,a)]. 49.96/49.83 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),A,one_one(quickcheck_code_int)),one_one(quickcheck_code_int))) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),A,hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),f265(B,quickcheck_code_int,C,D,E,A)),f266(B,quickcheck_code_int,C,D,E,A))),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),f267(B,quickcheck_code_int,C,D,E,A)),f268(B,quickcheck_code_int,C,D,E,A)))) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),E)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),A,hAPP(B,quickcheck_code_int,C,f269(B,quickcheck_code_int,C,D,E,A))),hAPP(B,quickcheck_code_int,D,f269(B,quickcheck_code_int,C,D,E,A)))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),A,hAPP(fun(B,bool),quickcheck_code_int,hAPP(fun(B,quickcheck_code_int),fun(fun(B,bool),quickcheck_code_int),big_co302546470etprod(B,quickcheck_code_int),C),E)),hAPP(fun(B,bool),quickcheck_code_int,hAPP(fun(B,quickcheck_code_int),fun(fun(B,bool),quickcheck_code_int),big_co302546470etprod(B,quickcheck_code_int),D),E))). [resolve(5587,a,5572,a)]. 49.96/49.83 Derived: hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B)),C) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),C)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),B),C)). [resolve(5587,a,5573,a)]. 49.96/49.83 Derived: ti(quickcheck_code_int,A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),one_one(quickcheck_code_int)). [resolve(5587,a,5574,a)]. 49.96/49.83 Derived: -dvd(quickcheck_code_int) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),D),C)),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),D),B))). [resolve(5587,a,5576,a)]. 49.96/49.83 Derived: -comm_monoid_add(quickcheck_code_int) | hAPP(fun(nat,bool),quickcheck_code_int,hAPP(fun(nat,quickcheck_code_int),fun(fun(nat,bool),quickcheck_code_int),big_co302546470etprod(nat,quickcheck_code_int),A),B) = hAPP(fun(int,bool),quickcheck_code_int,hAPP(fun(int,quickcheck_code_int),fun(fun(int,bool),quickcheck_code_int),big_co302546470etprod(int,quickcheck_code_int),hAPP(fun(int,nat),fun(int,quickcheck_code_int),hAPP(fun(nat,quickcheck_code_int),fun(fun(int,nat),fun(int,quickcheck_code_int)),combb(nat,quickcheck_code_int,int),A),nat_1)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B)). [resolve(5587,a,5577,a)]. 49.97/49.83 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),hAPP(quickcheck_code_int,fun(A,quickcheck_code_int),hAPP(fun(A,fun(quickcheck_code_int,quickcheck_code_int)),fun(quickcheck_code_int,fun(A,quickcheck_code_int)),combc(A,quickcheck_code_int,quickcheck_code_int),hAPP(fun(A,quickcheck_code_int),fun(A,fun(quickcheck_code_int,quickcheck_code_int)),hAPP(fun(A,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int))),fun(fun(A,quickcheck_code_int),fun(A,fun(quickcheck_code_int,quickcheck_code_int))),combs(A,quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),hAPP(fun(A,bool),fun(A,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int))),hAPP(fun(bool,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int))),fun(fun(A,bool),fun(A,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)))),combb(bool,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),A),if(quickcheck_code_int)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),E)),B) = hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),E),hAPP(fun(A,bool),nat,finite_card(A),B)). [resolve(5587,a,5579,a)]. 49.97/49.83 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),hAPP(quickcheck_code_int,fun(A,quickcheck_code_int),hAPP(fun(A,fun(quickcheck_code_int,quickcheck_code_int)),fun(quickcheck_code_int,fun(A,quickcheck_code_int)),combc(A,quickcheck_code_int,quickcheck_code_int),hAPP(fun(A,quickcheck_code_int),fun(A,fun(quickcheck_code_int,quickcheck_code_int)),hAPP(fun(A,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int))),fun(fun(A,quickcheck_code_int),fun(A,fun(quickcheck_code_int,quickcheck_code_int))),combs(A,quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),hAPP(fun(A,bool),fun(A,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int))),hAPP(fun(bool,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int))),fun(fun(A,bool),fun(A,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)))),combb(bool,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),A),if(quickcheck_code_int)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),E)),B) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(A,quickcheck_code_int,D,C)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),E),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(A,bool),nat,finite_card(A),B)),one_one(nat)))). [resolve(5587,a,5580,a)]. 49.97/49.83 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),hAPP(quickcheck_code_int,fun(A,quickcheck_code_int),hAPP(fun(A,fun(quickcheck_code_int,quickcheck_code_int)),fun(quickcheck_code_int,fun(A,quickcheck_code_int)),combc(A,quickcheck_code_int,quickcheck_code_int),hAPP(fun(A,quickcheck_code_int),fun(A,fun(quickcheck_code_int,quickcheck_code_int)),hAPP(fun(A,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int))),fun(fun(A,quickcheck_code_int),fun(A,fun(quickcheck_code_int,quickcheck_code_int))),combs(A,quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),hAPP(fun(A,bool),fun(A,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int))),hAPP(fun(bool,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int))),fun(fun(A,bool),fun(A,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)))),combb(bool,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),A),if(quickcheck_code_int)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),one_one(quickcheck_code_int))),B) = hAPP(A,quickcheck_code_int,D,C). [resolve(5587,a,5583,a)]. 50.37/50.29 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | one_one(quickcheck_code_int) = hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),hAPP(quickcheck_code_int,fun(A,quickcheck_code_int),hAPP(fun(A,fun(quickcheck_code_int,quickcheck_code_int)),fun(quickcheck_code_int,fun(A,quickcheck_code_int)),combc(A,quickcheck_code_int,quickcheck_code_int),hAPP(fun(A,quickcheck_code_int),fun(A,fun(quickcheck_code_int,quickcheck_code_int)),hAPP(fun(A,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int))),fun(fun(A,quickcheck_code_int),fun(A,fun(quickcheck_code_int,quickcheck_code_int))),combs(A,quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),hAPP(fun(A,bool),fun(A,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int))),hAPP(fun(bool,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int))),fun(fun(A,bool),fun(A,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)))),combb(bool,fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),A),if(quickcheck_code_int)),hAPP(A,fun(A,bool),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),C))),D)),one_one(quickcheck_code_int))),B). [resolve(5587,a,5584,a)]. 50.37/50.29 Derived: -comm_monoid_add(quickcheck_code_int) | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(fun(nat,bool),quickcheck_code_int,hAPP(fun(nat,quickcheck_code_int),fun(fun(nat,bool),quickcheck_code_int),big_co302546470etprod(nat,quickcheck_code_int),hAPP(fun(nat,int),fun(nat,quickcheck_code_int),hAPP(fun(int,quickcheck_code_int),fun(fun(nat,int),fun(nat,quickcheck_code_int)),combb(int,quickcheck_code_int,nat),B),semiring_1_of_nat(int))),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A)) = hAPP(fun(int,bool),quickcheck_code_int,hAPP(fun(int,quickcheck_code_int),fun(fun(int,bool),quickcheck_code_int),big_co302546470etprod(int,quickcheck_code_int),B),A). [resolve(5587,a,5585,a)]. 50.37/50.29 Derived: -comm_monoid_add(quickcheck_code_int) | hAPP(fun(nat,bool),quickcheck_code_int,hAPP(fun(nat,quickcheck_code_int),fun(fun(nat,bool),quickcheck_code_int),big_co1399186613setsum(nat,quickcheck_code_int),A),B) = hAPP(fun(int,bool),quickcheck_code_int,hAPP(fun(int,quickcheck_code_int),fun(fun(int,bool),quickcheck_code_int),big_co1399186613setsum(int,quickcheck_code_int),hAPP(fun(int,nat),fun(int,quickcheck_code_int),hAPP(fun(nat,quickcheck_code_int),fun(fun(int,nat),fun(int,quickcheck_code_int)),combb(nat,quickcheck_code_int,int),A),nat_1)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B)). [resolve(5587,a,5586,b)]. 50.37/50.29 5588 -comm_monoid_mult(A) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f630(B,A,C,D)),D)) | one_one(A) = hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),C),D) # label(fact_4797_setprod__1_H) # label(axiom). [clausify(3730)]. 50.37/50.29 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f630(A,nat,B,C)),C)) | one_one(nat) = hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),B),C). [resolve(5588,a,5550,a)]. 50.44/50.34 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f630(A,complex,B,C)),C)) | one_one(complex) = hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),B),C). [resolve(5588,a,5557,a)]. 50.44/50.34 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f630(A,rat,B,C)),C)) | one_one(rat) = hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),B),C). [resolve(5588,a,5559,a)]. 50.44/50.34 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f630(A,real,B,C)),C)) | one_one(real) = hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),B),C). [resolve(5588,a,5575,a)]. 50.44/50.34 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f630(A,code_code_numeral,B,C)),C)) | one_one(code_code_numeral) = hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),B),C). [resolve(5588,a,5578,a)]. 50.44/50.34 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f630(A,int,B,C)),C)) | one_one(int) = hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),B),C). [resolve(5588,a,5582,a)]. 50.44/50.34 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f630(A,quickcheck_code_int,B,C)),C)) | one_one(quickcheck_code_int) = hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),B),C). [resolve(5588,a,5587,a)]. 50.44/50.34 5589 -comm_monoid_mult(A) | one_one(A) != hAPP(B,A,C,f630(B,A,C,D)) | one_one(A) = hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),C),D) # label(fact_4797_setprod__1_H) # label(axiom). [clausify(3730)]. 50.44/50.34 Derived: one_one(nat) != hAPP(A,nat,B,f630(A,nat,B,C)) | one_one(nat) = hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),B),C). [resolve(5589,a,5550,a)]. 50.44/50.34 Derived: one_one(complex) != hAPP(A,complex,B,f630(A,complex,B,C)) | one_one(complex) = hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),B),C). [resolve(5589,a,5557,a)]. 50.44/50.34 Derived: one_one(rat) != hAPP(A,rat,B,f630(A,rat,B,C)) | one_one(rat) = hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),B),C). [resolve(5589,a,5559,a)]. 50.44/50.34 Derived: one_one(real) != hAPP(A,real,B,f630(A,real,B,C)) | one_one(real) = hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),B),C). [resolve(5589,a,5575,a)]. 50.44/50.34 Derived: one_one(code_code_numeral) != hAPP(A,code_code_numeral,B,f630(A,code_code_numeral,B,C)) | one_one(code_code_numeral) = hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),B),C). [resolve(5589,a,5578,a)]. 50.44/50.34 Derived: one_one(int) != hAPP(A,int,B,f630(A,int,B,C)) | one_one(int) = hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),B),C). [resolve(5589,a,5582,a)]. 50.44/50.34 Derived: one_one(quickcheck_code_int) != hAPP(A,quickcheck_code_int,B,f630(A,quickcheck_code_int,B,C)) | one_one(quickcheck_code_int) = hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),B),C). [resolve(5589,a,5587,a)]. 50.44/50.34 5590 -comm_monoid_mult(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | -hBOOL(hAPP(fun(B,bool),bool,hAPP(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),D),C)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f648(B,A,E,D,C)),hAPP(fun(B,bool),fun(B,bool),hAPP(fun(B,bool),fun(fun(B,bool),fun(B,bool)),minus_minus(fun(B,bool)),C),D))) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),E),C) = hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),E),D) # label(fact_3987_setprod__mono__one__right) # label(axiom). [clausify(3802)]. 50.44/50.35 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f648(A,nat,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),D),B) = hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),D),C). [resolve(5590,a,5550,a)]. 50.44/50.35 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f648(A,complex,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),D),B) = hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),D),C). [resolve(5590,a,5557,a)]. 50.44/50.35 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f648(A,rat,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),D),B) = hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),D),C). [resolve(5590,a,5559,a)]. 50.44/50.35 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f648(A,real,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),D),B) = hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),D),C). [resolve(5590,a,5575,a)]. 50.44/50.35 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f648(A,code_code_numeral,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),D),B) = hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),D),C). [resolve(5590,a,5578,a)]. 50.44/50.35 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f648(A,int,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),D),B) = hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),D),C). [resolve(5590,a,5582,a)]. 50.44/50.35 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f648(A,quickcheck_code_int,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),D),B) = hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),D),C). [resolve(5590,a,5587,a)]. 50.44/50.36 5591 -comm_monoid_mult(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | -hBOOL(hAPP(fun(B,bool),bool,hAPP(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),D),C)) | one_one(A) != hAPP(B,A,E,f648(B,A,E,D,C)) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),E),C) = hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),E),D) # label(fact_3987_setprod__mono__one__right) # label(axiom). [clausify(3802)]. 50.44/50.36 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | one_one(nat) != hAPP(A,nat,D,f648(A,nat,D,C,B)) | hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),D),B) = hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),D),C). [resolve(5591,a,5550,a)]. 50.44/50.36 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | one_one(complex) != hAPP(A,complex,D,f648(A,complex,D,C,B)) | hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),D),B) = hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),D),C). [resolve(5591,a,5557,a)]. 50.44/50.36 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | one_one(rat) != hAPP(A,rat,D,f648(A,rat,D,C,B)) | hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),D),B) = hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),D),C). [resolve(5591,a,5559,a)]. 50.44/50.36 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | one_one(real) != hAPP(A,real,D,f648(A,real,D,C,B)) | hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),D),B) = hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),D),C). [resolve(5591,a,5575,a)]. 50.44/50.36 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | one_one(code_code_numeral) != hAPP(A,code_code_numeral,D,f648(A,code_code_numeral,D,C,B)) | hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),D),B) = hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),D),C). [resolve(5591,a,5578,a)]. 50.44/50.36 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | one_one(int) != hAPP(A,int,D,f648(A,int,D,C,B)) | hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),D),B) = hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),D),C). [resolve(5591,a,5582,a)]. 50.44/50.36 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | one_one(quickcheck_code_int) != hAPP(A,quickcheck_code_int,D,f648(A,quickcheck_code_int,D,C,B)) | hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),D),B) = hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),D),C). [resolve(5591,a,5587,a)]. 50.74/50.60 5592 -semiring_1(A) | -comm_monoid_mult(A) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),hAPP(fun(B,nat),fun(B,A),hAPP(fun(nat,A),fun(fun(B,nat),fun(B,A)),combb(nat,A,B),semiring_1_of_nat(A)),C)),D) = hAPP(nat,A,semiring_1_of_nat(A),hAPP(fun(B,bool),nat,hAPP(fun(B,nat),fun(fun(B,bool),nat),big_co302546470etprod(B,nat),C),D)) # label(fact_3955_of__nat__setprod) # label(axiom). [clausify(4059)]. 50.74/50.60 Derived: -semiring_1(nat) | hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),hAPP(fun(A,nat),fun(A,nat),hAPP(fun(nat,nat),fun(fun(A,nat),fun(A,nat)),combb(nat,nat,A),semiring_1_of_nat(nat)),B)),C) = hAPP(nat,nat,semiring_1_of_nat(nat),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),B),C)). [resolve(5592,b,5550,a)]. 50.74/50.60 Derived: -semiring_1(complex) | hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),hAPP(fun(A,nat),fun(A,complex),hAPP(fun(nat,complex),fun(fun(A,nat),fun(A,complex)),combb(nat,complex,A),semiring_1_of_nat(complex)),B)),C) = hAPP(nat,complex,semiring_1_of_nat(complex),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),B),C)). [resolve(5592,b,5557,a)]. 50.74/50.60 Derived: -semiring_1(rat) | hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),hAPP(fun(A,nat),fun(A,rat),hAPP(fun(nat,rat),fun(fun(A,nat),fun(A,rat)),combb(nat,rat,A),semiring_1_of_nat(rat)),B)),C) = hAPP(nat,rat,semiring_1_of_nat(rat),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),B),C)). [resolve(5592,b,5559,a)]. 50.74/50.60 Derived: -semiring_1(real) | hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),hAPP(fun(A,nat),fun(A,real),hAPP(fun(nat,real),fun(fun(A,nat),fun(A,real)),combb(nat,real,A),semiring_1_of_nat(real)),B)),C) = hAPP(nat,real,semiring_1_of_nat(real),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),B),C)). [resolve(5592,b,5575,a)]. 50.74/50.60 Derived: -semiring_1(code_code_numeral) | hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),hAPP(fun(A,nat),fun(A,code_code_numeral),hAPP(fun(nat,code_code_numeral),fun(fun(A,nat),fun(A,code_code_numeral)),combb(nat,code_code_numeral,A),semiring_1_of_nat(code_code_numeral)),B)),C) = hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),B),C)). [resolve(5592,b,5578,a)]. 50.74/50.60 Derived: -semiring_1(int) | hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),hAPP(fun(A,nat),fun(A,int),hAPP(fun(nat,int),fun(fun(A,nat),fun(A,int)),combb(nat,int,A),semiring_1_of_nat(int)),B)),C) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),B),C)). [resolve(5592,b,5582,a)]. 50.74/50.60 Derived: -semiring_1(quickcheck_code_int) | hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),hAPP(fun(A,nat),fun(A,quickcheck_code_int),hAPP(fun(nat,quickcheck_code_int),fun(fun(A,nat),fun(A,quickcheck_code_int)),combb(nat,quickcheck_code_int,A),semiring_1_of_nat(quickcheck_code_int)),B)),C) = hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),B),C)). [resolve(5592,b,5587,a)]. 50.74/50.60 5593 -comm_monoid_mult(A) | -comm_monoid_add(A) | B != C | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),f709(A,D,E,C,B))) | hAPP(fun(int,bool),A,hAPP(fun(int,A),fun(fun(int,bool),A),big_co302546470etprod(int,A),E),B) = hAPP(fun(int,bool),A,hAPP(fun(int,A),fun(fun(int,bool),A),big_co302546470etprod(int,A),D),C) # label(fact_3989_transfer__nat__int__sum__prod__cong_I2_J) # label(axiom). [clausify(4122)]. 50.74/50.61 Derived: -comm_monoid_add(nat) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),f709(nat,C,D,B,A))) | hAPP(fun(int,bool),nat,hAPP(fun(int,nat),fun(fun(int,bool),nat),big_co302546470etprod(int,nat),D),A) = hAPP(fun(int,bool),nat,hAPP(fun(int,nat),fun(fun(int,bool),nat),big_co302546470etprod(int,nat),C),B). [resolve(5593,a,5550,a)]. 50.74/50.61 Derived: -comm_monoid_add(complex) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),f709(complex,C,D,B,A))) | hAPP(fun(int,bool),complex,hAPP(fun(int,complex),fun(fun(int,bool),complex),big_co302546470etprod(int,complex),D),A) = hAPP(fun(int,bool),complex,hAPP(fun(int,complex),fun(fun(int,bool),complex),big_co302546470etprod(int,complex),C),B). [resolve(5593,a,5557,a)]. 50.74/50.61 Derived: -comm_monoid_add(rat) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),f709(rat,C,D,B,A))) | hAPP(fun(int,bool),rat,hAPP(fun(int,rat),fun(fun(int,bool),rat),big_co302546470etprod(int,rat),D),A) = hAPP(fun(int,bool),rat,hAPP(fun(int,rat),fun(fun(int,bool),rat),big_co302546470etprod(int,rat),C),B). [resolve(5593,a,5559,a)]. 50.74/50.61 Derived: -comm_monoid_add(real) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),f709(real,C,D,B,A))) | hAPP(fun(int,bool),real,hAPP(fun(int,real),fun(fun(int,bool),real),big_co302546470etprod(int,real),D),A) = hAPP(fun(int,bool),real,hAPP(fun(int,real),fun(fun(int,bool),real),big_co302546470etprod(int,real),C),B). [resolve(5593,a,5575,a)]. 50.74/50.61 Derived: -comm_monoid_add(code_code_numeral) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),f709(code_code_numeral,C,D,B,A))) | hAPP(fun(int,bool),code_code_numeral,hAPP(fun(int,code_code_numeral),fun(fun(int,bool),code_code_numeral),big_co302546470etprod(int,code_code_numeral),D),A) = hAPP(fun(int,bool),code_code_numeral,hAPP(fun(int,code_code_numeral),fun(fun(int,bool),code_code_numeral),big_co302546470etprod(int,code_code_numeral),C),B). [resolve(5593,a,5578,a)]. 50.74/50.61 Derived: -comm_monoid_add(int) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),f709(int,C,D,B,A))) | hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),D),A) = hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),C),B). [resolve(5593,a,5582,a)]. 50.74/50.61 Derived: -comm_monoid_add(quickcheck_code_int) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),f709(quickcheck_code_int,C,D,B,A))) | hAPP(fun(int,bool),quickcheck_code_int,hAPP(fun(int,quickcheck_code_int),fun(fun(int,bool),quickcheck_code_int),big_co302546470etprod(int,quickcheck_code_int),D),A) = hAPP(fun(int,bool),quickcheck_code_int,hAPP(fun(int,quickcheck_code_int),fun(fun(int,bool),quickcheck_code_int),big_co302546470etprod(int,quickcheck_code_int),C),B). [resolve(5593,a,5587,a)]. 50.74/50.61 5594 -comm_monoid_mult(A) | -comm_monoid_add(A) | B != C | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,B)) | hAPP(int,A,D,f709(A,E,D,C,B)) != hAPP(int,A,E,f709(A,E,D,C,B)) | hAPP(fun(int,bool),A,hAPP(fun(int,A),fun(fun(int,bool),A),big_co302546470etprod(int,A),D),B) = hAPP(fun(int,bool),A,hAPP(fun(int,A),fun(fun(int,bool),A),big_co302546470etprod(int,A),E),C) # label(fact_3989_transfer__nat__int__sum__prod__cong_I2_J) # label(axiom). [clausify(4122)]. 50.74/50.61 Derived: -comm_monoid_add(nat) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(int,nat,C,f709(nat,D,C,B,A)) != hAPP(int,nat,D,f709(nat,D,C,B,A)) | hAPP(fun(int,bool),nat,hAPP(fun(int,nat),fun(fun(int,bool),nat),big_co302546470etprod(int,nat),C),A) = hAPP(fun(int,bool),nat,hAPP(fun(int,nat),fun(fun(int,bool),nat),big_co302546470etprod(int,nat),D),B). [resolve(5594,a,5550,a)]. 50.82/50.70 Derived: -comm_monoid_add(complex) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(int,complex,C,f709(complex,D,C,B,A)) != hAPP(int,complex,D,f709(complex,D,C,B,A)) | hAPP(fun(int,bool),complex,hAPP(fun(int,complex),fun(fun(int,bool),complex),big_co302546470etprod(int,complex),C),A) = hAPP(fun(int,bool),complex,hAPP(fun(int,complex),fun(fun(int,bool),complex),big_co302546470etprod(int,complex),D),B). [resolve(5594,a,5557,a)]. 50.82/50.70 Derived: -comm_monoid_add(rat) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(int,rat,C,f709(rat,D,C,B,A)) != hAPP(int,rat,D,f709(rat,D,C,B,A)) | hAPP(fun(int,bool),rat,hAPP(fun(int,rat),fun(fun(int,bool),rat),big_co302546470etprod(int,rat),C),A) = hAPP(fun(int,bool),rat,hAPP(fun(int,rat),fun(fun(int,bool),rat),big_co302546470etprod(int,rat),D),B). [resolve(5594,a,5559,a)]. 50.82/50.70 Derived: -comm_monoid_add(real) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(int,real,C,f709(real,D,C,B,A)) != hAPP(int,real,D,f709(real,D,C,B,A)) | hAPP(fun(int,bool),real,hAPP(fun(int,real),fun(fun(int,bool),real),big_co302546470etprod(int,real),C),A) = hAPP(fun(int,bool),real,hAPP(fun(int,real),fun(fun(int,bool),real),big_co302546470etprod(int,real),D),B). [resolve(5594,a,5575,a)]. 50.82/50.70 Derived: -comm_monoid_add(code_code_numeral) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(int,code_code_numeral,C,f709(code_code_numeral,D,C,B,A)) != hAPP(int,code_code_numeral,D,f709(code_code_numeral,D,C,B,A)) | hAPP(fun(int,bool),code_code_numeral,hAPP(fun(int,code_code_numeral),fun(fun(int,bool),code_code_numeral),big_co302546470etprod(int,code_code_numeral),C),A) = hAPP(fun(int,bool),code_code_numeral,hAPP(fun(int,code_code_numeral),fun(fun(int,bool),code_code_numeral),big_co302546470etprod(int,code_code_numeral),D),B). [resolve(5594,a,5578,a)]. 50.82/50.70 Derived: -comm_monoid_add(int) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(int,int,C,f709(int,D,C,B,A)) != hAPP(int,int,D,f709(int,D,C,B,A)) | hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),C),A) = hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),D),B). [resolve(5594,a,5582,a)]. 50.82/50.70 Derived: -comm_monoid_add(quickcheck_code_int) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(int,quickcheck_code_int,C,f709(quickcheck_code_int,D,C,B,A)) != hAPP(int,quickcheck_code_int,D,f709(quickcheck_code_int,D,C,B,A)) | hAPP(fun(int,bool),quickcheck_code_int,hAPP(fun(int,quickcheck_code_int),fun(fun(int,bool),quickcheck_code_int),big_co302546470etprod(int,quickcheck_code_int),C),A) = hAPP(fun(int,bool),quickcheck_code_int,hAPP(fun(int,quickcheck_code_int),fun(fun(int,bool),quickcheck_code_int),big_co302546470etprod(int,quickcheck_code_int),D),B). [resolve(5594,a,5587,a)]. 50.82/50.70 5595 -comm_monoid_mult(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f724(B,A,D,E,C)),C)) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),D),C) = hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),E),C) # label(fact_4794_setprod_Ocong) # label(axiom). [clausify(4224)]. 50.82/50.70 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f724(A,nat,C,D,B)),B)) | hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),C),B) = hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),D),B). [resolve(5595,a,5550,a)]. 50.82/50.70 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f724(A,complex,C,D,B)),B)) | hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),C),B) = hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),D),B). [resolve(5595,a,5557,a)]. 50.82/50.70 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f724(A,rat,C,D,B)),B)) | hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),C),B) = hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),D),B). [resolve(5595,a,5559,a)]. 50.82/50.70 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f724(A,real,C,D,B)),B)) | hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),C),B) = hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),D),B). [resolve(5595,a,5575,a)]. 50.82/50.70 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f724(A,code_code_numeral,C,D,B)),B)) | hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),C),B) = hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),D),B). [resolve(5595,a,5578,a)]. 50.82/50.70 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f724(A,int,C,D,B)),B)) | hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),C),B) = hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),D),B). [resolve(5595,a,5582,a)]. 50.82/50.70 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f724(A,quickcheck_code_int,C,D,B)),B)) | hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),C),B) = hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),D),B). [resolve(5595,a,5587,a)]. 50.82/50.70 5596 -comm_monoid_mult(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | hAPP(B,A,D,f724(B,A,D,E,C)) != hAPP(B,A,E,f724(B,A,D,E,C)) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),D),C) = hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),E),C) # label(fact_4794_setprod_Ocong) # label(axiom). [clausify(4224)]. 50.82/50.70 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hAPP(A,nat,C,f724(A,nat,C,D,B)) != hAPP(A,nat,D,f724(A,nat,C,D,B)) | hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),C),B) = hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),D),B). [resolve(5596,a,5550,a)]. 50.82/50.70 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hAPP(A,complex,C,f724(A,complex,C,D,B)) != hAPP(A,complex,D,f724(A,complex,C,D,B)) | hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),C),B) = hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),D),B). [resolve(5596,a,5557,a)]. 50.82/50.70 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hAPP(A,rat,C,f724(A,rat,C,D,B)) != hAPP(A,rat,D,f724(A,rat,C,D,B)) | hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),C),B) = hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),D),B). [resolve(5596,a,5559,a)]. 50.82/50.70 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hAPP(A,real,C,f724(A,real,C,D,B)) != hAPP(A,real,D,f724(A,real,C,D,B)) | hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),C),B) = hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),D),B). [resolve(5596,a,5575,a)]. 51.03/50.95 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hAPP(A,code_code_numeral,C,f724(A,code_code_numeral,C,D,B)) != hAPP(A,code_code_numeral,D,f724(A,code_code_numeral,C,D,B)) | hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),C),B) = hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),D),B). [resolve(5596,a,5578,a)]. 51.03/50.95 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hAPP(A,int,C,f724(A,int,C,D,B)) != hAPP(A,int,D,f724(A,int,C,D,B)) | hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),C),B) = hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),D),B). [resolve(5596,a,5582,a)]. 51.03/50.95 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hAPP(A,quickcheck_code_int,C,f724(A,quickcheck_code_int,C,D,B)) != hAPP(A,quickcheck_code_int,D,f724(A,quickcheck_code_int,C,D,B)) | hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),C),B) = hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),D),B). [resolve(5596,a,5587,a)]. 51.03/50.95 5597 -comm_monoid_mult(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),hAPP(A,fun(B,A),combk(A,B),D)),C) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),D),hAPP(fun(B,bool),nat,finite_card(B),C)) # label(fact_3986_setprod__constant) # label(axiom). [clausify(4564)]. 51.03/50.95 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),hAPP(nat,fun(A,nat),combk(nat,A),C)),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),C),hAPP(fun(A,bool),nat,finite_card(A),B)). [resolve(5597,a,5550,a)]. 51.03/50.95 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),hAPP(complex,fun(A,complex),combk(complex,A),C)),B) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),C),hAPP(fun(A,bool),nat,finite_card(A),B)). [resolve(5597,a,5557,a)]. 51.03/50.95 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),hAPP(rat,fun(A,rat),combk(rat,A),C)),B) = hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),C),hAPP(fun(A,bool),nat,finite_card(A),B)). [resolve(5597,a,5559,a)]. 51.03/50.95 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),hAPP(real,fun(A,real),combk(real,A),C)),B) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),C),hAPP(fun(A,bool),nat,finite_card(A),B)). [resolve(5597,a,5575,a)]. 51.03/50.95 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),hAPP(code_code_numeral,fun(A,code_code_numeral),combk(code_code_numeral,A),C)),B) = hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),C),hAPP(fun(A,bool),nat,finite_card(A),B)). [resolve(5597,a,5578,a)]. 51.03/50.95 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),hAPP(int,fun(A,int),combk(int,A),C)),B) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),C),hAPP(fun(A,bool),nat,finite_card(A),B)). [resolve(5597,a,5582,a)]. 51.03/50.95 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),hAPP(quickcheck_code_int,fun(A,quickcheck_code_int),combk(quickcheck_code_int,A),C)),B) = hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),C),hAPP(fun(A,bool),nat,finite_card(A),B)). [resolve(5597,a,5587,a)]. 51.34/51.24 5598 -dvd(A) | -comm_monoid_mult(A) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f831(B,A,C,D,E)),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),C),E)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),D),E))) # label(fact_4452_setprod__dvd__setprod) # label(axiom). [clausify(4902)]. 51.34/51.24 Derived: -dvd(nat) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f831(A,nat,B,C,D)),D)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),B),D)),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),C),D))). [resolve(5598,b,5550,a)]. 51.34/51.24 Derived: -dvd(complex) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f831(A,complex,B,C,D)),D)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),B),D)),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),C),D))). [resolve(5598,b,5557,a)]. 51.34/51.24 Derived: -dvd(rat) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f831(A,rat,B,C,D)),D)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),B),D)),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),C),D))). [resolve(5598,b,5559,a)]. 51.34/51.24 Derived: -dvd(real) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f831(A,real,B,C,D)),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),B),D)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),C),D))). [resolve(5598,b,5575,a)]. 51.34/51.24 Derived: -dvd(code_code_numeral) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f831(A,code_code_numeral,B,C,D)),D)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),B),D)),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),C),D))). [resolve(5598,b,5578,a)]. 51.34/51.24 Derived: -dvd(int) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f831(A,int,B,C,D)),D)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),B),D)),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),C),D))). [resolve(5598,b,5582,a)]. 51.34/51.24 Derived: -dvd(quickcheck_code_int) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f831(A,quickcheck_code_int,B,C,D)),D)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),B),D)),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),C),D))). [resolve(5598,b,5587,a)]. 51.34/51.24 5599 -dvd(A) | -comm_monoid_mult(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(B,A,C,f831(B,A,C,D,E))),hAPP(B,A,D,f831(B,A,C,D,E)))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),C),E)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),D),E))) # label(fact_4452_setprod__dvd__setprod) # label(axiom). [clausify(4902)]. 51.34/51.24 Derived: -dvd(nat) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(A,nat,B,f831(A,nat,B,C,D))),hAPP(A,nat,C,f831(A,nat,B,C,D)))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),B),D)),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),C),D))). [resolve(5599,b,5550,a)]. 51.34/51.24 Derived: -dvd(complex) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),hAPP(A,complex,B,f831(A,complex,B,C,D))),hAPP(A,complex,C,f831(A,complex,B,C,D)))) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),B),D)),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),C),D))). [resolve(5599,b,5557,a)]. 51.34/51.24 Derived: -dvd(rat) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),hAPP(A,rat,B,f831(A,rat,B,C,D))),hAPP(A,rat,C,f831(A,rat,B,C,D)))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),B),D)),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),C),D))). [resolve(5599,b,5559,a)]. 51.34/51.24 Derived: -dvd(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(A,real,B,f831(A,real,B,C,D))),hAPP(A,real,C,f831(A,real,B,C,D)))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),B),D)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),C),D))). [resolve(5599,b,5575,a)]. 51.34/51.24 Derived: -dvd(code_code_numeral) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),hAPP(A,code_code_numeral,B,f831(A,code_code_numeral,B,C,D))),hAPP(A,code_code_numeral,C,f831(A,code_code_numeral,B,C,D)))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),B),D)),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),C),D))). [resolve(5599,b,5578,a)]. 51.34/51.24 Derived: -dvd(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(A,int,B,f831(A,int,B,C,D))),hAPP(A,int,C,f831(A,int,B,C,D)))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),B),D)),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),C),D))). [resolve(5599,b,5582,a)]. 51.34/51.24 Derived: -dvd(quickcheck_code_int) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),hAPP(A,quickcheck_code_int,B,f831(A,quickcheck_code_int,B,C,D))),hAPP(A,quickcheck_code_int,C,f831(A,quickcheck_code_int,B,C,D)))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),B),D)),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),C),D))). [resolve(5599,b,5587,a)]. 51.34/51.24 5600 -comm_monoid_mult(A) | -comm_monoid_add(A) | B != C | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),f841(A,D,E,C,B))) | hAPP(fun(int,bool),A,hAPP(fun(int,A),fun(fun(int,bool),A),big_co1399186613setsum(int,A),E),B) = hAPP(fun(int,bool),A,hAPP(fun(int,A),fun(fun(int,bool),A),big_co1399186613setsum(int,A),D),C) # label(fact_4324_transfer__nat__int__sum__prod__cong_I1_J) # label(axiom). [clausify(4908)]. 51.34/51.25 Derived: -comm_monoid_add(nat) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),f841(nat,C,D,B,A))) | hAPP(fun(int,bool),nat,hAPP(fun(int,nat),fun(fun(int,bool),nat),big_co1399186613setsum(int,nat),D),A) = hAPP(fun(int,bool),nat,hAPP(fun(int,nat),fun(fun(int,bool),nat),big_co1399186613setsum(int,nat),C),B). [resolve(5600,a,5550,a)]. 51.34/51.25 Derived: -comm_monoid_add(complex) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),f841(complex,C,D,B,A))) | hAPP(fun(int,bool),complex,hAPP(fun(int,complex),fun(fun(int,bool),complex),big_co1399186613setsum(int,complex),D),A) = hAPP(fun(int,bool),complex,hAPP(fun(int,complex),fun(fun(int,bool),complex),big_co1399186613setsum(int,complex),C),B). [resolve(5600,a,5557,a)]. 51.34/51.25 Derived: -comm_monoid_add(rat) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),f841(rat,C,D,B,A))) | hAPP(fun(int,bool),rat,hAPP(fun(int,rat),fun(fun(int,bool),rat),big_co1399186613setsum(int,rat),D),A) = hAPP(fun(int,bool),rat,hAPP(fun(int,rat),fun(fun(int,bool),rat),big_co1399186613setsum(int,rat),C),B). [resolve(5600,a,5559,a)]. 51.34/51.25 Derived: -comm_monoid_add(real) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),f841(real,C,D,B,A))) | hAPP(fun(int,bool),real,hAPP(fun(int,real),fun(fun(int,bool),real),big_co1399186613setsum(int,real),D),A) = hAPP(fun(int,bool),real,hAPP(fun(int,real),fun(fun(int,bool),real),big_co1399186613setsum(int,real),C),B). [resolve(5600,a,5575,a)]. 51.34/51.25 Derived: -comm_monoid_add(code_code_numeral) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),f841(code_code_numeral,C,D,B,A))) | hAPP(fun(int,bool),code_code_numeral,hAPP(fun(int,code_code_numeral),fun(fun(int,bool),code_code_numeral),big_co1399186613setsum(int,code_code_numeral),D),A) = hAPP(fun(int,bool),code_code_numeral,hAPP(fun(int,code_code_numeral),fun(fun(int,bool),code_code_numeral),big_co1399186613setsum(int,code_code_numeral),C),B). [resolve(5600,a,5578,a)]. 51.34/51.25 Derived: -comm_monoid_add(int) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),f841(int,C,D,B,A))) | hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co1399186613setsum(int,int),D),A) = hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co1399186613setsum(int,int),C),B). [resolve(5600,a,5582,a)]. 51.34/51.25 Derived: -comm_monoid_add(quickcheck_code_int) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),f841(quickcheck_code_int,C,D,B,A))) | hAPP(fun(int,bool),quickcheck_code_int,hAPP(fun(int,quickcheck_code_int),fun(fun(int,bool),quickcheck_code_int),big_co1399186613setsum(int,quickcheck_code_int),D),A) = hAPP(fun(int,bool),quickcheck_code_int,hAPP(fun(int,quickcheck_code_int),fun(fun(int,bool),quickcheck_code_int),big_co1399186613setsum(int,quickcheck_code_int),C),B). [resolve(5600,a,5587,a)]. 51.34/51.25 5601 -comm_monoid_mult(A) | -comm_monoid_add(A) | B != C | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,B)) | hAPP(int,A,D,f841(A,E,D,C,B)) != hAPP(int,A,E,f841(A,E,D,C,B)) | hAPP(fun(int,bool),A,hAPP(fun(int,A),fun(fun(int,bool),A),big_co1399186613setsum(int,A),D),B) = hAPP(fun(int,bool),A,hAPP(fun(int,A),fun(fun(int,bool),A),big_co1399186613setsum(int,A),E),C) # label(fact_4324_transfer__nat__int__sum__prod__cong_I1_J) # label(axiom). [clausify(4908)]. 53.06/52.91 Derived: -comm_monoid_add(nat) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(int,nat,C,f841(nat,D,C,B,A)) != hAPP(int,nat,D,f841(nat,D,C,B,A)) | hAPP(fun(int,bool),nat,hAPP(fun(int,nat),fun(fun(int,bool),nat),big_co1399186613setsum(int,nat),C),A) = hAPP(fun(int,bool),nat,hAPP(fun(int,nat),fun(fun(int,bool),nat),big_co1399186613setsum(int,nat),D),B). [resolve(5601,a,5550,a)]. 53.06/52.91 Derived: -comm_monoid_add(complex) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(int,complex,C,f841(complex,D,C,B,A)) != hAPP(int,complex,D,f841(complex,D,C,B,A)) | hAPP(fun(int,bool),complex,hAPP(fun(int,complex),fun(fun(int,bool),complex),big_co1399186613setsum(int,complex),C),A) = hAPP(fun(int,bool),complex,hAPP(fun(int,complex),fun(fun(int,bool),complex),big_co1399186613setsum(int,complex),D),B). [resolve(5601,a,5557,a)]. 53.06/52.91 Derived: -comm_monoid_add(rat) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(int,rat,C,f841(rat,D,C,B,A)) != hAPP(int,rat,D,f841(rat,D,C,B,A)) | hAPP(fun(int,bool),rat,hAPP(fun(int,rat),fun(fun(int,bool),rat),big_co1399186613setsum(int,rat),C),A) = hAPP(fun(int,bool),rat,hAPP(fun(int,rat),fun(fun(int,bool),rat),big_co1399186613setsum(int,rat),D),B). [resolve(5601,a,5559,a)]. 53.06/52.91 Derived: -comm_monoid_add(real) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(int,real,C,f841(real,D,C,B,A)) != hAPP(int,real,D,f841(real,D,C,B,A)) | hAPP(fun(int,bool),real,hAPP(fun(int,real),fun(fun(int,bool),real),big_co1399186613setsum(int,real),C),A) = hAPP(fun(int,bool),real,hAPP(fun(int,real),fun(fun(int,bool),real),big_co1399186613setsum(int,real),D),B). [resolve(5601,a,5575,a)]. 53.06/52.91 Derived: -comm_monoid_add(code_code_numeral) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(int,code_code_numeral,C,f841(code_code_numeral,D,C,B,A)) != hAPP(int,code_code_numeral,D,f841(code_code_numeral,D,C,B,A)) | hAPP(fun(int,bool),code_code_numeral,hAPP(fun(int,code_code_numeral),fun(fun(int,bool),code_code_numeral),big_co1399186613setsum(int,code_code_numeral),C),A) = hAPP(fun(int,bool),code_code_numeral,hAPP(fun(int,code_code_numeral),fun(fun(int,bool),code_code_numeral),big_co1399186613setsum(int,code_code_numeral),D),B). [resolve(5601,a,5578,a)]. 53.06/52.91 Derived: -comm_monoid_add(int) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(int,int,C,f841(int,D,C,B,A)) != hAPP(int,int,D,f841(int,D,C,B,A)) | hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co1399186613setsum(int,int),C),A) = hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co1399186613setsum(int,int),D),B). [resolve(5601,a,5582,a)]. 53.06/52.91 Derived: -comm_monoid_add(quickcheck_code_int) | A != B | -hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A)) | hAPP(int,quickcheck_code_int,C,f841(quickcheck_code_int,D,C,B,A)) != hAPP(int,quickcheck_code_int,D,f841(quickcheck_code_int,D,C,B,A)) | hAPP(fun(int,bool),quickcheck_code_int,hAPP(fun(int,quickcheck_code_int),fun(fun(int,bool),quickcheck_code_int),big_co1399186613setsum(int,quickcheck_code_int),C),A) = hAPP(fun(int,bool),quickcheck_code_int,hAPP(fun(int,quickcheck_code_int),fun(fun(int,bool),quickcheck_code_int),big_co1399186613setsum(int,quickcheck_code_int),D),B). [resolve(5601,a,5587,a)]. 53.06/52.91 5602 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C))) # label(fact_438_split__mult__pos__le) # label(axiom). [clausify(867)]. 53.06/52.91 5603 ordered_ring(rat) # label(arity_Rat_Orat___Rings_Oordered__ring) # label(axiom). [assumption]. 53.06/52.91 5604 ordered_ring(real) # label(arity_RealDef_Oreal___Rings_Oordered__ring) # label(axiom). [assumption]. 53.06/52.91 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B))). [resolve(5602,a,5603,a)]. 53.08/52.93 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B))). [resolve(5602,a,5604,a)]. 53.08/52.93 5605 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C))) # label(fact_438_split__mult__pos__le) # label(axiom). [clausify(867)]. 53.08/52.93 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),zero_zero(rat))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B))). [resolve(5605,a,5603,a)]. 53.08/52.93 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B))). [resolve(5605,a,5604,a)]. 53.08/52.93 5606 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),C),D)),E)),F))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),E)),B)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),E)),F))) # label(fact_774_less__add__iff2) # label(axiom). [clausify(909)]. 53.08/52.93 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),B),C)),D)),E))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),D)),A)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),D)),E))). [resolve(5606,a,5603,a)]. 53.08/52.93 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),C)),D)),E))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),D)),A)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),D)),E))). [resolve(5606,a,5604,a)]. 53.08/52.93 5607 -ordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),C),D)),E)),F))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),E)),B)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),E)),F))) # label(fact_774_less__add__iff2) # label(axiom). [clausify(909)]. 53.08/52.93 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),B),C)),D)),E))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),D)),A)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),D)),E))). [resolve(5607,a,5603,a)]. 53.08/52.97 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),C)),D)),E))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),D)),A)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),D)),E))). [resolve(5607,a,5604,a)]. 53.08/52.97 5608 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),C),D)),E)),F))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),E)),B)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),E)),F))) # label(fact_772_le__add__iff2) # label(axiom). [clausify(1065)]. 53.08/52.97 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),B),C)),D)),E))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),D)),A)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),D)),E))). [resolve(5608,a,5603,a)]. 53.08/52.97 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),C)),D)),E))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),D)),A)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),D)),E))). [resolve(5608,a,5604,a)]. 53.08/52.97 5609 -ordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),C),D)),E)),F))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),E)),B)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),E)),F))) # label(fact_772_le__add__iff2) # label(axiom). [clausify(1065)]. 53.08/52.97 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),B),C)),D)),E))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),D)),A)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),D)),E))). [resolve(5609,a,5603,a)]. 53.08/52.97 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),C)),D)),E))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),D)),A)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),D)),E))). [resolve(5609,a,5604,a)]. 53.46/53.34 5610 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),D)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),E),C)),F))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),E)),C)),D)),F)) # label(fact_773_le__add__iff1) # label(axiom). [clausify(1711)]. 53.46/53.34 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),C)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),D),B)),E))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),D)),B)),C)),E)). [resolve(5610,a,5603,a)]. 53.46/53.34 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),C)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),D),B)),E))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),D)),B)),C)),E)). [resolve(5610,a,5604,a)]. 53.46/53.34 5611 -ordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),D)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),E),C)),F))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),E)),C)),D)),F)) # label(fact_773_le__add__iff1) # label(axiom). [clausify(1711)]. 53.46/53.34 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),C)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),D),B)),E))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),D)),B)),C)),E)). [resolve(5611,a,5603,a)]. 53.46/53.34 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),C)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),D),B)),E))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),D)),B)),C)),E)). [resolve(5611,a,5604,a)]. 53.46/53.34 5612 ordered_ring(int) # label(arity_Int_Oint___Rings_Oordered__ring) # label(axiom). [assumption]. 53.46/53.34 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B))). [resolve(5612,a,5602,a)]. 53.46/53.34 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B))). [resolve(5612,a,5605,a)]. 53.66/53.53 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),C)),D)),E))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),D)),A)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),D)),E))). [resolve(5612,a,5606,a)]. 53.66/53.53 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),C)),D)),E))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),D)),A)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),D)),E))). [resolve(5612,a,5607,a)]. 53.66/53.53 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),C)),D)),E))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),D)),A)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),D)),E))). [resolve(5612,a,5608,a)]. 53.66/53.53 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),C)),D)),E))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),D)),A)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),D)),E))). [resolve(5612,a,5609,a)]. 53.66/53.53 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),C)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),D),B)),E))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),D)),B)),C)),E)). [resolve(5612,a,5610,a)]. 53.66/53.53 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),C)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),D),B)),E))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),D)),B)),C)),E)). [resolve(5612,a,5611,a)]. 53.66/53.53 5613 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),B))) # label(fact_435_mult__left__mono__neg) # label(axiom). [clausify(2784)]. 53.66/53.53 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),C),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),A))). [resolve(5613,a,5603,a)]. 54.53/54.40 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),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),B)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),A))). [resolve(5613,a,5604,a)]. 54.53/54.40 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),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),B)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),A))). [resolve(5613,a,5612,a)]. 54.53/54.40 5614 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C))) # label(fact_430_mult__nonpos__nonpos) # label(axiom). [clausify(2795)]. 54.53/54.40 5615 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D))) # label(fact_434_mult__right__mono__neg) # label(axiom). [clausify(3346)]. 54.53/54.40 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),C),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),C)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),C))). [resolve(5615,a,5603,a)]. 54.53/54.40 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),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),C)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),C))). [resolve(5615,a,5604,a)]. 54.53/54.40 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),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),C))). [resolve(5615,a,5612,a)]. 54.53/54.40 5616 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C)),D)),E)),F)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)),E)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D)),F))) # label(fact_775_less__add__iff1) # label(axiom). [clausify(4025)]. 54.53/54.40 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B)),C)),D)),E)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),C)),D)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),C)),E))). [resolve(5616,a,5603,a)]. 54.53/54.40 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B)),C)),D)),E)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),C)),D)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),C)),E))). [resolve(5616,a,5604,a)]. 56.86/56.74 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B)),C)),D)),E)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),C)),D)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C)),E))). [resolve(5616,a,5612,a)]. 56.86/56.74 5617 -ordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C)),D)),E)),F)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)),E)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D)),F))) # label(fact_775_less__add__iff1) # label(axiom). [clausify(4025)]. 56.86/56.74 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B)),C)),D)),E)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),C)),D)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),C)),E))). [resolve(5617,a,5603,a)]. 56.86/56.74 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B)),C)),D)),E)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),C)),D)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),C)),E))). [resolve(5617,a,5604,a)]. 56.86/56.74 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B)),C)),D)),E)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),C)),D)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C)),E))). [resolve(5617,a,5612,a)]. 56.86/56.74 5618 ordere236663937imp_le(quickcheck_code_int) # label(arity_Quickcheck__Narrowing_Ocode__int___Groups_Oordered__ab__semigroup__add__im) # label(axiom). [assumption]. 56.86/56.74 5619 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) # label(fact_209_add__le__imp__le__left) # label(axiom). [clausify(53)]. 56.86/56.74 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),C))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),B),C)). [resolve(5618,a,5619,a)]. 56.93/56.86 5620 ordere236663937imp_le(rat) # label(arity_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 56.93/56.86 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),C))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),C)). [resolve(5620,a,5619,a)]. 56.93/56.86 5621 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),D))) # label(fact_331_add__less__cancel__right) # label(axiom). [clausify(957)]. 56.93/56.86 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),C)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),B),C))). [resolve(5621,a,5618,a)]. 56.93/56.86 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),C)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),B),C))). [resolve(5621,a,5620,a)]. 56.93/56.86 5622 -ordere236663937imp_le(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),D))) # label(fact_331_add__less__cancel__right) # label(axiom). [clausify(957)]. 56.93/56.86 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),C)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),B),C))). [resolve(5622,a,5618,a)]. 56.93/56.86 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),C)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),B),C))). [resolve(5622,a,5620,a)]. 56.93/56.86 5623 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),D),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),D)) # label(fact_210_add__le__imp__le__right) # label(axiom). [clausify(1033)]. 56.93/56.86 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),C),B))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),C)). [resolve(5623,a,5618,a)]. 56.93/56.86 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),C),B))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),C)). [resolve(5623,a,5620,a)]. 56.93/56.86 5624 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),D),B)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),D),C))) # label(fact_214_add__le__cancel__left) # label(axiom). [clausify(1140)]. 57.09/57.00 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),C),A)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),C),B))). [resolve(5624,a,5618,a)]. 57.09/57.00 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),C),A)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),C),B))). [resolve(5624,a,5620,a)]. 57.09/57.00 5625 -ordere236663937imp_le(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),D),B)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),D),C))) # label(fact_214_add__le__cancel__left) # label(axiom). [clausify(1140)]. 57.09/57.00 5626 ordere236663937imp_le(int) # label(arity_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 57.09/57.00 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),C))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)). [resolve(5626,a,5619,a)]. 57.09/57.00 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),C)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),C))). [resolve(5626,a,5621,a)]. 57.09/57.00 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),C)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),C))). [resolve(5626,a,5622,a)]. 57.09/57.00 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),C),B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),C)). [resolve(5626,a,5623,a)]. 57.09/57.00 5627 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) # label(fact_330_add__less__cancel__left) # label(axiom). [clausify(1621)]. 57.09/57.00 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),C))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),B),C)). [resolve(5627,a,5618,a)]. 57.09/57.00 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),C))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),C)). [resolve(5627,a,5620,a)]. 57.09/57.00 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),C))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)). [resolve(5627,a,5626,a)]. 57.83/57.73 5628 -ordere236663937imp_le(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),D))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) # label(fact_330_add__less__cancel__left) # label(axiom). [clausify(1621)]. 57.83/57.73 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),C))) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),B),C)). [resolve(5628,a,5618,a)]. 57.83/57.73 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),C))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),C)). [resolve(5628,a,5620,a)]. 57.83/57.73 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),C))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)). [resolve(5628,a,5626,a)]. 57.83/57.73 5629 ordere236663937imp_le(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 57.83/57.73 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),C))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),C)). [resolve(5629,a,5619,a)]. 57.83/57.73 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),C)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),C))). [resolve(5629,a,5621,a)]. 57.83/57.73 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),C)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),C))). [resolve(5629,a,5622,a)]. 57.83/57.73 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),C),B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),C)). [resolve(5629,a,5623,a)]. 57.83/57.73 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),C),A)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),C),B))). [resolve(5629,a,5624,a)]. 57.83/57.73 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),C))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),C)). [resolve(5629,a,5627,a)]. 57.83/57.73 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),C))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),C)). [resolve(5629,a,5628,a)]. 57.83/57.73 5630 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) # label(fact_325_add__less__imp__less__left) # label(axiom). [clausify(3054)]. 57.83/57.73 5631 ordere236663937imp_le(code_code_numeral) # label(arity_Code__Numeral_Ocode__numeral___Groups_Oordered__ab__semigroup__add__imp__l) # label(axiom). [assumption]. 58.53/58.39 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),C))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),B),C)). [resolve(5631,a,5619,a)]. 58.53/58.39 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),C)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),B),C))). [resolve(5631,a,5621,a)]. 58.53/58.39 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),C)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),B),C))). [resolve(5631,a,5622,a)]. 58.53/58.39 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),C),B))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),C)). [resolve(5631,a,5623,a)]. 58.53/58.39 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),C),A)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),C),B))). [resolve(5631,a,5624,a)]. 58.53/58.39 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),C))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),B),C)). [resolve(5631,a,5627,a)]. 58.53/58.39 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),C))) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),B),C)). [resolve(5631,a,5628,a)]. 58.53/58.39 5632 ordere236663937imp_le(nat) # label(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 58.57/58.46 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)). [resolve(5632,a,5619,a)]. 58.57/58.46 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,hAPP(nat,fun(nat,nat),plus_plus(nat),A),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),C))). [resolve(5632,a,5621,a)]. 58.57/58.46 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,hAPP(nat,fun(nat,nat),plus_plus(nat),A),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),C))). [resolve(5632,a,5622,a)]. 58.57/58.46 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),C),B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),C)). [resolve(5632,a,5623,a)]. 58.57/58.46 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),C),A)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),C),B))). [resolve(5632,a,5624,a)]. 58.57/58.46 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)). [resolve(5632,a,5627,a)]. 58.57/58.46 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),C))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)). [resolve(5632,a,5628,a)]. 58.57/58.46 5633 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),D))) # label(fact_215_add__le__cancel__right) # label(axiom). [clausify(4254)]. 58.57/58.46 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),C)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),B),C))). [resolve(5633,a,5618,a)]. 58.57/58.46 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),C)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),B),C))). [resolve(5633,a,5620,a)]. 58.57/58.46 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),C)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),C))). [resolve(5633,a,5626,a)]. 58.57/58.46 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),C)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),C))). [resolve(5633,a,5629,a)]. 58.57/58.46 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),C)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),B),C))). [resolve(5633,a,5631,a)]. 58.95/58.86 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,hAPP(nat,fun(nat,nat),plus_plus(nat),A),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),C))). [resolve(5633,a,5632,a)]. 58.95/58.86 5634 -ordere236663937imp_le(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),D))) # label(fact_215_add__le__cancel__right) # label(axiom). [clausify(4254)]. 58.95/58.86 5635 -ordere236663937imp_le(A) | hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),D),C)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),D)),C) # label(fact_4841_min__add__distrib__left) # label(axiom). [clausify(4628)]. 58.95/58.86 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),C),B)) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),C)),B). [resolve(5635,a,5618,a)]. 58.95/58.86 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),C),B)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),C)),B). [resolve(5635,a,5620,a)]. 58.95/58.86 Derived: hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),C),B)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),C)),B). [resolve(5635,a,5626,a)]. 58.95/58.86 Derived: hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),C),B)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),C)),B). [resolve(5635,a,5629,a)]. 58.95/58.86 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),C),B)) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),C)),B). [resolve(5635,a,5631,a)]. 58.95/58.86 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),C),B)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),C)),B). [resolve(5635,a,5632,a)]. 58.95/58.86 5636 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),D),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),D)) # label(fact_326_add__less__imp__less__right) # label(axiom). [clausify(4777)]. 61.67/61.55 5637 ordered_ab_group_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__group__add) # label(axiom). [assumption]. 61.67/61.55 5638 -ordered_ab_group_add(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f17(B,A,D,E,F,C)),C)) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),F),C) != E | -hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),D),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(B,A,F,D)),E)) # label(fact_4326_setsum__nonneg__leq__bound) # label(axiom). [clausify(54)]. 61.67/61.55 5639 -ordered_ab_group_add(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(B,A,D,f17(B,A,E,F,D,C)))) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),D),C) != F | -hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),E),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(B,A,D,E)),F)) # label(fact_4326_setsum__nonneg__leq__bound) # label(axiom). [clausify(54)]. 61.67/61.55 5640 -ordered_ab_group_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(A,A,uminus_uminus(A),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),hAPP(A,A,uminus_uminus(A),B))) # label(fact_1930_less__minus__iff) # label(axiom). [clausify(264)]. 61.67/61.55 5641 -ordered_ab_group_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(A,A,uminus_uminus(A),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),hAPP(A,A,uminus_uminus(A),B))) # label(fact_1930_less__minus__iff) # label(axiom). [clausify(264)]. 61.67/61.55 5642 -ordered_ab_group_add(A) | hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C) != hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),D),E) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_737_diff__eq__diff__less__eq) # label(axiom). [clausify(364)]. 61.67/61.55 5643 -ordered_ab_group_add(A) | hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C) != hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),D),E) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_737_diff__eq__diff__less__eq) # label(axiom). [clausify(364)]. 61.67/61.55 5644 -ordered_ab_group_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,uminus_uminus(A),C)),hAPP(A,A,uminus_uminus(A),B))) # label(fact_1932_neg__less__iff__less) # label(axiom). [clausify(711)]. 61.67/61.55 5645 -ordered_ab_group_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,uminus_uminus(A),C)),hAPP(A,A,uminus_uminus(A),B))) # label(fact_1932_neg__less__iff__less) # label(axiom). [clausify(711)]. 61.67/61.55 5646 -ordered_ab_group_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,uminus_uminus(A),B))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) # label(fact_1951_neg__0__le__iff__le) # label(axiom). [clausify(1089)]. 61.67/61.55 5647 -ordered_ab_group_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,uminus_uminus(A),B))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) # label(fact_1951_neg__0__le__iff__le) # label(axiom). [clausify(1089)]. 61.67/61.55 5648 -ordered_ab_group_add(A) | hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,A),fun(fun(A,bool),fun(A,bool)),image(A,A),uminus_uminus(A)),hAPP(A,fun(A,bool),hAPP(A,fun(A,fun(A,bool)),ord_gr676655987ssThan(A),B),C)) = hAPP(A,fun(A,bool),hAPP(A,fun(A,fun(A,bool)),ord_gr676655987ssThan(A),hAPP(A,A,uminus_uminus(A),C)),hAPP(A,A,uminus_uminus(A),B)) # label(fact_3696_image__uminus__greaterThanLessThan) # label(axiom). [clausify(1440)]. 61.67/61.55 5649 -ordered_ab_group_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C)),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) # label(fact_752_less__iff__diff__less__0) # label(axiom). [clausify(1510)]. 61.67/61.55 5650 -ordered_ab_group_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C)),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) # label(fact_752_less__iff__diff__less__0) # label(axiom). [clausify(1510)]. 61.67/61.55 5651 -ordered_ab_group_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,uminus_uminus(A),B)),hAPP(A,A,uminus_uminus(A),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),B)) # label(fact_1900_neg__le__iff__le) # label(axiom). [clausify(1550)]. 61.67/61.55 5652 -ordered_ab_group_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,uminus_uminus(A),B)),hAPP(A,A,uminus_uminus(A),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),B)) # label(fact_1900_neg__le__iff__le) # label(axiom). [clausify(1550)]. 61.67/61.55 5653 -ordered_ab_group_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C)),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_751_le__iff__diff__le__0) # label(axiom). [clausify(1710)]. 61.67/61.55 5654 -ordered_ab_group_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C)),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_751_le__iff__diff__le__0) # label(axiom). [clausify(1710)]. 61.67/61.55 5655 -ordered_ab_group_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),hAPP(A,A,uminus_uminus(A),B)),zero_zero(A))) # label(fact_1954_neg__less__0__iff__less) # label(axiom). [clausify(1774)]. 61.67/61.55 5656 -ordered_ab_group_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),hAPP(A,A,uminus_uminus(A),B)),zero_zero(A))) # label(fact_1954_neg__less__0__iff__less) # label(axiom). [clausify(1774)]. 61.67/61.55 5657 -ordered_ab_group_add(A) | hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),D),C)) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),D)),C) # label(fact_4840_min__diff__distrib__left) # label(axiom). [clausify(2341)]. 61.67/61.55 5658 -ordered_ab_group_add(A) | -hBOOL(hAPP(fun(nat,A),bool,monoseq(A),B)) | hBOOL(hAPP(fun(nat,A),bool,monoseq(A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(A,A),fun(fun(nat,A),fun(nat,A)),combb(A,A,nat),uminus_uminus(A)),B))) # label(fact_3948_monoseq__minus) # label(axiom). [clausify(2449)]. 61.67/61.55 5659 -ordered_ab_group_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,uminus_uminus(A),B))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) # label(fact_1953_neg__0__less__iff__less) # label(axiom). [clausify(2464)]. 61.67/61.55 5660 -ordered_ab_group_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,uminus_uminus(A),B))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) # label(fact_1953_neg__0__less__iff__less) # label(axiom). [clausify(2464)]. 61.67/61.55 5661 -ordered_ab_group_add(A) | ord_gr676655987ssThan(A) = ti(fun(A,fun(A,fun(A,bool))),ord_gr676655987ssThan(A)) # label(tsy_c_SetInterval_Oord__class_OgreaterThanLessThan_2_res) # label(axiom). [clausify(2702)]. 61.67/61.55 5662 -ordered_ab_group_add(A) | hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C) != hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),D),E) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),E)) # label(fact_738_diff__eq__diff__less) # label(axiom). [clausify(2713)]. 61.67/61.55 5663 -ordered_ab_group_add(A) | hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C) != hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),D),E) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),E)) # label(fact_738_diff__eq__diff__less) # label(axiom). [clausify(2713)]. 61.67/61.56 5664 -ordered_ab_group_add(A) | hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,A),fun(fun(A,bool),fun(A,bool)),image(A,A),uminus_uminus(A)),hAPP(A,fun(A,bool),hAPP(A,fun(A,fun(A,bool)),ord_atLeastAtMost(A),B),C)) = hAPP(A,fun(A,bool),hAPP(A,fun(A,fun(A,bool)),ord_atLeastAtMost(A),hAPP(A,A,uminus_uminus(A),C)),hAPP(A,A,uminus_uminus(A),B)) # label(fact_4551_image__uminus__atLeastAtMost) # label(axiom). [clausify(2815)]. 61.67/61.56 5665 -ordered_ab_group_add(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f489(B,A,D,E,C)),C)) | zero_zero(A) != hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),E),C) | -hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),D),C)) | zero_zero(A) = hAPP(B,A,E,D) # label(fact_4327_setsum__nonneg__0) # label(axiom). [clausify(2915)]. 61.67/61.56 5666 -ordered_ab_group_add(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(B,A,D,f489(B,A,E,D,C)))) | zero_zero(A) != hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),D),C) | -hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),E),C)) | zero_zero(A) = hAPP(B,A,D,E) # label(fact_4327_setsum__nonneg__0) # label(axiom). [clausify(2915)]. 61.67/61.56 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f17(A,real,C,D,E,B)),B)) | hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),E),B) != D | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(A,real,E,C)),D)). [resolve(5637,a,5638,a)]. 61.67/61.56 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(A,real,C,f17(A,real,D,E,C,B)))) | hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),C),B) != E | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),D),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(A,real,C,D)),E)). [resolve(5637,a,5639,a)]. 61.67/61.56 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(real,real,uminus_uminus(real),B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),hAPP(real,real,uminus_uminus(real),A))). [resolve(5637,a,5640,a)]. 61.67/61.56 Derived: hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B) != hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),C),D) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)). [resolve(5637,a,5642,a)]. 61.67/61.56 Derived: hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B) != hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),C),D) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),D)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)). [resolve(5637,a,5643,a)]. 61.67/61.56 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),B)),hAPP(real,real,uminus_uminus(real),A))). [resolve(5637,a,5644,a)]. 61.67/61.56 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),B)),hAPP(real,real,uminus_uminus(real),A))). [resolve(5637,a,5645,a)]. 61.67/61.56 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,uminus_uminus(real),A))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))). [resolve(5637,a,5646,a)]. 61.67/61.56 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,uminus_uminus(real),A))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))). [resolve(5637,a,5647,a)]. 61.67/61.58 Derived: hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,real),fun(fun(real,bool),fun(real,bool)),image(real,real),uminus_uminus(real)),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_gr676655987ssThan(real),A),B)) = hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_gr676655987ssThan(real),hAPP(real,real,uminus_uminus(real),B)),hAPP(real,real,uminus_uminus(real),A)). [resolve(5637,a,5648,a)]. 61.67/61.58 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B)),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)). [resolve(5637,a,5649,a)]. 61.67/61.58 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B)),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)). [resolve(5637,a,5650,a)]. 61.67/61.58 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),A)),hAPP(real,real,uminus_uminus(real),B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),A)). [resolve(5637,a,5651,a)]. 61.67/61.58 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),A)),hAPP(real,real,uminus_uminus(real),B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),A)). [resolve(5637,a,5652,a)]. 61.67/61.58 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),A)),zero_zero(real))). [resolve(5637,a,5655,a)]. 61.67/61.58 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),A)),zero_zero(real))). [resolve(5637,a,5656,a)]. 61.67/61.58 Derived: hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),C),B)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),C)),B). [resolve(5637,a,5657,a)]. 61.67/61.58 Derived: -hBOOL(hAPP(fun(nat,real),bool,monoseq(real),A)) | hBOOL(hAPP(fun(nat,real),bool,monoseq(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),uminus_uminus(real)),A))). [resolve(5637,a,5658,a)]. 61.67/61.58 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,uminus_uminus(real),A))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))). [resolve(5637,a,5659,a)]. 61.67/61.58 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,uminus_uminus(real),A))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))). [resolve(5637,a,5660,a)]. 61.67/61.58 Derived: ord_gr676655987ssThan(real) = ti(fun(real,fun(real,fun(real,bool))),ord_gr676655987ssThan(real)). [resolve(5637,a,5661,a)]. 61.67/61.58 Derived: hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B) != hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),C),D) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),D)). [resolve(5637,a,5662,a)]. 61.67/61.58 Derived: hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B) != hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),C),D) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),D)). [resolve(5637,a,5663,a)]. 61.67/61.58 Derived: hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,real),fun(fun(real,bool),fun(real,bool)),image(real,real),uminus_uminus(real)),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastAtMost(real),A),B)) = hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastAtMost(real),hAPP(real,real,uminus_uminus(real),B)),hAPP(real,real,uminus_uminus(real),A)). [resolve(5637,a,5664,a)]. 61.85/61.75 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f489(A,real,C,D,B)),B)) | zero_zero(real) != hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),D),B) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | zero_zero(real) = hAPP(A,real,D,C). [resolve(5637,a,5665,a)]. 61.85/61.75 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(A,real,C,f489(A,real,D,C,B)))) | zero_zero(real) != hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),C),B) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),D),B)) | zero_zero(real) = hAPP(A,real,C,D). [resolve(5637,a,5666,a)]. 61.85/61.75 5667 ordered_ab_group_add(rat) # label(arity_Rat_Orat___Groups_Oordered__ab__group__add) # label(axiom). [assumption]. 61.85/61.75 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f17(A,rat,C,D,E,B)),B)) | hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),E),B) != D | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(A,rat,E,C)),D)). [resolve(5667,a,5638,a)]. 61.85/61.75 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(A,rat,C,f17(A,rat,D,E,C,B)))) | hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),C),B) != E | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),D),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(A,rat,C,D)),E)). [resolve(5667,a,5639,a)]. 61.85/61.75 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(rat,rat,uminus_uminus(rat),B))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),hAPP(rat,rat,uminus_uminus(rat),A))). [resolve(5667,a,5640,a)]. 61.85/61.75 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B) != hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),C),D) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),C),D)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)). [resolve(5667,a,5642,a)]. 61.85/61.75 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B) != hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),C),D) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),C),D)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)). [resolve(5667,a,5643,a)]. 61.85/61.75 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,uminus_uminus(rat),B)),hAPP(rat,rat,uminus_uminus(rat),A))). [resolve(5667,a,5644,a)]. 61.85/61.75 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,uminus_uminus(rat),B)),hAPP(rat,rat,uminus_uminus(rat),A))). [resolve(5667,a,5645,a)]. 61.85/61.75 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(rat,rat,uminus_uminus(rat),A))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),zero_zero(rat))). [resolve(5667,a,5646,a)]. 61.85/61.75 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(rat,rat,uminus_uminus(rat),A))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),zero_zero(rat))). [resolve(5667,a,5647,a)]. 61.85/61.75 Derived: hAPP(fun(rat,bool),fun(rat,bool),hAPP(fun(rat,rat),fun(fun(rat,bool),fun(rat,bool)),image(rat,rat),uminus_uminus(rat)),hAPP(rat,fun(rat,bool),hAPP(rat,fun(rat,fun(rat,bool)),ord_gr676655987ssThan(rat),A),B)) = hAPP(rat,fun(rat,bool),hAPP(rat,fun(rat,fun(rat,bool)),ord_gr676655987ssThan(rat),hAPP(rat,rat,uminus_uminus(rat),B)),hAPP(rat,rat,uminus_uminus(rat),A)). [resolve(5667,a,5648,a)]. 61.85/61.76 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B)),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)). [resolve(5667,a,5649,a)]. 61.85/61.76 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B)),zero_zero(rat))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)). [resolve(5667,a,5650,a)]. 61.85/61.76 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,uminus_uminus(rat),A)),hAPP(rat,rat,uminus_uminus(rat),B))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),A)). [resolve(5667,a,5651,a)]. 61.85/61.76 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,uminus_uminus(rat),A)),hAPP(rat,rat,uminus_uminus(rat),B))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),A)). [resolve(5667,a,5652,a)]. 61.85/61.76 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B)),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)). [resolve(5667,a,5653,a)]. 61.85/61.76 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B)),zero_zero(rat))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)). [resolve(5667,a,5654,a)]. 61.85/61.76 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,uminus_uminus(rat),A)),zero_zero(rat))). [resolve(5667,a,5655,a)]. 61.85/61.76 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,uminus_uminus(rat),A)),zero_zero(rat))). [resolve(5667,a,5656,a)]. 61.85/61.76 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),C),B)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),C)),B). [resolve(5667,a,5657,a)]. 61.85/61.76 Derived: -hBOOL(hAPP(fun(nat,rat),bool,monoseq(rat),A)) | hBOOL(hAPP(fun(nat,rat),bool,monoseq(rat),hAPP(fun(nat,rat),fun(nat,rat),hAPP(fun(rat,rat),fun(fun(nat,rat),fun(nat,rat)),combb(rat,rat,nat),uminus_uminus(rat)),A))). [resolve(5667,a,5658,a)]. 61.85/61.76 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(rat,rat,uminus_uminus(rat),A))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),zero_zero(rat))). [resolve(5667,a,5659,a)]. 61.85/61.76 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(rat,rat,uminus_uminus(rat),A))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),zero_zero(rat))). [resolve(5667,a,5660,a)]. 61.85/61.76 Derived: ord_gr676655987ssThan(rat) = ti(fun(rat,fun(rat,fun(rat,bool))),ord_gr676655987ssThan(rat)). [resolve(5667,a,5661,a)]. 61.85/61.76 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B) != hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),C),D) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),C),D)). [resolve(5667,a,5662,a)]. 61.85/61.76 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B) != hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),C),D) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),C),D)). [resolve(5667,a,5663,a)]. 61.85/61.76 Derived: hAPP(fun(rat,bool),fun(rat,bool),hAPP(fun(rat,rat),fun(fun(rat,bool),fun(rat,bool)),image(rat,rat),uminus_uminus(rat)),hAPP(rat,fun(rat,bool),hAPP(rat,fun(rat,fun(rat,bool)),ord_atLeastAtMost(rat),A),B)) = hAPP(rat,fun(rat,bool),hAPP(rat,fun(rat,fun(rat,bool)),ord_atLeastAtMost(rat),hAPP(rat,rat,uminus_uminus(rat),B)),hAPP(rat,rat,uminus_uminus(rat),A)). [resolve(5667,a,5664,a)]. 62.57/62.43 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f489(A,rat,C,D,B)),B)) | zero_zero(rat) != hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),D),B) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | zero_zero(rat) = hAPP(A,rat,D,C). [resolve(5667,a,5665,a)]. 62.57/62.43 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(A,rat,C,f489(A,rat,D,C,B)))) | zero_zero(rat) != hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),C),B) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),D),B)) | zero_zero(rat) = hAPP(A,rat,C,D). [resolve(5667,a,5666,a)]. 62.57/62.43 5668 -ordered_ab_group_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,uminus_uminus(A),B)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,uminus_uminus(A),C)),B)) # label(fact_1931_minus__less__iff) # label(axiom). [clausify(4061)]. 62.57/62.43 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),A)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),B)),A)). [resolve(5668,a,5637,a)]. 62.57/62.43 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,uminus_uminus(rat),A)),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,uminus_uminus(rat),B)),A)). [resolve(5668,a,5667,a)]. 62.57/62.43 5669 -ordered_ab_group_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,uminus_uminus(A),B)),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,uminus_uminus(A),C)),B)) # label(fact_1931_minus__less__iff) # label(axiom). [clausify(4061)]. 62.57/62.43 5670 ordered_ab_group_add(int) # label(arity_Int_Oint___Groups_Oordered__ab__group__add) # label(axiom). [assumption]. 62.57/62.43 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f17(A,int,C,D,E,B)),B)) | hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),E),B) != D | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(A,int,E,C)),D)). [resolve(5670,a,5638,a)]. 62.57/62.43 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(A,int,C,f17(A,int,D,E,C,B)))) | hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),C),B) != E | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),D),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(A,int,C,D)),E)). [resolve(5670,a,5639,a)]. 62.57/62.43 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(int,int,uminus_uminus(int),B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),hAPP(int,int,uminus_uminus(int),A))). [resolve(5670,a,5640,a)]. 62.57/62.43 Derived: hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),C),D) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),D)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(5670,a,5642,a)]. 62.57/62.43 Derived: hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),C),D) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),D)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(5670,a,5643,a)]. 62.57/62.43 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),hAPP(int,int,uminus_uminus(int),B)),hAPP(int,int,uminus_uminus(int),A))). [resolve(5670,a,5644,a)]. 62.57/62.43 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),hAPP(int,int,uminus_uminus(int),B)),hAPP(int,int,uminus_uminus(int),A))). [resolve(5670,a,5645,a)]. 62.57/62.44 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,uminus_uminus(int),A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))). [resolve(5670,a,5646,a)]. 62.57/62.44 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,uminus_uminus(int),A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))). [resolve(5670,a,5647,a)]. 62.57/62.44 Derived: hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,int),fun(fun(int,bool),fun(int,bool)),image(int,int),uminus_uminus(int)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_gr676655987ssThan(int),A),B)) = hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_gr676655987ssThan(int),hAPP(int,int,uminus_uminus(int),B)),hAPP(int,int,uminus_uminus(int),A)). [resolve(5670,a,5648,a)]. 62.57/62.44 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,uminus_uminus(int),A)),hAPP(int,int,uminus_uminus(int),B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),A)). [resolve(5670,a,5651,a)]. 62.57/62.44 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,uminus_uminus(int),A)),hAPP(int,int,uminus_uminus(int),B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),A)). [resolve(5670,a,5652,a)]. 62.57/62.44 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B)),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(5670,a,5653,a)]. 62.57/62.44 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B)),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(5670,a,5654,a)]. 62.57/62.44 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(int),hAPP(int,int,uminus_uminus(int),A)),zero_zero(int))). [resolve(5670,a,5655,a)]. 62.57/62.44 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(int),hAPP(int,int,uminus_uminus(int),A)),zero_zero(int))). [resolve(5670,a,5656,a)]. 62.57/62.44 Derived: hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),C),B)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),C)),B). [resolve(5670,a,5657,a)]. 62.57/62.44 Derived: -hBOOL(hAPP(fun(nat,int),bool,monoseq(int),A)) | hBOOL(hAPP(fun(nat,int),bool,monoseq(int),hAPP(fun(nat,int),fun(nat,int),hAPP(fun(int,int),fun(fun(nat,int),fun(nat,int)),combb(int,int,nat),uminus_uminus(int)),A))). [resolve(5670,a,5658,a)]. 62.57/62.44 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,uminus_uminus(int),A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))). [resolve(5670,a,5659,a)]. 62.57/62.44 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,uminus_uminus(int),A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))). [resolve(5670,a,5660,a)]. 62.57/62.44 Derived: ord_gr676655987ssThan(int) = ti(fun(int,fun(int,fun(int,bool))),ord_gr676655987ssThan(int)). [resolve(5670,a,5661,a)]. 62.57/62.44 Derived: hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),C),D) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),D)). [resolve(5670,a,5662,a)]. 62.57/62.44 Derived: hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),C),D) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),D)). [resolve(5670,a,5663,a)]. 62.75/62.67 Derived: hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,int),fun(fun(int,bool),fun(int,bool)),image(int,int),uminus_uminus(int)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),A),B)) = hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),hAPP(int,int,uminus_uminus(int),B)),hAPP(int,int,uminus_uminus(int),A)). [resolve(5670,a,5664,a)]. 62.75/62.67 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f489(A,int,C,D,B)),B)) | zero_zero(int) != hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),D),B) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | zero_zero(int) = hAPP(A,int,D,C). [resolve(5670,a,5665,a)]. 62.75/62.67 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(A,int,C,f489(A,int,D,C,B)))) | zero_zero(int) != hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),C),B) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),D),B)) | zero_zero(int) = hAPP(A,int,C,D). [resolve(5670,a,5666,a)]. 62.75/62.67 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,uminus_uminus(int),A)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,uminus_uminus(int),B)),A)). [resolve(5670,a,5668,a)]. 62.75/62.67 5671 -ordered_ab_group_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),hAPP(A,A,uminus_uminus(A),B)),zero_zero(A))) # label(fact_1949_neg__le__0__iff__le) # label(axiom). [clausify(4236)]. 62.75/62.67 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),hAPP(real,real,uminus_uminus(real),A)),zero_zero(real))). [resolve(5671,a,5637,a)]. 62.75/62.67 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,uminus_uminus(rat),A)),zero_zero(rat))). [resolve(5671,a,5667,a)]. 62.75/62.67 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),hAPP(int,int,uminus_uminus(int),A)),zero_zero(int))). [resolve(5671,a,5670,a)]. 62.75/62.67 5672 -ordered_ab_group_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),hAPP(A,A,uminus_uminus(A),B)),zero_zero(A))) # label(fact_1949_neg__le__0__iff__le) # label(axiom). [clausify(4236)]. 62.75/62.67 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),hAPP(real,real,uminus_uminus(real),A)),zero_zero(real))). [resolve(5672,a,5637,a)]. 62.75/62.67 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,uminus_uminus(rat),A)),zero_zero(rat))). [resolve(5672,a,5667,a)]. 62.75/62.67 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),hAPP(int,int,uminus_uminus(int),A)),zero_zero(int))). [resolve(5672,a,5670,a)]. 62.75/62.67 5673 -ordered_ab_group_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,uminus_uminus(A),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),hAPP(A,A,uminus_uminus(A),B))) # label(fact_1902_le__minus__iff) # label(axiom). [clausify(4369)]. 62.75/62.67 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),hAPP(real,real,uminus_uminus(real),B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),hAPP(real,real,uminus_uminus(real),A))). [resolve(5673,a,5637,a)]. 62.75/62.67 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),hAPP(rat,rat,uminus_uminus(rat),B))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),hAPP(rat,rat,uminus_uminus(rat),A))). [resolve(5673,a,5667,a)]. 67.05/66.97 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(int,int,uminus_uminus(int),B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),hAPP(int,int,uminus_uminus(int),A))). [resolve(5673,a,5670,a)]. 67.05/66.97 5674 -ordered_ab_group_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,uminus_uminus(A),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),hAPP(A,A,uminus_uminus(A),B))) # label(fact_1902_le__minus__iff) # label(axiom). [clausify(4369)]. 67.05/66.97 5675 -ordered_ab_group_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,uminus_uminus(A),C)),hAPP(A,A,uminus_uminus(A),B))) # label(fact_1899_le__imp__neg__le) # label(axiom). [clausify(4438)]. 67.05/66.97 5676 -ordered_ab_group_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,uminus_uminus(A),B)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,uminus_uminus(A),C)),B)) # label(fact_1901_minus__le__iff) # label(axiom). [clausify(4605)]. 67.05/66.97 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),A)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),B)),A)). [resolve(5676,a,5637,a)]. 67.05/66.97 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,uminus_uminus(rat),A)),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,uminus_uminus(rat),B)),A)). [resolve(5676,a,5667,a)]. 67.05/66.97 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,uminus_uminus(int),A)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,uminus_uminus(int),B)),A)). [resolve(5676,a,5670,a)]. 67.05/66.97 5677 -ordered_ab_group_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,uminus_uminus(A),B)),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,uminus_uminus(A),C)),B)) # label(fact_1901_minus__le__iff) # label(axiom). [clausify(4605)]. 67.05/66.97 5678 -cancel146912293up_add(A) | hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C) != hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),D) | ti(A,C) = ti(A,D) # label(fact_291_add__imp__eq) # label(axiom). [clausify(4893)]. 67.05/66.97 5679 cancel146912293up_add(real) # label(arity_RealDef_Oreal___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 67.05/66.97 5680 cancel146912293up_add(int) # label(arity_Int_Oint___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 67.05/66.97 5681 cancel146912293up_add(complex) # label(arity_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 67.05/66.97 5682 cancel146912293up_add(nat) # label(arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 67.05/66.97 5683 cancel146912293up_add(rat) # label(arity_Rat_Orat___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 67.05/66.97 5684 cancel146912293up_add(code_code_numeral) # label(arity_Code__Numeral_Ocode__numeral___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 67.05/66.97 5685 cancel146912293up_add(quickcheck_code_int) # label(arity_Quickcheck__Narrowing_Ocode__int___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 67.05/66.97 Derived: hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B) != hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),C) | ti(real,B) = ti(real,C). [resolve(5678,a,5679,a)]. 67.05/66.97 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B) != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),C) | ti(int,B) = ti(int,C). [resolve(5678,a,5680,a)]. 67.05/66.97 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),B) != hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),C) | ti(complex,B) = ti(complex,C). [resolve(5678,a,5681,a)]. 67.05/66.97 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),B) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),C) | ti(nat,B) = ti(nat,C). [resolve(5678,a,5682,a)]. 68.94/68.82 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B) != hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),C) | ti(rat,B) = ti(rat,C). [resolve(5678,a,5683,a)]. 68.94/68.82 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),B) != hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),C) | ti(code_code_numeral,B) = ti(code_code_numeral,C). [resolve(5678,a,5684,a)]. 68.94/68.82 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),B) != hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),C) | ti(quickcheck_code_int,B) = ti(quickcheck_code_int,C). [resolve(5678,a,5685,a)]. 68.94/68.82 5686 ordere779506340up_add(code_code_numeral) # label(arity_Code__Numeral_Ocode__numeral___Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 68.94/68.82 5687 -comm_monoid_add(A) | -ordere779506340up_add(A) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f18(B,A,C,D)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),C),D))) # label(fact_4336_setsum__nonneg) # label(axiom). [clausify(61)]. 68.94/68.82 5688 -comm_monoid_add(A) | -ordere779506340up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(B,A,C,f18(B,A,C,D)))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),C),D))) # label(fact_4336_setsum__nonneg) # label(axiom). [clausify(61)]. 68.94/68.82 Derived: -comm_monoid_add(code_code_numeral) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f18(A,code_code_numeral,B,C)),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),B),C))). [resolve(5686,a,5687,b)]. 68.98/68.82 Derived: -comm_monoid_add(code_code_numeral) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),hAPP(A,code_code_numeral,B,f18(A,code_code_numeral,B,C)))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),B),C))). [resolve(5686,a,5688,b)]. 68.98/68.82 5689 ordere779506340up_add(quickcheck_code_int) # label(arity_Quickcheck__Narrowing_Ocode__int___Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 68.98/68.82 Derived: -comm_monoid_add(quickcheck_code_int) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f18(A,quickcheck_code_int,B,C)),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),B),C))). [resolve(5689,a,5687,b)]. 68.98/68.82 Derived: -comm_monoid_add(quickcheck_code_int) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(A,quickcheck_code_int,B,f18(A,quickcheck_code_int,B,C)))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),B),C))). [resolve(5689,a,5688,b)]. 69.15/69.00 5690 ordere779506340up_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 69.15/69.00 Derived: -comm_monoid_add(real) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f18(A,real,B,C)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),B),C))). [resolve(5690,a,5687,b)]. 69.15/69.00 Derived: -comm_monoid_add(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(A,real,B,f18(A,real,B,C)))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),B),C))). [resolve(5690,a,5688,b)]. 69.15/69.00 5691 -ordere779506340up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),E))) # label(fact_211_add__mono) # label(axiom). [clausify(1439)]. 69.15/69.00 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),C),D)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),C)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),B),D))). [resolve(5691,a,5686,a)]. 69.15/69.00 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),C),D)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),C)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),B),D))). [resolve(5691,a,5689,a)]. 69.15/69.00 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),C)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),D))). [resolve(5691,a,5690,a)]. 69.15/69.00 5692 -ordere779506340up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),D),B)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),D),C))) # label(fact_212_add__left__mono) # label(axiom). [clausify(1746)]. 69.15/69.00 5693 -ordere779506340up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),D))) # label(fact_213_add__right__mono) # label(axiom). [clausify(1901)]. 69.15/69.00 5694 -ordere779506340up_add(A) | -semiring_1(A) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f324(B,A,C,D,E)),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),C),E)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(nat,A,semiring_1_of_nat(A),hAPP(fun(B,bool),nat,finite_card(B),E))),D))) # label(fact_4323_setsum__bounded) # label(axiom). [clausify(1964)]. 69.15/69.01 Derived: -semiring_1(code_code_numeral) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f324(A,code_code_numeral,B,C,D)),D)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),B),D)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),hAPP(fun(A,bool),nat,finite_card(A),D))),C))). [resolve(5694,a,5686,a)]. 69.15/69.01 Derived: -semiring_1(quickcheck_code_int) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f324(A,quickcheck_code_int,B,C,D)),D)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),B),D)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),hAPP(fun(A,bool),nat,finite_card(A),D))),C))). [resolve(5694,a,5689,a)]. 69.15/69.01 Derived: -semiring_1(real) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f324(A,real,B,C,D)),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),B),D)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,semiring_1_of_nat(real),hAPP(fun(A,bool),nat,finite_card(A),D))),C))). [resolve(5694,a,5690,a)]. 69.15/69.01 5695 -ordere779506340up_add(A) | -semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(B,A,C,f324(B,A,C,D,E))),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),C),E)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(nat,A,semiring_1_of_nat(A),hAPP(fun(B,bool),nat,finite_card(B),E))),D))) # label(fact_4323_setsum__bounded) # label(axiom). [clausify(1964)]. 69.15/69.01 Derived: -semiring_1(code_code_numeral) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(A,code_code_numeral,B,f324(A,code_code_numeral,B,C,D))),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),B),D)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),hAPP(fun(A,bool),nat,finite_card(A),D))),C))). [resolve(5695,a,5686,a)]. 69.15/69.01 Derived: -semiring_1(quickcheck_code_int) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(A,quickcheck_code_int,B,f324(A,quickcheck_code_int,B,C,D))),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),B),D)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),hAPP(fun(A,bool),nat,finite_card(A),D))),C))). [resolve(5695,a,5689,a)]. 69.26/69.12 Derived: -semiring_1(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(A,real,B,f324(A,real,B,C,D))),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),B),D)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,semiring_1_of_nat(real),hAPP(fun(A,bool),nat,finite_card(A),D))),C))). [resolve(5695,a,5690,a)]. 69.26/69.12 5696 ordere779506340up_add(int) # label(arity_Int_Oint___Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 69.26/69.12 Derived: -comm_monoid_add(int) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f18(A,int,B,C)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),B),C))). [resolve(5696,a,5687,b)]. 69.26/69.12 Derived: -comm_monoid_add(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(A,int,B,f18(A,int,B,C)))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),B),C))). [resolve(5696,a,5688,b)]. 69.26/69.12 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),C),D)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),C)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),D))). [resolve(5696,a,5691,a)]. 69.26/69.12 Derived: -semiring_1(int) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f324(A,int,B,C,D)),D)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),B),D)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(A,bool),nat,finite_card(A),D))),C))). [resolve(5696,a,5694,a)]. 69.26/69.12 Derived: -semiring_1(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(A,int,B,f324(A,int,B,C,D))),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),B),D)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(A,bool),nat,finite_card(A),D))),C))). [resolve(5696,a,5695,a)]. 69.26/69.12 5697 -comm_monoid_add(A) | -ordere779506340up_add(A) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f375(B,A,C,D,E)),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),C),E)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),D),E))) # label(fact_4370_setsum__mono) # label(axiom). [clausify(2239)]. 69.26/69.12 Derived: -comm_monoid_add(code_code_numeral) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f375(A,code_code_numeral,B,C,D)),D)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),B),D)),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),C),D))). [resolve(5697,b,5686,a)]. 69.26/69.12 Derived: -comm_monoid_add(quickcheck_code_int) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f375(A,quickcheck_code_int,B,C,D)),D)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),B),D)),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),C),D))). [resolve(5697,b,5689,a)]. 69.28/69.16 Derived: -comm_monoid_add(real) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f375(A,real,B,C,D)),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),B),D)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),C),D))). [resolve(5697,b,5690,a)]. 69.28/69.16 Derived: -comm_monoid_add(int) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f375(A,int,B,C,D)),D)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),B),D)),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),C),D))). [resolve(5697,b,5696,a)]. 69.28/69.16 5698 -comm_monoid_add(A) | -ordere779506340up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(B,A,C,f375(B,A,C,D,E))),hAPP(B,A,D,f375(B,A,C,D,E)))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),C),E)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),D),E))) # label(fact_4370_setsum__mono) # label(axiom). [clausify(2239)]. 69.28/69.16 Derived: -comm_monoid_add(code_code_numeral) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(A,code_code_numeral,B,f375(A,code_code_numeral,B,C,D))),hAPP(A,code_code_numeral,C,f375(A,code_code_numeral,B,C,D)))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),B),D)),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),C),D))). [resolve(5698,b,5686,a)]. 69.28/69.16 Derived: -comm_monoid_add(quickcheck_code_int) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(A,quickcheck_code_int,B,f375(A,quickcheck_code_int,B,C,D))),hAPP(A,quickcheck_code_int,C,f375(A,quickcheck_code_int,B,C,D)))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),B),D)),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),C),D))). [resolve(5698,b,5689,a)]. 69.28/69.16 Derived: -comm_monoid_add(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(A,real,B,f375(A,real,B,C,D))),hAPP(A,real,C,f375(A,real,B,C,D)))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),B),D)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),C),D))). [resolve(5698,b,5690,a)]. 69.28/69.16 Derived: -comm_monoid_add(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(A,int,B,f375(A,int,B,C,D))),hAPP(A,int,C,f375(A,int,B,C,D)))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),B),D)),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),C),D))). [resolve(5698,b,5696,a)]. 69.28/69.16 5699 ordere779506340up_add(nat) # label(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 69.28/69.16 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),D))). [resolve(5699,a,5691,a)]. 69.58/69.44 Derived: -semiring_1(nat) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f324(A,nat,B,C,D)),D)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co1399186613setsum(A,nat),B),D)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,semiring_1_of_nat(nat),hAPP(fun(A,bool),nat,finite_card(A),D))),C))). [resolve(5699,a,5694,a)]. 69.58/69.44 Derived: -semiring_1(nat) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(A,nat,B,f324(A,nat,B,C,D))),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co1399186613setsum(A,nat),B),D)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,semiring_1_of_nat(nat),hAPP(fun(A,bool),nat,finite_card(A),D))),C))). [resolve(5699,a,5695,a)]. 69.58/69.44 Derived: -comm_monoid_add(nat) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f375(A,nat,B,C,D)),D)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co1399186613setsum(A,nat),B),D)),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co1399186613setsum(A,nat),C),D))). [resolve(5699,a,5697,b)]. 69.58/69.44 Derived: -comm_monoid_add(nat) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(A,nat,B,f375(A,nat,B,C,D))),hAPP(A,nat,C,f375(A,nat,B,C,D)))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co1399186613setsum(A,nat),B),D)),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co1399186613setsum(A,nat),C),D))). [resolve(5699,a,5698,b)]. 69.58/69.44 5700 -ordere779506340up_add(A) | -comm_monoid_add(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | -hBOOL(hAPP(fun(B,bool),bool,hAPP(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),D),C)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f488(B,A,E,D,C)),hAPP(fun(B,bool),fun(B,bool),hAPP(fun(B,bool),fun(fun(B,bool),fun(B,bool)),minus_minus(fun(B,bool)),C),D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),E),D)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),E),C))) # label(fact_4319_setsum__mono3) # label(axiom). [clausify(2913)]. 69.58/69.44 Derived: -comm_monoid_add(code_code_numeral) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f488(A,code_code_numeral,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),D),C)),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),D),B))). [resolve(5700,a,5686,a)]. 69.58/69.44 Derived: -comm_monoid_add(quickcheck_code_int) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f488(A,quickcheck_code_int,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),D),C)),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),D),B))). [resolve(5700,a,5689,a)]. 69.58/69.45 Derived: -comm_monoid_add(real) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f488(A,real,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),D),C)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),D),B))). [resolve(5700,a,5690,a)]. 69.58/69.45 Derived: -comm_monoid_add(int) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f488(A,int,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),D),C)),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),D),B))). [resolve(5700,a,5696,a)]. 69.58/69.45 Derived: -comm_monoid_add(nat) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f488(A,nat,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co1399186613setsum(A,nat),D),C)),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co1399186613setsum(A,nat),D),B))). [resolve(5700,a,5699,a)]. 69.58/69.45 5701 -ordere779506340up_add(A) | -comm_monoid_add(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | -hBOOL(hAPP(fun(B,bool),bool,hAPP(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),D),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(B,A,E,f488(B,A,E,D,C)))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),E),D)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),E),C))) # label(fact_4319_setsum__mono3) # label(axiom). [clausify(2913)]. 69.58/69.45 Derived: -comm_monoid_add(code_code_numeral) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),hAPP(A,code_code_numeral,D,f488(A,code_code_numeral,D,C,B)))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),D),C)),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),D),B))). [resolve(5701,a,5686,a)]. 69.58/69.45 Derived: -comm_monoid_add(quickcheck_code_int) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(A,quickcheck_code_int,D,f488(A,quickcheck_code_int,D,C,B)))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),D),C)),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),D),B))). [resolve(5701,a,5689,a)]. 69.97/69.84 Derived: -comm_monoid_add(real) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(A,real,D,f488(A,real,D,C,B)))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),D),C)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),D),B))). [resolve(5701,a,5690,a)]. 69.97/69.84 Derived: -comm_monoid_add(int) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(A,int,D,f488(A,int,D,C,B)))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),D),C)),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),D),B))). [resolve(5701,a,5696,a)]. 69.97/69.84 Derived: -comm_monoid_add(nat) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),hAPP(A,nat,D,f488(A,nat,D,C,B)))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co1399186613setsum(A,nat),D),C)),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co1399186613setsum(A,nat),D),B))). [resolve(5701,a,5699,a)]. 69.97/69.84 5702 ordere779506340up_add(rat) # label(arity_Rat_Orat___Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 69.97/69.84 Derived: -comm_monoid_add(rat) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f18(A,rat,B,C)),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),B),C))). [resolve(5702,a,5687,b)]. 69.97/69.84 Derived: -comm_monoid_add(rat) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(A,rat,B,f18(A,rat,B,C)))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),B),C))). [resolve(5702,a,5688,b)]. 69.97/69.84 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),C),D)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),C)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),B),D))). [resolve(5702,a,5691,a)]. 69.97/69.84 Derived: -semiring_1(rat) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f324(A,rat,B,C,D)),D)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),B),D)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(nat,rat,semiring_1_of_nat(rat),hAPP(fun(A,bool),nat,finite_card(A),D))),C))). [resolve(5702,a,5694,a)]. 69.97/69.84 Derived: -semiring_1(rat) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(A,rat,B,f324(A,rat,B,C,D))),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),B),D)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(nat,rat,semiring_1_of_nat(rat),hAPP(fun(A,bool),nat,finite_card(A),D))),C))). [resolve(5702,a,5695,a)]. 70.38/70.26 Derived: -comm_monoid_add(rat) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f375(A,rat,B,C,D)),D)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),B),D)),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),C),D))). [resolve(5702,a,5697,b)]. 70.38/70.26 Derived: -comm_monoid_add(rat) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(A,rat,B,f375(A,rat,B,C,D))),hAPP(A,rat,C,f375(A,rat,B,C,D)))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),B),D)),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),C),D))). [resolve(5702,a,5698,b)]. 70.38/70.26 Derived: -comm_monoid_add(rat) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f488(A,rat,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),D),C)),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),D),B))). [resolve(5702,a,5700,a)]. 70.38/70.26 Derived: -comm_monoid_add(rat) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(A,rat,D,f488(A,rat,D,C,B)))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),D),C)),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),D),B))). [resolve(5702,a,5701,a)]. 70.38/70.26 5703 -comm_monoid_add(A) | -ordere779506340up_add(A) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f697(B,A,C,D)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),C),D)),zero_zero(A))) # label(fact_4335_setsum__nonpos) # label(axiom). [clausify(4072)]. 70.38/70.26 Derived: -comm_monoid_add(code_code_numeral) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f697(A,code_code_numeral,B,C)),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),B),C)),zero_zero(code_code_numeral))). [resolve(5703,b,5686,a)]. 70.38/70.26 Derived: -comm_monoid_add(quickcheck_code_int) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f697(A,quickcheck_code_int,B,C)),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),B),C)),zero_zero(quickcheck_code_int))). [resolve(5703,b,5689,a)]. 70.38/70.26 Derived: -comm_monoid_add(real) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f697(A,real,B,C)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),B),C)),zero_zero(real))). [resolve(5703,b,5690,a)]. 70.38/70.26 Derived: -comm_monoid_add(int) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f697(A,int,B,C)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),B),C)),zero_zero(int))). [resolve(5703,b,5696,a)]. 72.89/72.76 Derived: -comm_monoid_add(nat) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f697(A,nat,B,C)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co1399186613setsum(A,nat),B),C)),zero_zero(nat))). [resolve(5703,b,5699,a)]. 72.89/72.76 Derived: -comm_monoid_add(rat) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f697(A,rat,B,C)),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),B),C)),zero_zero(rat))). [resolve(5703,b,5702,a)]. 72.89/72.76 5704 -comm_monoid_add(A) | -ordere779506340up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(B,A,C,f697(B,A,C,D))),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),C),D)),zero_zero(A))) # label(fact_4335_setsum__nonpos) # label(axiom). [clausify(4072)]. 72.89/72.76 Derived: -comm_monoid_add(code_code_numeral) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(A,code_code_numeral,B,f697(A,code_code_numeral,B,C))),zero_zero(code_code_numeral))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),B),C)),zero_zero(code_code_numeral))). [resolve(5704,b,5686,a)]. 72.89/72.76 Derived: -comm_monoid_add(quickcheck_code_int) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(A,quickcheck_code_int,B,f697(A,quickcheck_code_int,B,C))),zero_zero(quickcheck_code_int))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),B),C)),zero_zero(quickcheck_code_int))). [resolve(5704,b,5689,a)]. 72.89/72.76 Derived: -comm_monoid_add(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(A,real,B,f697(A,real,B,C))),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),B),C)),zero_zero(real))). [resolve(5704,b,5690,a)]. 72.89/72.76 Derived: -comm_monoid_add(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(A,int,B,f697(A,int,B,C))),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),B),C)),zero_zero(int))). [resolve(5704,b,5696,a)]. 72.89/72.76 Derived: -comm_monoid_add(nat) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(A,nat,B,f697(A,nat,B,C))),zero_zero(nat))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co1399186613setsum(A,nat),B),C)),zero_zero(nat))). [resolve(5704,b,5699,a)]. 72.89/72.76 Derived: -comm_monoid_add(rat) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(A,rat,B,f697(A,rat,B,C))),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),B),C)),zero_zero(rat))). [resolve(5704,b,5702,a)]. 72.89/72.76 5705 number_semiring(real) # label(arity_RealDef_Oreal___Int_Onumber__semiring) # label(axiom). [assumption]. 72.89/72.76 5706 -number_semiring(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(int,A,number_number_of(A),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),B) # label(fact_546_semiring__mult__2) # label(axiom). [clausify(71)]. 73.03/72.96 5707 -number_semiring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) | hAPP(int,A,number_number_of(A),hAPP(int,int,succ,B)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(int,A,number_number_of(A),B)),one_one(A)) # label(fact_270_semiring__number__of__add__1) # label(axiom). [clausify(851)]. 73.03/72.96 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | hAPP(int,real,number_number_of(real),hAPP(int,int,succ,A)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,number_number_of(real),A)),one_one(real)). [resolve(5705,a,5707,a)]. 73.03/72.96 5708 -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)) | hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(int,A,number_number_of(A),B)),hAPP(int,A,number_number_of(A),C)) # label(fact_422_semiring__mult__number__of) # label(axiom). [clausify(1001)]. 73.03/72.96 5709 -number_semiring(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(int,A,number_number_of(A),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),B) # label(fact_544_semiring__mult__2__right) # label(axiom). [clausify(1058)]. 73.03/72.96 5710 -number_semiring(A) | hAPP(nat,A,semiring_1_of_nat(A),B) = hAPP(int,A,number_number_of(A),hAPP(nat,int,semiring_1_of_nat(int),B)) # label(fact_97_number__of__int) # label(axiom). [clausify(1130)]. 73.03/72.96 Derived: hAPP(nat,real,semiring_1_of_nat(real),A) = hAPP(int,real,number_number_of(real),hAPP(nat,int,semiring_1_of_nat(int),A)). [resolve(5710,a,5705,a)]. 73.03/72.96 5711 -number_semiring(A) | one_one(A) = hAPP(int,A,number_number_of(A),hAPP(int,int,bit1,pls)) # label(fact_104_semiring__numeral__1__eq__1) # label(axiom). [clausify(1634)]. 73.03/72.96 5712 number_semiring(rat) # label(arity_Rat_Orat___Int_Onumber__semiring) # label(axiom). [assumption]. 73.03/72.96 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | hAPP(int,rat,number_number_of(rat),hAPP(int,int,succ,A)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,number_number_of(rat),A)),one_one(rat)). [resolve(5712,a,5707,a)]. 73.03/72.96 Derived: hAPP(nat,rat,semiring_1_of_nat(rat),A) = hAPP(int,rat,number_number_of(rat),hAPP(nat,int,semiring_1_of_nat(int),A)). [resolve(5712,a,5710,a)]. 73.03/72.96 5713 -number_semiring(A) | zero_zero(A) = hAPP(int,A,number_number_of(A),pls) # label(fact_84_semiring__numeral__0__eq__0) # label(axiom). [clausify(1777)]. 73.03/72.96 5714 -number_semiring(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(int,A,number_number_of(A),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B)),C)) # label(fact_575_power2__sum) # label(axiom). [clausify(1788)]. 73.03/72.96 Derived: hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A)),B)). [resolve(5714,a,5705,a)]. 73.75/73.64 Derived: hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(int,rat,number_number_of(rat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A)),B)). [resolve(5714,a,5712,a)]. 73.75/73.64 5715 -number_semiring(A) | hAPP(int,A,number_number_of(A),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)) # label(fact_9_semiring__one__add__one__is__two) # label(axiom). [clausify(2393)]. 73.75/73.64 5716 number_semiring(nat) # label(arity_Nat_Onat___Int_Onumber__semiring) # label(axiom). [assumption]. 73.75/73.64 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | hAPP(int,nat,number_number_of(nat),hAPP(int,int,succ,A)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),A)),one_one(nat)). [resolve(5716,a,5707,a)]. 73.75/73.64 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)) | hAPP(int,nat,number_number_of(nat),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),A)),hAPP(int,nat,number_number_of(nat),B)). [resolve(5716,a,5708,a)]. 73.75/73.64 Derived: hAPP(nat,nat,semiring_1_of_nat(nat),A) = hAPP(int,nat,number_number_of(nat),hAPP(nat,int,semiring_1_of_nat(int),A)). [resolve(5716,a,5710,a)]. 73.75/73.64 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A)),B)). [resolve(5716,a,5714,a)]. 73.75/73.64 Derived: hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)). [resolve(5716,a,5715,a)]. 73.75/73.64 5717 number_semiring(int) # label(arity_Int_Oint___Int_Onumber__semiring) # label(axiom). [assumption]. 73.75/73.64 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | hAPP(int,int,number_number_of(int),hAPP(int,int,succ,A)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),A)),one_one(int)). [resolve(5717,a,5707,a)]. 73.75/73.64 Derived: hAPP(nat,int,semiring_1_of_nat(int),A) = hAPP(int,int,number_number_of(int),hAPP(nat,int,semiring_1_of_nat(int),A)). [resolve(5717,a,5710,a)]. 73.75/73.64 Derived: hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A)),B)). [resolve(5717,a,5714,a)]. 77.14/76.99 5718 number_semiring(complex) # label(arity_Complex_Ocomplex___Int_Onumber__semiring) # label(axiom). [assumption]. 77.14/76.99 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | hAPP(int,complex,number_number_of(complex),hAPP(int,int,succ,A)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(int,complex,number_number_of(complex),A)),one_one(complex)). [resolve(5718,a,5707,a)]. 77.14/76.99 Derived: hAPP(nat,complex,semiring_1_of_nat(complex),A) = hAPP(int,complex,number_number_of(complex),hAPP(nat,int,semiring_1_of_nat(int),A)). [resolve(5718,a,5710,a)]. 77.14/76.99 Derived: hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A)),B)). [resolve(5718,a,5714,a)]. 77.14/76.99 5719 -number_semiring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) | hAPP(int,A,number_number_of(A),hAPP(int,int,succ,B)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),one_one(A)),hAPP(int,A,number_number_of(A),B)) # label(fact_271_semiring__1__add__number__of) # label(axiom). [clausify(3876)]. 77.14/76.99 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | hAPP(int,nat,number_number_of(nat),hAPP(int,int,succ,A)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),hAPP(int,nat,number_number_of(nat),A)). [resolve(5719,a,5716,a)]. 77.14/76.99 5720 -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)) | hAPP(int,A,number_number_of(A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(int,A,number_number_of(A),B)),hAPP(int,A,number_number_of(A),C)) # label(fact_305_semiring__add__number__of) # label(axiom). [clausify(4377)]. 77.14/76.99 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)) | hAPP(int,nat,number_number_of(nat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),A)),hAPP(int,nat,number_number_of(nat),B)). [resolve(5720,a,5716,a)]. 77.14/76.99 5721 -ordere453448008miring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C))) # label(fact_426_mult__nonneg__nonneg) # label(axiom). [clausify(1790)]. 77.14/76.99 5722 ordere453448008miring(nat) # label(arity_Nat_Onat___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 77.14/76.99 5723 ordere453448008miring(rat) # label(arity_Rat_Orat___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 77.14/76.99 5724 ordere453448008miring(real) # label(arity_RealDef_Oreal___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 77.14/76.99 5725 ordere453448008miring(int) # label(arity_Int_Oint___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 77.94/77.86 5726 -ordere453448008miring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),B)),zero_zero(A))) # label(fact_428_mult__nonneg__nonpos2) # label(axiom). [clausify(2394)]. 77.94/77.86 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),zero_zero(nat))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),A)),zero_zero(nat))). [resolve(5726,a,5722,a)]. 77.94/77.86 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),A)),zero_zero(rat))). [resolve(5726,a,5723,a)]. 77.94/77.86 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),A)),zero_zero(real))). [resolve(5726,a,5724,a)]. 77.94/77.86 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),A)),zero_zero(int))). [resolve(5726,a,5725,a)]. 77.94/77.86 5727 ordere453448008miring(quickcheck_code_int) # label(arity_Quickcheck__Narrowing_Ocode__int___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 77.94/77.86 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),A)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B))). [resolve(5727,a,5721,a)]. 77.94/77.86 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),A)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),B),zero_zero(quickcheck_code_int))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),B),A)),zero_zero(quickcheck_code_int))). [resolve(5727,a,5726,a)]. 77.94/77.86 5728 -ordere453448008miring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),B)),zero_zero(A))) # label(fact_439_split__mult__neg__le) # label(axiom). [clausify(3574)]. 77.94/77.86 5729 -ordere453448008miring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),zero_zero(A))) # label(fact_439_split__mult__neg__le) # label(axiom). [clausify(3574)]. 78.63/78.47 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),zero_zero(nat))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),zero_zero(nat))). [resolve(5729,a,5722,a)]. 78.63/78.47 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),zero_zero(rat))). [resolve(5729,a,5723,a)]. 78.63/78.47 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),zero_zero(real))). [resolve(5729,a,5724,a)]. 78.63/78.47 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),zero_zero(int))). [resolve(5729,a,5725,a)]. 78.63/78.47 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),A)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),B),zero_zero(quickcheck_code_int))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B)),zero_zero(quickcheck_code_int))). [resolve(5729,a,5727,a)]. 78.63/78.47 5730 -ordere453448008miring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),zero_zero(A))) # label(fact_429_mult__nonpos__nonneg) # label(axiom). [clausify(3713)]. 78.63/78.47 5731 ordere453448008miring(code_code_numeral) # label(arity_Code__Numeral_Ocode__numeral___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 78.63/78.47 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),A)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B))). [resolve(5731,a,5721,a)]. 78.63/78.47 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),A)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),B),zero_zero(code_code_numeral))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),B),A)),zero_zero(code_code_numeral))). [resolve(5731,a,5726,a)]. 78.63/78.47 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),A)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),B),zero_zero(code_code_numeral))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B)),zero_zero(code_code_numeral))). [resolve(5731,a,5729,a)]. 80.76/80.61 5732 -ordere453448008miring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),zero_zero(A))) # label(fact_427_mult__nonneg__nonpos) # label(axiom). [clausify(4900)]. 80.76/80.61 5733 -comm_semiring_1(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | zero_zero(A) != hAPP(B,A,D,E) | -hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),E),C)) | zero_zero(A) = hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),D),C) # label(fact_3976_setprod__zero) # label(axiom). [clausify(79)]. 80.76/80.61 5734 comm_semiring_1(code_code_numeral) # label(arity_Code__Numeral_Ocode__numeral___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 80.76/80.61 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | zero_zero(code_code_numeral) != hAPP(A,code_code_numeral,C,D) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),D),B)) | zero_zero(code_code_numeral) = hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),C),B). [resolve(5733,a,5734,a)]. 80.76/80.61 5735 -comm_semiring_1(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),one_one(A))),C) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)) # label(fact_463_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom). [clausify(112)]. 80.76/80.61 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),one_one(code_code_numeral))),B) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),B),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B)). [resolve(5735,a,5734,a)]. 80.76/80.61 5736 -comm_semiring_1(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)) # label(fact_382_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom). [clausify(213)]. 80.76/80.61 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),B),C)) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),C)). [resolve(5736,a,5734,a)]. 80.76/80.61 5737 comm_semiring_1(int) # label(arity_Int_Oint___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 80.76/80.61 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | zero_zero(int) != hAPP(A,int,C,D) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),D),B)) | zero_zero(int) = hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),C),B). [resolve(5737,a,5733,a)]. 80.77/80.69 Derived: hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),one_one(int))),B) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)). [resolve(5737,a,5735,a)]. 80.77/80.69 5738 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(nat,A,hAPP(A,fun(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,hAPP(A,fun(nat,A),power_power(A),B),E)),D)) # label(fact_748_power__le__dvd) # label(axiom). [clausify(356)]. 80.77/80.69 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B)),C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),D),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),D)),C)). [resolve(5738,a,5734,a)]. 80.77/80.69 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,hAPP(int,fun(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,hAPP(int,fun(nat,int),power_power(int),A),D)),C)). [resolve(5738,a,5737,a)]. 80.77/80.69 5739 -comm_semiring_1(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(nat,nat,suc,C)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C)) # label(fact_1367_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) # label(axiom). [clausify(440)]. 80.77/80.69 Derived: hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),hAPP(nat,nat,suc,B)) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B)). [resolve(5739,a,5734,a)]. 80.77/80.69 Derived: hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),hAPP(nat,nat,suc,B)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),B)). [resolve(5739,a,5737,a)]. 80.77/80.69 5740 -comm_semiring_1(A) | hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),B) # label(fact_301_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom). [clausify(450)]. 80.77/80.69 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),B) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),B),A). [resolve(5740,a,5734,a)]. 80.77/80.69 5741 -comm_semiring_1(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),zero_zero(A)) # label(fact_319_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom). [clausify(589)]. 80.77/80.69 Derived: ti(code_code_numeral,A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),zero_zero(code_code_numeral)). [resolve(5741,a,5734,a)]. 80.77/80.69 5742 -comm_semiring_1(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),one_one(A)) # label(fact_391_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom). [clausify(755)]. 80.77/80.69 5743 -comm_semiring_1(A) | one_one(A) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),zero_zero(nat)) # label(fact_196_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom). [clausify(805)]. 80.77/80.69 Derived: one_one(code_code_numeral) = hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),zero_zero(nat)). [resolve(5743,a,5734,a)]. 80.88/80.74 Derived: one_one(int) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),zero_zero(nat)). [resolve(5743,a,5737,a)]. 80.88/80.74 5744 -comm_semiring_1(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),B) # label(fact_548_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom). [clausify(882)]. 80.88/80.74 Derived: hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),A). [resolve(5744,a,5734,a)]. 80.88/80.74 Derived: hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),A). [resolve(5744,a,5737,a)]. 80.88/80.74 5745 -comm_semiring_1(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D)) # label(fact_360_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom). [clausify(893)]. 80.88/80.74 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B)),C) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),B),C)). [resolve(5745,a,5734,a)]. 80.88/80.74 Derived: hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),C) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C)). [resolve(5745,a,5737,a)]. 80.88/80.74 5746 -comm_semiring_1(A) | hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),D),E)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),E)) # label(fact_293_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom). [clausify(1037)]. 80.88/80.74 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),C),D)) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),C)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),B),D)). [resolve(5746,a,5734,a)]. 80.88/80.74 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),C),D)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),C)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),D)). [resolve(5746,a,5737,a)]. 80.88/80.74 5747 -comm_semiring_1(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),B))) # label(fact_728_dvd__triv__right) # label(axiom). [clausify(1044)]. 80.97/80.86 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),B),A))). [resolve(5747,a,5734,a)]. 80.97/80.86 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),A))). [resolve(5747,a,5737,a)]. 80.97/80.86 5748 -comm_semiring_1(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),one_one(A)),B)) # label(fact_731_one__dvd) # label(axiom). [clausify(1210)]. 80.97/80.86 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),one_one(code_code_numeral)),A)). [resolve(5748,a,5734,a)]. 80.97/80.86 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),one_one(int)),A)). [resolve(5748,a,5737,a)]. 80.97/80.86 5749 -comm_semiring_1(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A))),B) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),B) # label(fact_462_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom). [clausify(1372)]. 80.97/80.86 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),one_one(code_code_numeral)),one_one(code_code_numeral))),A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),A). [resolve(5749,a,5734,a)]. 80.97/80.86 Derived: hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),one_one(int))),A) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),A). [resolve(5749,a,5737,a)]. 80.97/80.86 5750 -comm_semiring_1(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),C))) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C))) # label(fact_1504_comm__semiring__1__class_Onormalizing__semiring__rules_I37_J) # label(axiom). [clausify(1472)]. 80.97/80.86 Derived: hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B))). [resolve(5750,a,5734,a)]. 80.97/80.86 Derived: hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),B)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),B))). [resolve(5750,a,5737,a)]. 80.97/80.86 5751 -comm_semiring_1(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),zero_zero(A)),B) # label(fact_323_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom). [clausify(1517)]. 80.97/80.86 Derived: ti(code_code_numeral,A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),zero_zero(code_code_numeral)),A). [resolve(5751,a,5734,a)]. 81.08/80.94 5752 -comm_semiring_1(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),E)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),E))) # label(fact_364_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom). [clausify(1539)]. 81.08/80.94 5753 -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,hAPP(A,fun(nat,A),power_power(A),D),B)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),D),C))) # label(fact_750_le__imp__power__dvd) # label(axiom). [clausify(1640)]. 81.08/80.94 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),C),A)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),C),B))). [resolve(5753,a,5734,a)]. 81.08/80.94 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,hAPP(int,fun(nat,int),power_power(int),C),A)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),C),B))). [resolve(5753,a,5737,a)]. 81.08/80.94 5754 -comm_semiring_1(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),E)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),E)) # label(fact_366_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom). [clausify(1691)]. 81.08/80.94 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),C),D)) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),C)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),B),D)). [resolve(5754,a,5734,a)]. 81.08/80.94 Derived: hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),D)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),C)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),D)). [resolve(5754,a,5737,a)]. 81.08/80.94 5755 -comm_semiring_1(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(nat,A,semiring_1_of_nat(A),B)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(nat,A,semiring_1_of_nat(A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),B),one_one(nat)))),D)))) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A))),hAPP(fun(nat,bool),A,hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(A,A),fun(fun(nat,A),fun(nat,A)),combb(A,A,nat),hAPP(A,fun(A,A),plus_plus(A),C)),hAPP(A,fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(A,fun(nat,A)),combc(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),semiring_1_of_nat(A))),D))),hAPP(nat,fun(nat,bool),ord_lessThan(nat),B))) # label(fact_4351_arith__series__general) # label(axiom). [clausify(1826)]. 81.08/80.98 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),A)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),B),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),B),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A),one_one(nat)))),C)))) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),one_one(code_code_numeral)),one_one(code_code_numeral))),hAPP(fun(nat,bool),code_code_numeral,hAPP(fun(nat,code_code_numeral),fun(fun(nat,bool),code_code_numeral),big_co1399186613setsum(nat,code_code_numeral),hAPP(fun(nat,code_code_numeral),fun(nat,code_code_numeral),hAPP(fun(code_code_numeral,code_code_numeral),fun(fun(nat,code_code_numeral),fun(nat,code_code_numeral)),combb(code_code_numeral,code_code_numeral,nat),hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),B)),hAPP(code_code_numeral,fun(nat,code_code_numeral),hAPP(fun(nat,fun(code_code_numeral,code_code_numeral)),fun(code_code_numeral,fun(nat,code_code_numeral)),combc(nat,code_code_numeral,code_code_numeral),hAPP(fun(nat,code_code_numeral),fun(nat,fun(code_code_numeral,code_code_numeral)),hAPP(fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)),fun(fun(nat,code_code_numeral),fun(nat,fun(code_code_numeral,code_code_numeral))),combb(code_code_numeral,fun(code_code_numeral,code_code_numeral),nat),times_times(code_code_numeral)),semiring_1_of_nat(code_code_numeral))),C))),hAPP(nat,fun(nat,bool),ord_lessThan(nat),A))). [resolve(5755,a,5734,a)]. 81.08/80.98 Derived: hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),A)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A),one_one(nat)))),C)))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),one_one(int))),hAPP(fun(nat,bool),int,hAPP(fun(nat,int),fun(fun(nat,bool),int),big_co1399186613setsum(nat,int),hAPP(fun(nat,int),fun(nat,int),hAPP(fun(int,int),fun(fun(nat,int),fun(nat,int)),combb(int,int,nat),hAPP(int,fun(int,int),plus_plus(int),B)),hAPP(int,fun(nat,int),hAPP(fun(nat,fun(int,int)),fun(int,fun(nat,int)),combc(nat,int,int),hAPP(fun(nat,int),fun(nat,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(nat,int),fun(nat,fun(int,int))),combb(int,fun(int,int),nat),times_times(int)),semiring_1_of_nat(int))),C))),hAPP(nat,fun(nat,bool),ord_lessThan(nat),A))). [resolve(5755,a,5737,a)]. 81.08/80.98 5756 -comm_semiring_1(A) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),zero_zero(A)) # label(fact_379_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom). [clausify(1928)]. 81.08/80.98 Derived: zero_zero(code_code_numeral) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),zero_zero(code_code_numeral)). [resolve(5756,a,5734,a)]. 81.08/80.98 Derived: zero_zero(int) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),zero_zero(int)). [resolve(5756,a,5737,a)]. 81.08/80.98 5757 -comm_semiring_1(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D)) # label(fact_384_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom). [clausify(1962)]. 81.20/81.11 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),B)),C) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),C)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),B),C)). [resolve(5757,a,5734,a)]. 81.20/81.11 Derived: hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)),C) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),C)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C)). [resolve(5757,a,5737,a)]. 81.20/81.11 5758 -comm_semiring_1(A) | hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),D)) # label(fact_298_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom). [clausify(2033)]. 81.20/81.11 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),B)),C) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),B),C)). [resolve(5758,a,5734,a)]. 81.20/81.11 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)),C) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),C)). [resolve(5758,a,5737,a)]. 81.20/81.11 5759 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),D)) # label(fact_722_dvd__mult__right) # label(axiom). [clausify(2163)]. 81.20/81.11 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B)),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),B),C)). [resolve(5759,a,5734,a)]. 81.20/81.11 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),C)). [resolve(5759,a,5737,a)]. 81.20/81.11 5760 -comm_semiring_1(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(nat,nat,suc,C)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C)),B) # label(fact_1369_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) # label(axiom). [clausify(2304)]. 81.20/81.11 Derived: hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),hAPP(nat,nat,suc,B)) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B)),A). [resolve(5760,a,5734,a)]. 81.29/81.16 Derived: hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),hAPP(nat,nat,suc,B)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),B)),A). [resolve(5760,a,5737,a)]. 81.29/81.16 5761 -comm_semiring_1(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)),C) # label(fact_363_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom). [clausify(2306)]. 81.29/81.16 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B)),C) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),C)),B). [resolve(5761,a,5734,a)]. 81.29/81.16 Derived: hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),C) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),C)),B). [resolve(5761,a,5737,a)]. 81.29/81.16 5762 -comm_semiring_1(A) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f393(B,A,C,D,E)),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),D),E))) # label(fact_4798_dvd__setsum) # label(axiom). [clausify(2330)]. 81.29/81.16 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f393(A,code_code_numeral,B,C,D)),D)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),B),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),C),D))). [resolve(5762,a,5734,a)]. 81.29/81.16 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f393(A,int,B,C,D)),D)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),C),D))). [resolve(5762,a,5737,a)]. 81.29/81.16 5763 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(C,A,D,f393(C,A,B,D,E)))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(fun(C,bool),A,hAPP(fun(C,A),fun(fun(C,bool),A),big_co1399186613setsum(C,A),D),E))) # label(fact_4798_dvd__setsum) # label(axiom). [clausify(2330)]. 81.29/81.16 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),hAPP(B,code_code_numeral,C,f393(B,code_code_numeral,A,C,D)))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),hAPP(fun(B,bool),code_code_numeral,hAPP(fun(B,code_code_numeral),fun(fun(B,bool),code_code_numeral),big_co1399186613setsum(B,code_code_numeral),C),D))). [resolve(5763,a,5734,a)]. 81.29/81.16 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(B,int,C,f393(B,int,A,C,D)))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(fun(B,bool),int,hAPP(fun(B,int),fun(fun(B,bool),int),big_co1399186613setsum(B,int),C),D))). [resolve(5763,a,5737,a)]. 81.29/81.16 5764 comm_semiring_1(rat) # label(arity_Rat_Orat___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 81.29/81.16 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | zero_zero(rat) != hAPP(A,rat,C,D) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),D),B)) | zero_zero(rat) = hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),C),B). [resolve(5764,a,5733,a)]. 81.29/81.16 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),one_one(rat))),B) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)). [resolve(5764,a,5735,a)]. 81.29/81.17 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),B),C)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),C)). [resolve(5764,a,5736,a)]. 81.29/81.17 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B)),C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),D),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),D)),C)). [resolve(5764,a,5738,a)]. 81.29/81.17 Derived: hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(nat,nat,suc,B)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B)). [resolve(5764,a,5739,a)]. 81.29/81.17 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),B),A). [resolve(5764,a,5740,a)]. 81.29/81.17 Derived: ti(rat,A) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),zero_zero(rat)). [resolve(5764,a,5741,a)]. 81.29/81.17 Derived: one_one(rat) = hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),zero_zero(nat)). [resolve(5764,a,5743,a)]. 81.29/81.17 Derived: hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),A). [resolve(5764,a,5744,a)]. 81.29/81.17 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),C) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),C)). [resolve(5764,a,5745,a)]. 81.29/81.17 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),C),D)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),C)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),B),D)). [resolve(5764,a,5746,a)]. 81.29/81.17 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),A))). [resolve(5764,a,5747,a)]. 81.29/81.17 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),one_one(rat)),A)). [resolve(5764,a,5748,a)]. 81.29/81.17 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),one_one(rat)),one_one(rat))),A) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),A). [resolve(5764,a,5749,a)]. 81.29/81.17 Derived: hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B))). [resolve(5764,a,5750,a)]. 81.29/81.17 Derived: ti(rat,A) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),zero_zero(rat)),A). [resolve(5764,a,5751,a)]. 81.29/81.17 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),C),A)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),C),B))). [resolve(5764,a,5753,a)]. 81.29/81.17 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),D)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),C)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),D)). [resolve(5764,a,5754,a)]. 81.29/81.17 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(nat,rat,semiring_1_of_nat(rat),A)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(nat,rat,semiring_1_of_nat(rat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A),one_one(nat)))),C)))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),one_one(rat)),one_one(rat))),hAPP(fun(nat,bool),rat,hAPP(fun(nat,rat),fun(fun(nat,bool),rat),big_co1399186613setsum(nat,rat),hAPP(fun(nat,rat),fun(nat,rat),hAPP(fun(rat,rat),fun(fun(nat,rat),fun(nat,rat)),combb(rat,rat,nat),hAPP(rat,fun(rat,rat),plus_plus(rat),B)),hAPP(rat,fun(nat,rat),hAPP(fun(nat,fun(rat,rat)),fun(rat,fun(nat,rat)),combc(nat,rat,rat),hAPP(fun(nat,rat),fun(nat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,fun(rat,rat))),combb(rat,fun(rat,rat),nat),times_times(rat)),semiring_1_of_nat(rat))),C))),hAPP(nat,fun(nat,bool),ord_lessThan(nat),A))). [resolve(5764,a,5755,a)]. 81.37/81.26 Derived: zero_zero(rat) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),zero_zero(rat)). [resolve(5764,a,5756,a)]. 81.37/81.26 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B)),C) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),C)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),C)). [resolve(5764,a,5757,a)]. 81.37/81.26 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B)),C) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),B),C)). [resolve(5764,a,5758,a)]. 81.37/81.26 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),B),C)). [resolve(5764,a,5759,a)]. 81.37/81.26 Derived: hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(nat,nat,suc,B)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B)),A). [resolve(5764,a,5760,a)]. 81.37/81.26 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),C) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),C)),B). [resolve(5764,a,5761,a)]. 81.37/81.26 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f393(A,rat,B,C,D)),D)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),B),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),C),D))). [resolve(5764,a,5762,a)]. 81.37/81.26 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),A),hAPP(B,rat,C,f393(B,rat,A,C,D)))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),A),hAPP(fun(B,bool),rat,hAPP(fun(B,rat),fun(fun(B,bool),rat),big_co1399186613setsum(B,rat),C),D))). [resolve(5764,a,5763,a)]. 81.37/81.26 5765 -comm_semiring_1(A) | hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),D)) # label(fact_299_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom). [clausify(2472)]. 81.37/81.26 5766 -comm_semiring_1(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),D)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),D)) # label(fact_395_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom). [clausify(2513)]. 81.37/81.26 5767 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),E))) # label(fact_725_mult__dvd__mono) # label(axiom). [clausify(2590)]. 81.46/81.33 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),C),D)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),C)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),B),D))). [resolve(5767,a,5734,a)]. 81.46/81.33 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),C),D)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),C)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),D))). [resolve(5767,a,5737,a)]. 81.46/81.33 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),C),D)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),C)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),D))). [resolve(5767,a,5764,a)]. 81.46/81.33 5768 -comm_semiring_1(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),B)) # label(fact_713_dvd__refl) # label(axiom). [clausify(2617)]. 81.46/81.33 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),A)). [resolve(5768,a,5734,a)]. 81.46/81.33 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),A)). [resolve(5768,a,5737,a)]. 81.46/81.33 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),A),A)). [resolve(5768,a,5764,a)]. 81.46/81.33 5769 -comm_semiring_1(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | -hBOOL(hAPP(fun(B,bool),bool,hAPP(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),D),C)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f462(B,A,E,F,D,C)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),E),D)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),F),C))) # label(fact_3992_setprod__dvd__setprod__subset2) # label(axiom). [clausify(2752)]. 81.46/81.33 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f462(A,code_code_numeral,D,E,C,B)),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),D),C)),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),E),B))). [resolve(5769,a,5734,a)]. 81.46/81.33 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f462(A,int,D,E,C,B)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),D),C)),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),E),B))). [resolve(5769,a,5737,a)]. 81.46/81.33 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f462(A,rat,D,E,C,B)),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),D),C)),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),E),B))). [resolve(5769,a,5764,a)]. 81.46/81.34 5770 -comm_semiring_1(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | -hBOOL(hAPP(fun(B,bool),bool,hAPP(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),D),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(B,A,E,f462(B,A,E,F,D,C))),hAPP(B,A,F,f462(B,A,E,F,D,C)))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),E),D)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),F),C))) # label(fact_3992_setprod__dvd__setprod__subset2) # label(axiom). [clausify(2752)]. 81.46/81.34 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),hAPP(A,code_code_numeral,D,f462(A,code_code_numeral,D,E,C,B))),hAPP(A,code_code_numeral,E,f462(A,code_code_numeral,D,E,C,B)))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),D),C)),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),E),B))). [resolve(5770,a,5734,a)]. 81.46/81.34 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(A,int,D,f462(A,int,D,E,C,B))),hAPP(A,int,E,f462(A,int,D,E,C,B)))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),D),C)),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),E),B))). [resolve(5770,a,5737,a)]. 81.46/81.34 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),hAPP(A,rat,D,f462(A,rat,D,E,C,B))),hAPP(A,rat,E,f462(A,rat,D,E,C,B)))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),D),C)),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),E),B))). [resolve(5770,a,5764,a)]. 81.46/81.34 5771 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),D))) # label(fact_730_dvd__add) # label(axiom). [clausify(2758)]. 81.46/81.34 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),B),C))). [resolve(5771,a,5734,a)]. 81.46/81.34 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),C))). [resolve(5771,a,5737,a)]. 81.46/81.34 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),A),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),B),C))). [resolve(5771,a,5764,a)]. 81.55/81.39 5772 -comm_semiring_1(A) | hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),D)) # label(fact_300_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom). [clausify(2787)]. 81.55/81.39 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),B),C)) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),B),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),C)). [resolve(5772,a,5734,a)]. 81.55/81.39 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),C)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),C)). [resolve(5772,a,5737,a)]. 81.55/81.39 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),B),C)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),C)). [resolve(5772,a,5764,a)]. 81.55/81.39 5773 -comm_semiring_1(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),zero_zero(A))) # label(fact_706_dvd__0__right) # label(axiom). [clausify(2811)]. 81.55/81.39 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),zero_zero(code_code_numeral))). [resolve(5773,a,5734,a)]. 81.55/81.39 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),zero_zero(int))). [resolve(5773,a,5737,a)]. 81.55/81.39 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),A),zero_zero(rat))). [resolve(5773,a,5764,a)]. 81.55/81.39 5774 -comm_semiring_1(A) | ti(A,B) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),one_one(nat)) # label(fact_335_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom). [clausify(2827)]. 81.55/81.39 Derived: ti(code_code_numeral,A) = hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),one_one(nat)). [resolve(5774,a,5734,a)]. 81.55/81.39 Derived: ti(int,A) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),one_one(nat)). [resolve(5774,a,5737,a)]. 81.55/81.39 Derived: ti(rat,A) = hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),one_one(nat)). [resolve(5774,a,5764,a)]. 81.55/81.39 5775 comm_semiring_1(complex) # label(arity_Complex_Ocomplex___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 81.55/81.39 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | zero_zero(complex) != hAPP(A,complex,C,D) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),D),B)) | zero_zero(complex) = hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),C),B). [resolve(5775,a,5733,a)]. 81.55/81.39 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),one_one(complex))),B) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),B),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)). [resolve(5775,a,5735,a)]. 81.55/81.39 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),B)),C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),D),B)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),D)),C)). [resolve(5775,a,5738,a)]. 81.55/81.39 Derived: hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),hAPP(nat,nat,suc,B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),B)). [resolve(5775,a,5739,a)]. 81.57/81.41 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),B) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),B),A). [resolve(5775,a,5740,a)]. 81.57/81.41 Derived: ti(complex,A) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),zero_zero(complex)). [resolve(5775,a,5741,a)]. 81.57/81.41 Derived: one_one(complex) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),zero_zero(nat)). [resolve(5775,a,5743,a)]. 81.57/81.41 Derived: hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),A). [resolve(5775,a,5744,a)]. 81.57/81.41 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)),C) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),B),C)). [resolve(5775,a,5745,a)]. 81.57/81.41 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),B)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),C),D)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),C)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),B),D)). [resolve(5775,a,5746,a)]. 81.57/81.41 Derived: hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),A),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),B),A))). [resolve(5775,a,5747,a)]. 81.57/81.41 Derived: hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),one_one(complex)),A)). [resolve(5775,a,5748,a)]. 81.57/81.41 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),one_one(complex)),one_one(complex))),A) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),A). [resolve(5775,a,5749,a)]. 81.57/81.41 Derived: hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),B)),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),B))). [resolve(5775,a,5750,a)]. 81.57/81.41 Derived: ti(complex,A) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),zero_zero(complex)),A). [resolve(5775,a,5751,a)]. 81.57/81.41 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),C),A)),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),C),B))). [resolve(5775,a,5753,a)]. 81.57/81.41 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),D)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),C)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),B),D)). [resolve(5775,a,5754,a)]. 81.57/81.41 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(nat,complex,semiring_1_of_nat(complex),A)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),B),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),B),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(nat,complex,semiring_1_of_nat(complex),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A),one_one(nat)))),C)))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),one_one(complex)),one_one(complex))),hAPP(fun(nat,bool),complex,hAPP(fun(nat,complex),fun(fun(nat,bool),complex),big_co1399186613setsum(nat,complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(complex,complex),fun(fun(nat,complex),fun(nat,complex)),combb(complex,complex,nat),hAPP(complex,fun(complex,complex),plus_plus(complex),B)),hAPP(complex,fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(complex,fun(nat,complex)),combc(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),semiring_1_of_nat(complex))),C))),hAPP(nat,fun(nat,bool),ord_lessThan(nat),A))). [resolve(5775,a,5755,a)]. 81.59/81.42 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),B)),C) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),B),C)). [resolve(5775,a,5758,a)]. 81.59/81.42 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)),C)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),B),C)). [resolve(5775,a,5759,a)]. 81.59/81.42 Derived: hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),hAPP(nat,nat,suc,B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),B)),A). [resolve(5775,a,5760,a)]. 81.59/81.42 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)),C) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),C)),B). [resolve(5775,a,5761,a)]. 81.59/81.42 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f393(A,complex,B,C,D)),D)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),B),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co1399186613setsum(A,complex),C),D))). [resolve(5775,a,5762,a)]. 81.59/81.42 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),A),hAPP(B,complex,C,f393(B,complex,A,C,D)))) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),A),hAPP(fun(B,bool),complex,hAPP(fun(B,complex),fun(fun(B,bool),complex),big_co1399186613setsum(B,complex),C),D))). [resolve(5775,a,5763,a)]. 81.59/81.42 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),A),B)) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),C),D)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),C)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),B),D))). [resolve(5775,a,5767,a)]. 81.59/81.42 Derived: hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),A),A)). [resolve(5775,a,5768,a)]. 81.59/81.42 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f462(A,complex,D,E,C,B)),C)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),D),C)),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),E),B))). [resolve(5775,a,5769,a)]. 81.67/81.51 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),hAPP(A,complex,D,f462(A,complex,D,E,C,B))),hAPP(A,complex,E,f462(A,complex,D,E,C,B)))) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),D),C)),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),E),B))). [resolve(5775,a,5770,a)]. 81.67/81.51 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),A),B)) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),A),C)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),A),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),B),C))). [resolve(5775,a,5771,a)]. 81.67/81.51 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),B),C)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),B),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),C)). [resolve(5775,a,5772,a)]. 81.67/81.51 Derived: hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),A),zero_zero(complex))). [resolve(5775,a,5773,a)]. 81.67/81.51 Derived: ti(complex,A) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),one_one(nat)). [resolve(5775,a,5774,a)]. 81.67/81.51 5776 comm_semiring_1(quickcheck_code_int) # label(arity_Quickcheck__Narrowing_Ocode__int___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 81.67/81.51 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | zero_zero(quickcheck_code_int) != hAPP(A,quickcheck_code_int,C,D) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),D),B)) | zero_zero(quickcheck_code_int) = hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),C),B). [resolve(5776,a,5733,a)]. 81.67/81.51 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),one_one(quickcheck_code_int))),B) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),B),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B)). [resolve(5776,a,5735,a)]. 81.67/81.51 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),B),C)) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),C)). [resolve(5776,a,5736,a)]. 81.67/81.51 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B)),C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),D),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),D)),C)). [resolve(5776,a,5738,a)]. 81.67/81.52 Derived: hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),hAPP(nat,nat,suc,B)) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B)). [resolve(5776,a,5739,a)]. 81.67/81.52 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),B) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),B),A). [resolve(5776,a,5740,a)]. 81.67/81.52 Derived: ti(quickcheck_code_int,A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),zero_zero(quickcheck_code_int)). [resolve(5776,a,5741,a)]. 81.67/81.52 Derived: one_one(quickcheck_code_int) = hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),zero_zero(nat)). [resolve(5776,a,5743,a)]. 81.67/81.52 Derived: hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),A). [resolve(5776,a,5744,a)]. 81.67/81.52 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B)),C) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),B),C)). [resolve(5776,a,5745,a)]. 81.67/81.52 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),C),D)) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),C)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),B),D)). [resolve(5776,a,5746,a)]. 81.67/81.52 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),B),A))). [resolve(5776,a,5747,a)]. 81.67/81.52 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),one_one(quickcheck_code_int)),A)). [resolve(5776,a,5748,a)]. 81.67/81.52 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),one_one(quickcheck_code_int)),one_one(quickcheck_code_int))),A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),A). [resolve(5776,a,5749,a)]. 81.67/81.52 Derived: hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B))). [resolve(5776,a,5750,a)]. 81.67/81.52 Derived: ti(quickcheck_code_int,A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),zero_zero(quickcheck_code_int)),A). [resolve(5776,a,5751,a)]. 81.67/81.52 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),C),A)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),C),B))). [resolve(5776,a,5753,a)]. 81.67/81.52 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),C),D)) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),C)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),B),D)). [resolve(5776,a,5754,a)]. 81.67/81.52 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),A)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),B),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),B),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A),one_one(nat)))),C)))) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),one_one(quickcheck_code_int)),one_one(quickcheck_code_int))),hAPP(fun(nat,bool),quickcheck_code_int,hAPP(fun(nat,quickcheck_code_int),fun(fun(nat,bool),quickcheck_code_int),big_co1399186613setsum(nat,quickcheck_code_int),hAPP(fun(nat,quickcheck_code_int),fun(nat,quickcheck_code_int),hAPP(fun(quickcheck_code_int,quickcheck_code_int),fun(fun(nat,quickcheck_code_int),fun(nat,quickcheck_code_int)),combb(quickcheck_code_int,quickcheck_code_int,nat),hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),B)),hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),hAPP(fun(nat,fun(quickcheck_code_int,quickcheck_code_int)),fun(quickcheck_code_int,fun(nat,quickcheck_code_int)),combc(nat,quickcheck_code_int,quickcheck_code_int),hAPP(fun(nat,quickcheck_code_int),fun(nat,fun(quickcheck_code_int,quickcheck_code_int)),hAPP(fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),fun(fun(nat,quickcheck_code_int),fun(nat,fun(quickcheck_code_int,quickcheck_code_int))),combb(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),nat),times_times(quickcheck_code_int)),semiring_1_of_nat(quickcheck_code_int))),C))),hAPP(nat,fun(nat,bool),ord_lessThan(nat),A))). [resolve(5776,a,5755,a)]. 81.67/81.53 Derived: zero_zero(quickcheck_code_int) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),zero_zero(quickcheck_code_int)). [resolve(5776,a,5756,a)]. 81.67/81.53 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),B)),C) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),C)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),B),C)). [resolve(5776,a,5757,a)]. 81.67/81.53 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),B)),C) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),B),C)). [resolve(5776,a,5758,a)]. 81.67/81.53 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B)),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),B),C)). [resolve(5776,a,5759,a)]. 81.67/81.53 Derived: hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),hAPP(nat,nat,suc,B)) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B)),A). [resolve(5776,a,5760,a)]. 81.67/81.53 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B)),C) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),C)),B). [resolve(5776,a,5761,a)]. 81.67/81.54 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f393(A,quickcheck_code_int,B,C,D)),D)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),B),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),C),D))). [resolve(5776,a,5762,a)]. 81.67/81.54 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),hAPP(B,quickcheck_code_int,C,f393(B,quickcheck_code_int,A,C,D)))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),hAPP(fun(B,bool),quickcheck_code_int,hAPP(fun(B,quickcheck_code_int),fun(fun(B,bool),quickcheck_code_int),big_co1399186613setsum(B,quickcheck_code_int),C),D))). [resolve(5776,a,5763,a)]. 81.67/81.54 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),C),D)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),C)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),B),D))). [resolve(5776,a,5767,a)]. 81.67/81.54 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),A)). [resolve(5776,a,5768,a)]. 81.67/81.54 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f462(A,quickcheck_code_int,D,E,C,B)),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),D),C)),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),E),B))). [resolve(5776,a,5769,a)]. 81.67/81.54 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),hAPP(A,quickcheck_code_int,D,f462(A,quickcheck_code_int,D,E,C,B))),hAPP(A,quickcheck_code_int,E,f462(A,quickcheck_code_int,D,E,C,B)))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),D),C)),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),E),B))). [resolve(5776,a,5770,a)]. 81.67/81.54 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),B),C))). [resolve(5776,a,5771,a)]. 81.76/81.59 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),B),C)) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),B),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),C)). [resolve(5776,a,5772,a)]. 81.76/81.59 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),zero_zero(quickcheck_code_int))). [resolve(5776,a,5773,a)]. 81.76/81.59 Derived: ti(quickcheck_code_int,A) = hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),one_one(nat)). [resolve(5776,a,5774,a)]. 81.76/81.59 5777 -comm_semiring_1(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D)) # label(fact_385_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom). [clausify(3017)]. 81.76/81.59 5778 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) # label(fact_723_dvd__mult__left) # label(axiom). [clausify(3046)]. 81.76/81.59 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B)),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),C)). [resolve(5778,a,5734,a)]. 81.76/81.59 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),C)). [resolve(5778,a,5737,a)]. 81.76/81.59 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),A),C)). [resolve(5778,a,5764,a)]. 81.76/81.59 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)),C)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),A),C)). [resolve(5778,a,5775,a)]. 81.76/81.59 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B)),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),C)). [resolve(5778,a,5776,a)]. 81.76/81.59 5779 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D))) # label(fact_727_dvd__mult2) # label(axiom). [clausify(3099)]. 81.76/81.59 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),B),C))). [resolve(5779,a,5734,a)]. 81.85/81.68 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C))). [resolve(5779,a,5737,a)]. 81.85/81.68 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),A),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),C))). [resolve(5779,a,5764,a)]. 81.85/81.68 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),A),B)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),A),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),B),C))). [resolve(5779,a,5775,a)]. 81.85/81.68 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),B),C))). [resolve(5779,a,5776,a)]. 81.85/81.68 5780 -comm_semiring_1(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(nat,A,semiring_1_of_nat(A),B)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(nat,A,semiring_1_of_nat(A),B)),one_one(A))) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A))),hAPP(fun(nat,bool),A,hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),semiring_1_of_nat(A)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),one_one(nat)),B))) # label(fact_4642_gauss__sum) # label(axiom). [clausify(3280)]. 81.85/81.68 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),A)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),A)),one_one(code_code_numeral))) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),one_one(code_code_numeral)),one_one(code_code_numeral))),hAPP(fun(nat,bool),code_code_numeral,hAPP(fun(nat,code_code_numeral),fun(fun(nat,bool),code_code_numeral),big_co1399186613setsum(nat,code_code_numeral),semiring_1_of_nat(code_code_numeral)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),one_one(nat)),A))). [resolve(5780,a,5734,a)]. 81.85/81.68 Derived: hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),A)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,semiring_1_of_nat(int),A)),one_one(int))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),one_one(int))),hAPP(fun(nat,bool),int,hAPP(fun(nat,int),fun(fun(nat,bool),int),big_co1399186613setsum(nat,int),semiring_1_of_nat(int)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),one_one(nat)),A))). [resolve(5780,a,5737,a)]. 81.85/81.68 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(nat,rat,semiring_1_of_nat(rat),A)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(nat,rat,semiring_1_of_nat(rat),A)),one_one(rat))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),one_one(rat)),one_one(rat))),hAPP(fun(nat,bool),rat,hAPP(fun(nat,rat),fun(fun(nat,bool),rat),big_co1399186613setsum(nat,rat),semiring_1_of_nat(rat)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),one_one(nat)),A))). [resolve(5780,a,5764,a)]. 81.85/81.70 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(nat,complex,semiring_1_of_nat(complex),A)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(nat,complex,semiring_1_of_nat(complex),A)),one_one(complex))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),one_one(complex)),one_one(complex))),hAPP(fun(nat,bool),complex,hAPP(fun(nat,complex),fun(fun(nat,bool),complex),big_co1399186613setsum(nat,complex),semiring_1_of_nat(complex)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),one_one(nat)),A))). [resolve(5780,a,5775,a)]. 81.85/81.70 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),A)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),A)),one_one(quickcheck_code_int))) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),one_one(quickcheck_code_int)),one_one(quickcheck_code_int))),hAPP(fun(nat,bool),quickcheck_code_int,hAPP(fun(nat,quickcheck_code_int),fun(fun(nat,bool),quickcheck_code_int),big_co1399186613setsum(nat,quickcheck_code_int),semiring_1_of_nat(quickcheck_code_int)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),one_one(nat)),A))). [resolve(5780,a,5776,a)]. 81.85/81.70 5781 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),C))) # label(fact_726_dvd__mult) # label(axiom). [clausify(3283)]. 81.85/81.70 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),C),B))). [resolve(5781,a,5734,a)]. 81.85/81.70 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),B))). [resolve(5781,a,5737,a)]. 81.85/81.70 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),A),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),B))). [resolve(5781,a,5764,a)]. 81.85/81.70 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),A),B)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),A),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),B))). [resolve(5781,a,5775,a)]. 81.85/81.70 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),C),B))). [resolve(5781,a,5776,a)]. 81.85/81.70 5782 -comm_semiring_1(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),B) # label(fact_358_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom). [clausify(3307)]. 81.91/81.75 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),B),A). [resolve(5782,a,5734,a)]. 81.91/81.75 Derived: hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),A). [resolve(5782,a,5737,a)]. 81.91/81.75 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),A). [resolve(5782,a,5764,a)]. 81.91/81.75 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),B),A). [resolve(5782,a,5775,a)]. 81.91/81.75 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),B),A). [resolve(5782,a,5776,a)]. 81.91/81.75 5783 -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,hAPP(A,fun(nat,A),power_power(A),B),D)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),D))) # label(fact_732_dvd__power__same) # label(axiom). [clausify(3323)]. 81.91/81.75 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),C)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),B),C))). [resolve(5783,a,5734,a)]. 81.91/81.75 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),C)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),C))). [resolve(5783,a,5737,a)]. 81.91/81.75 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),A),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),C)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),B),C))). [resolve(5783,a,5764,a)]. 81.91/81.75 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),A),B)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),C)),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),B),C))). [resolve(5783,a,5775,a)]. 81.91/81.75 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),C)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),B),C))). [resolve(5783,a,5776,a)]. 81.91/81.75 5784 -comm_semiring_1(A) | ti(A,B) != one_one(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C))) # label(fact_785_dvd__power) # label(axiom). [clausify(3365)]. 81.91/81.75 Derived: ti(code_code_numeral,A) != one_one(code_code_numeral) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B))). [resolve(5784,a,5734,a)]. 81.91/81.75 Derived: ti(int,A) != one_one(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),B))). [resolve(5784,a,5737,a)]. 82.05/81.93 Derived: ti(rat,A) != one_one(rat) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),A),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B))). [resolve(5784,a,5764,a)]. 82.05/81.93 Derived: ti(complex,A) != one_one(complex) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),A),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),B))). [resolve(5784,a,5775,a)]. 82.05/81.93 Derived: ti(quickcheck_code_int,A) != one_one(quickcheck_code_int) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B))). [resolve(5784,a,5776,a)]. 82.05/81.93 5785 -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,hAPP(A,fun(nat,A),power_power(A),C),B))) # label(fact_785_dvd__power) # label(axiom). [clausify(3365)]. 82.05/81.93 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),B),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),B),A))). [resolve(5785,a,5734,a)]. 82.05/81.93 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,hAPP(int,fun(nat,int),power_power(int),B),A))). [resolve(5785,a,5737,a)]. 82.05/81.93 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),B),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),B),A))). [resolve(5785,a,5764,a)]. 82.05/81.93 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),B),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),B),A))). [resolve(5785,a,5775,a)]. 82.05/81.93 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),B),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),B),A))). [resolve(5785,a,5776,a)]. 82.05/81.93 5786 -comm_semiring_1(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C)),D) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),D)) # label(fact_607_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom). [clausify(3480)]. 82.05/81.93 Derived: hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B)),C) = hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C)). [resolve(5786,a,5734,a)]. 82.05/81.93 Derived: hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B)),C) = hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C)). [resolve(5786,a,5764,a)]. 82.05/81.93 Derived: hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),B)),C) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C)). [resolve(5786,a,5775,a)]. 82.05/81.93 Derived: hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B)),C) = hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C)). [resolve(5786,a,5776,a)]. 82.28/82.14 5787 -comm_semiring_1(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),C)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C)) # label(fact_473_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom). [clausify(3708)]. 82.28/82.14 Derived: hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B)) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B)). [resolve(5787,a,5734,a)]. 82.28/82.14 Derived: hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),B)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),B)). [resolve(5787,a,5737,a)]. 82.28/82.14 Derived: hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B)). [resolve(5787,a,5764,a)]. 82.28/82.14 Derived: hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),B)),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),B)). [resolve(5787,a,5775,a)]. 82.28/82.14 Derived: hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B)) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B)). [resolve(5787,a,5776,a)]. 82.28/82.14 5788 -comm_semiring_1(A) | hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),D)),C) # label(fact_296_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom). [clausify(3817)]. 82.28/82.14 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),B)),C) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),C)),B). [resolve(5788,a,5734,a)]. 82.35/82.19 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)),C) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),C)),B). [resolve(5788,a,5737,a)]. 82.35/82.19 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B)),C) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),C)),B). [resolve(5788,a,5764,a)]. 82.35/82.19 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),B)),C) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),C)),B). [resolve(5788,a,5775,a)]. 82.35/82.19 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),B)),C) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),C)),B). [resolve(5788,a,5776,a)]. 82.35/82.19 5789 -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_712_dvd__trans) # label(axiom). [clausify(3871)]. 82.35/82.19 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),B),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),C)). [resolve(5789,a,5734,a)]. 82.35/82.19 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(5789,a,5737,a)]. 82.35/82.19 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),B),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),A),C)). [resolve(5789,a,5764,a)]. 82.35/82.19 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),A),B)) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),B),C)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),A),C)). [resolve(5789,a,5775,a)]. 82.35/82.19 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),B),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),C)). [resolve(5789,a,5776,a)]. 82.35/82.19 5790 -comm_semiring_1(A) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),zero_zero(A)),B) # label(fact_378_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom). [clausify(3898)]. 82.35/82.19 Derived: zero_zero(code_code_numeral) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),zero_zero(code_code_numeral)),A). [resolve(5790,a,5734,a)]. 82.35/82.19 Derived: zero_zero(int) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),zero_zero(int)),A). [resolve(5790,a,5737,a)]. 82.35/82.21 Derived: zero_zero(rat) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),zero_zero(rat)),A). [resolve(5790,a,5764,a)]. 82.35/82.21 Derived: zero_zero(quickcheck_code_int) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),zero_zero(quickcheck_code_int)),A). [resolve(5790,a,5776,a)]. 82.35/82.21 5791 -comm_semiring_1(A) | -no_zero_divisors(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | zero_zero(A) != hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),D),C) | zero_zero(A) = hAPP(B,A,D,f664(B,A,D,C)) # label(fact_3977_setprod__zero__iff) # label(axiom). [clausify(3913)]. 82.35/82.21 Derived: -no_zero_divisors(code_code_numeral) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | zero_zero(code_code_numeral) != hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),C),B) | zero_zero(code_code_numeral) = hAPP(A,code_code_numeral,C,f664(A,code_code_numeral,C,B)). [resolve(5791,a,5734,a)]. 82.35/82.21 Derived: -no_zero_divisors(int) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | zero_zero(int) != hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),C),B) | zero_zero(int) = hAPP(A,int,C,f664(A,int,C,B)). [resolve(5791,a,5737,a)]. 82.35/82.21 Derived: -no_zero_divisors(rat) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | zero_zero(rat) != hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),C),B) | zero_zero(rat) = hAPP(A,rat,C,f664(A,rat,C,B)). [resolve(5791,a,5764,a)]. 82.35/82.21 Derived: -no_zero_divisors(complex) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | zero_zero(complex) != hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),C),B) | zero_zero(complex) = hAPP(A,complex,C,f664(A,complex,C,B)). [resolve(5791,a,5775,a)]. 82.35/82.21 Derived: -no_zero_divisors(quickcheck_code_int) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | zero_zero(quickcheck_code_int) != hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),C),B) | zero_zero(quickcheck_code_int) = hAPP(A,quickcheck_code_int,C,f664(A,quickcheck_code_int,C,B)). [resolve(5791,a,5776,a)]. 82.35/82.21 5792 -comm_semiring_1(A) | -no_zero_divisors(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | zero_zero(A) != hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),D),C) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f664(B,A,D,C)),C)) # label(fact_3977_setprod__zero__iff) # label(axiom). [clausify(3913)]. 82.35/82.21 Derived: -no_zero_divisors(code_code_numeral) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | zero_zero(code_code_numeral) != hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),C),B) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f664(A,code_code_numeral,C,B)),B)). [resolve(5792,a,5734,a)]. 82.35/82.21 Derived: -no_zero_divisors(int) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | zero_zero(int) != hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),C),B) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f664(A,int,C,B)),B)). [resolve(5792,a,5737,a)]. 82.35/82.21 Derived: -no_zero_divisors(rat) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | zero_zero(rat) != hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),C),B) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f664(A,rat,C,B)),B)). [resolve(5792,a,5764,a)]. 82.35/82.21 Derived: -no_zero_divisors(complex) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | zero_zero(complex) != hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),C),B) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f664(A,complex,C,B)),B)). [resolve(5792,a,5775,a)]. 82.35/82.27 Derived: -no_zero_divisors(quickcheck_code_int) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | zero_zero(quickcheck_code_int) != hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),C),B) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f664(A,quickcheck_code_int,C,B)),B)). [resolve(5792,a,5776,a)]. 82.35/82.27 5793 -comm_semiring_1(A) | -no_zero_divisors(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | zero_zero(A) = hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),D),C) | zero_zero(A) != hAPP(B,A,D,E) | -hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),E),C)) # label(fact_3977_setprod__zero__iff) # label(axiom). [clausify(3913)]. 82.35/82.27 5794 -comm_semiring_1(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),E)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),E)) # label(fact_365_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom). [clausify(3992)]. 82.35/82.27 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),C),D)) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),C),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B)),D)). [resolve(5794,a,5734,a)]. 82.35/82.27 Derived: hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),D)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),D)). [resolve(5794,a,5737,a)]. 82.35/82.27 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),D)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),D)). [resolve(5794,a,5764,a)]. 82.35/82.27 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),D)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)),D)). [resolve(5794,a,5775,a)]. 82.35/82.27 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),C),D)) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),C),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B)),D)). [resolve(5794,a,5776,a)]. 82.53/82.35 5795 -comm_semiring_1(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C))) # label(fact_729_dvd__triv__left) # label(axiom). [clausify(4019)]. 82.53/82.35 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B))). [resolve(5795,a,5734,a)]. 82.53/82.35 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B))). [resolve(5795,a,5737,a)]. 82.53/82.35 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B))). [resolve(5795,a,5764,a)]. 82.53/82.35 Derived: hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),A),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B))). [resolve(5795,a,5775,a)]. 82.53/82.35 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B))). [resolve(5795,a,5776,a)]. 82.53/82.35 5796 -comm_semiring_1(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),D)) # label(fact_466_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom). [clausify(4080)]. 82.53/82.35 Derived: hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),C)) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),C)). [resolve(5796,a,5734,a)]. 82.53/82.35 Derived: hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),C)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),C)). [resolve(5796,a,5764,a)]. 82.53/82.35 Derived: hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),C)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),B)),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),C)). [resolve(5796,a,5775,a)]. 82.53/82.35 Derived: hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),C)) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),C)). [resolve(5796,a,5776,a)]. 82.53/82.35 5797 -comm_semiring_1(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D)) # label(fact_362_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom). [clausify(4082)]. 82.61/82.44 5798 -comm_semiring_1(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),one_one(A)),B) # label(fact_388_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom). [clausify(4118)]. 82.61/82.44 Derived: ti(int,A) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),one_one(int)),A). [resolve(5798,a,5737,a)]. 82.61/82.44 5799 -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,hAPP(A,fun(nat,A),power_power(A),B),D)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),E))) # label(fact_749_dvd__power__le) # label(axiom). [clausify(4163)]. 82.61/82.44 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),C)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),B),D))). [resolve(5799,a,5734,a)]. 82.61/82.44 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,hAPP(int,fun(nat,int),power_power(int),A),C)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),D))). [resolve(5799,a,5737,a)]. 82.61/82.44 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),C)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),B),D))). [resolve(5799,a,5764,a)]. 82.61/82.44 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),C)),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),B),D))). [resolve(5799,a,5775,a)]. 82.61/82.44 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),C)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),B),D))). [resolve(5799,a,5776,a)]. 82.61/82.44 5800 -comm_semiring_1(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),one_one(A))),C) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),C) # label(fact_464_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom). [clausify(4196)]. 82.61/82.44 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),one_one(code_code_numeral))),B) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B)),B). [resolve(5800,a,5734,a)]. 82.75/82.59 Derived: hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),one_one(int))),B) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),B). [resolve(5800,a,5737,a)]. 82.75/82.59 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),one_one(rat))),B) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),B). [resolve(5800,a,5764,a)]. 82.75/82.59 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),one_one(complex))),B) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)),B). [resolve(5800,a,5775,a)]. 82.75/82.59 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),one_one(quickcheck_code_int))),B) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B)),B). [resolve(5800,a,5776,a)]. 82.75/82.59 5801 comm_semiring_1(real) # label(arity_RealDef_Oreal___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 82.75/82.59 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | zero_zero(real) != hAPP(A,real,C,D) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),D),B)) | zero_zero(real) = hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),C),B). [resolve(5801,a,5733,a)]. 82.75/82.59 Derived: hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),one_one(real))),B) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)). [resolve(5801,a,5735,a)]. 82.75/82.59 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(nat,real,hAPP(real,fun(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,hAPP(real,fun(nat,real),power_power(real),A),D)),C)). [resolve(5801,a,5738,a)]. 82.75/82.59 Derived: hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(nat,nat,suc,B)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B)). [resolve(5801,a,5739,a)]. 82.75/82.59 Derived: hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),A). [resolve(5801,a,5740,a)]. 82.75/82.59 Derived: ti(real,A) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),zero_zero(real)). [resolve(5801,a,5741,a)]. 82.75/82.59 Derived: one_one(real) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),zero_zero(nat)). [resolve(5801,a,5743,a)]. 82.75/82.59 Derived: hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),A). [resolve(5801,a,5744,a)]. 82.75/82.59 Derived: hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),C) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),C)). [resolve(5801,a,5745,a)]. 82.75/82.59 Derived: hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),C),D)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),C)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),D)). [resolve(5801,a,5746,a)]. 82.75/82.60 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),A))). [resolve(5801,a,5747,a)]. 82.75/82.60 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),one_one(real)),A)). [resolve(5801,a,5748,a)]. 82.75/82.60 Derived: hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),one_one(real))),A) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),A). [resolve(5801,a,5749,a)]. 82.75/82.60 Derived: hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B))). [resolve(5801,a,5750,a)]. 82.75/82.60 Derived: ti(real,A) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),zero_zero(real)),A). [resolve(5801,a,5751,a)]. 82.75/82.60 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,hAPP(real,fun(nat,real),power_power(real),C),A)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),C),B))). [resolve(5801,a,5753,a)]. 82.75/82.60 Derived: hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),D)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),C)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),D)). [resolve(5801,a,5754,a)]. 82.75/82.60 Derived: hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,semiring_1_of_nat(real),A)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,semiring_1_of_nat(real),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A),one_one(nat)))),C)))) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),one_one(real))),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),plus_plus(real),B)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),semiring_1_of_nat(real))),C))),hAPP(nat,fun(nat,bool),ord_lessThan(nat),A))). [resolve(5801,a,5755,a)]. 82.75/82.60 Derived: hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B)),C) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),C)). [resolve(5801,a,5758,a)]. 82.75/82.60 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),B),C)). [resolve(5801,a,5759,a)]. 82.75/82.60 Derived: hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(nat,nat,suc,B)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B)),A). [resolve(5801,a,5760,a)]. 82.75/82.60 Derived: hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),C) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),C)),B). [resolve(5801,a,5761,a)]. 82.75/82.61 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f393(A,real,B,C,D)),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),B),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),C),D))). [resolve(5801,a,5762,a)]. 82.75/82.61 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(B,real,C,f393(B,real,A,C,D)))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(fun(B,bool),real,hAPP(fun(B,real),fun(fun(B,bool),real),big_co1399186613setsum(B,real),C),D))). [resolve(5801,a,5763,a)]. 82.75/82.61 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),C),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),C)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),D))). [resolve(5801,a,5767,a)]. 82.75/82.61 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),A)). [resolve(5801,a,5768,a)]. 82.75/82.61 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f462(A,real,D,E,C,B)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),D),C)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),E),B))). [resolve(5801,a,5769,a)]. 82.75/82.61 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(A,real,D,f462(A,real,D,E,C,B))),hAPP(A,real,E,f462(A,real,D,E,C,B)))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),D),C)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),E),B))). [resolve(5801,a,5770,a)]. 82.75/82.61 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),C))). [resolve(5801,a,5771,a)]. 82.75/82.61 Derived: hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),C)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),C)). [resolve(5801,a,5772,a)]. 82.75/82.61 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),zero_zero(real))). [resolve(5801,a,5773,a)]. 82.75/82.61 Derived: ti(real,A) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),one_one(nat)). [resolve(5801,a,5774,a)]. 82.75/82.61 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),C)). [resolve(5801,a,5778,a)]. 82.75/82.61 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),C))). [resolve(5801,a,5779,a)]. 82.75/82.61 Derived: hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,semiring_1_of_nat(real),A)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,semiring_1_of_nat(real),A)),one_one(real))) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),one_one(real))),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),semiring_1_of_nat(real)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),one_one(nat)),A))). [resolve(5801,a,5780,a)]. 82.79/82.62 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),B))). [resolve(5801,a,5781,a)]. 82.79/82.62 Derived: hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),A). [resolve(5801,a,5782,a)]. 82.79/82.62 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,hAPP(real,fun(nat,real),power_power(real),A),C)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),C))). [resolve(5801,a,5783,a)]. 82.79/82.62 Derived: ti(real,A) != one_one(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B))). [resolve(5801,a,5784,a)]. 82.79/82.62 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,hAPP(real,fun(nat,real),power_power(real),B),A))). [resolve(5801,a,5785,a)]. 82.79/82.62 Derived: hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B)),C) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C)). [resolve(5801,a,5786,a)]. 82.79/82.62 Derived: hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B)). [resolve(5801,a,5787,a)]. 82.79/82.62 Derived: hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B)),C) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),C)),B). [resolve(5801,a,5788,a)]. 82.79/82.62 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(5801,a,5789,a)]. 82.79/82.62 Derived: -no_zero_divisors(real) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | zero_zero(real) != hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),C),B) | zero_zero(real) = hAPP(A,real,C,f664(A,real,C,B)). [resolve(5801,a,5791,a)]. 82.79/82.62 Derived: -no_zero_divisors(real) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | zero_zero(real) != hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),C),B) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f664(A,real,C,B)),B)). [resolve(5801,a,5792,a)]. 82.79/82.62 Derived: hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),D)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),D)). [resolve(5801,a,5794,a)]. 82.79/82.62 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B))). [resolve(5801,a,5795,a)]. 82.79/82.62 Derived: hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),C)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),C)). [resolve(5801,a,5796,a)]. 82.83/82.72 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,hAPP(real,fun(nat,real),power_power(real),A),C)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),D))). [resolve(5801,a,5799,a)]. 82.83/82.72 Derived: hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),one_one(real))),B) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),B). [resolve(5801,a,5800,a)]. 82.83/82.72 5802 -comm_semiring_1(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | -hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),D),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(B,A,E,D)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co302546470etprod(B,A),E),C))) # label(fact_3980_dvd__setprod) # label(axiom). [clausify(4473)]. 82.83/82.72 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),hAPP(A,code_code_numeral,D,C)),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co302546470etprod(A,code_code_numeral),D),B))). [resolve(5802,a,5734,a)]. 82.83/82.72 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(A,int,D,C)),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co302546470etprod(A,int),D),B))). [resolve(5802,a,5737,a)]. 82.83/82.72 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),hAPP(A,rat,D,C)),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co302546470etprod(A,rat),D),B))). [resolve(5802,a,5764,a)]. 82.83/82.72 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),hAPP(A,complex,D,C)),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co302546470etprod(A,complex),D),B))). [resolve(5802,a,5775,a)]. 82.83/82.72 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),hAPP(A,quickcheck_code_int,D,C)),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co302546470etprod(A,quickcheck_code_int),D),B))). [resolve(5802,a,5776,a)]. 82.83/82.72 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(A,real,D,C)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co302546470etprod(A,real),D),B))). [resolve(5802,a,5801,a)]. 82.83/82.72 5803 comm_semiring_1(nat) # label(arity_Nat_Onat___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 82.83/82.72 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | zero_zero(nat) != hAPP(A,nat,C,D) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),D),B)) | zero_zero(nat) = hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),C),B). [resolve(5803,a,5733,a)]. 82.83/82.72 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),one_one(nat))),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)). [resolve(5803,a,5735,a)]. 82.83/82.72 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),C)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),C)). [resolve(5803,a,5736,a)]. 82.83/82.73 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(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,hAPP(nat,fun(nat,nat),power_power(nat),A),D)),C)). [resolve(5803,a,5738,a)]. 82.83/82.73 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),hAPP(nat,nat,suc,B)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),B)). [resolve(5803,a,5739,a)]. 82.83/82.73 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),A). [resolve(5803,a,5740,a)]. 82.83/82.73 Derived: ti(nat,A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),zero_zero(nat)). [resolve(5803,a,5741,a)]. 82.83/82.73 Derived: one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),zero_zero(nat)). [resolve(5803,a,5743,a)]. 82.83/82.73 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),A). [resolve(5803,a,5744,a)]. 82.83/82.73 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),C) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C)). [resolve(5803,a,5745,a)]. 82.83/82.73 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),C),D)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),D)). [resolve(5803,a,5746,a)]. 82.83/82.73 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),A))). [resolve(5803,a,5747,a)]. 82.83/82.73 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),one_one(nat)),A)). [resolve(5803,a,5748,a)]. 82.83/82.73 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat))),A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),A). [resolve(5803,a,5749,a)]. 82.83/82.73 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),B))). [resolve(5803,a,5750,a)]. 82.83/82.73 Derived: ti(nat,A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),A). [resolve(5803,a,5751,a)]. 82.83/82.73 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),C),A)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),C),B))). [resolve(5803,a,5753,a)]. 82.83/82.73 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),D)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),D)). [resolve(5803,a,5754,a)]. 82.83/82.73 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,semiring_1_of_nat(nat),A)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,semiring_1_of_nat(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A),one_one(nat)))),C)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat))),hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),hAPP(nat,fun(nat,nat),plus_plus(nat),B)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),times_times(nat)),semiring_1_of_nat(nat))),C))),hAPP(nat,fun(nat,bool),ord_lessThan(nat),A))). [resolve(5803,a,5755,a)]. 82.92/82.73 Derived: zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),zero_zero(nat)). [resolve(5803,a,5756,a)]. 82.92/82.73 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),B)),C) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C)). [resolve(5803,a,5757,a)]. 82.92/82.73 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),B)),C) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),C)). [resolve(5803,a,5758,a)]. 82.92/82.73 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),C)). [resolve(5803,a,5759,a)]. 82.92/82.73 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),hAPP(nat,nat,suc,B)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),B)),A). [resolve(5803,a,5760,a)]. 82.92/82.73 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),C) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),C)),B). [resolve(5803,a,5761,a)]. 82.92/82.73 Derived: hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f393(A,nat,B,C,D)),D)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co1399186613setsum(A,nat),C),D))). [resolve(5803,a,5762,a)]. 82.92/82.73 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(B,nat,C,f393(B,nat,A,C,D)))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(fun(B,bool),nat,hAPP(fun(B,nat),fun(fun(B,bool),nat),big_co1399186613setsum(B,nat),C),D))). [resolve(5803,a,5763,a)]. 82.92/82.73 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),C),D)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),D))). [resolve(5803,a,5767,a)]. 82.92/82.73 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),A)). [resolve(5803,a,5768,a)]. 82.92/82.73 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f462(A,nat,D,E,C,B)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),D),C)),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),E),B))). [resolve(5803,a,5769,a)]. 82.92/82.73 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(A,nat,D,f462(A,nat,D,E,C,B))),hAPP(A,nat,E,f462(A,nat,D,E,C,B)))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),D),C)),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),E),B))). [resolve(5803,a,5770,a)]. 82.92/82.74 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),C))). [resolve(5803,a,5771,a)]. 82.92/82.74 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),C)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),C)). [resolve(5803,a,5772,a)]. 82.92/82.74 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),zero_zero(nat))). [resolve(5803,a,5773,a)]. 82.92/82.74 Derived: ti(nat,A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),one_one(nat)). [resolve(5803,a,5774,a)]. 82.92/82.74 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),C)). [resolve(5803,a,5778,a)]. 82.92/82.74 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C))). [resolve(5803,a,5779,a)]. 82.92/82.74 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,semiring_1_of_nat(nat),A)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,semiring_1_of_nat(nat),A)),one_one(nat))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat))),hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),semiring_1_of_nat(nat)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),one_one(nat)),A))). [resolve(5803,a,5780,a)]. 82.92/82.74 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),B))). [resolve(5803,a,5781,a)]. 82.92/82.74 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),A). [resolve(5803,a,5782,a)]. 82.92/82.74 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,hAPP(nat,fun(nat,nat),power_power(nat),A),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),C))). [resolve(5803,a,5783,a)]. 82.92/82.74 Derived: ti(nat,A) != one_one(nat) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),B))). [resolve(5803,a,5784,a)]. 82.92/82.74 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,hAPP(nat,fun(nat,nat),power_power(nat),B),A))). [resolve(5803,a,5785,a)]. 82.92/82.74 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),B)),C) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C)). [resolve(5803,a,5786,a)]. 82.92/82.74 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),B)). [resolve(5803,a,5787,a)]. 82.92/82.74 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),B)),C) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),C)),B). [resolve(5803,a,5788,a)]. 82.92/82.77 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),C)). [resolve(5803,a,5789,a)]. 82.92/82.77 Derived: zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),zero_zero(nat)),A). [resolve(5803,a,5790,a)]. 82.92/82.77 Derived: -no_zero_divisors(nat) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | zero_zero(nat) != hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),C),B) | zero_zero(nat) = hAPP(A,nat,C,f664(A,nat,C,B)). [resolve(5803,a,5791,a)]. 82.92/82.77 Derived: -no_zero_divisors(nat) | -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | zero_zero(nat) != hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),C),B) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f664(A,nat,C,B)),B)). [resolve(5803,a,5792,a)]. 82.92/82.77 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),D)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),D)). [resolve(5803,a,5794,a)]. 82.92/82.77 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B))). [resolve(5803,a,5795,a)]. 82.92/82.77 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),C)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),C)). [resolve(5803,a,5796,a)]. 82.92/82.77 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),D))). [resolve(5803,a,5799,a)]. 82.92/82.77 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),one_one(nat))),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),B). [resolve(5803,a,5800,a)]. 82.92/82.77 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),C),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(A,nat,D,C)),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co302546470etprod(A,nat),D),B))). [resolve(5803,a,5802,a)]. 82.92/82.77 5804 -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_721_dvd__0__left) # label(axiom). [clausify(4552)]. 82.92/82.77 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),zero_zero(code_code_numeral)),A)) | ti(code_code_numeral,A) = zero_zero(code_code_numeral). [resolve(5804,a,5734,a)]. 82.92/82.77 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),zero_zero(int)),A)) | ti(int,A) = zero_zero(int). [resolve(5804,a,5737,a)]. 82.92/82.77 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),zero_zero(rat)),A)) | ti(rat,A) = zero_zero(rat). [resolve(5804,a,5764,a)]. 82.92/82.77 Derived: -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),zero_zero(complex)),A)) | ti(complex,A) = zero_zero(complex). [resolve(5804,a,5775,a)]. 82.92/82.77 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),zero_zero(quickcheck_code_int)),A)) | ti(quickcheck_code_int,A) = zero_zero(quickcheck_code_int). [resolve(5804,a,5776,a)]. 82.92/82.77 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),zero_zero(real)),A)) | ti(real,A) = zero_zero(real). [resolve(5804,a,5801,a)]. 82.92/82.77 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A)) | ti(nat,A) = zero_zero(nat). [resolve(5804,a,5803,a)]. 85.19/85.06 5805 -comm_semiring_1(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)) # label(fact_359_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom). [clausify(4754)]. 85.19/85.06 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),B),C)) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),B),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),C)). [resolve(5805,a,5734,a)]. 85.19/85.06 Derived: hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),C)). [resolve(5805,a,5737,a)]. 85.19/85.06 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),C)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),C)). [resolve(5805,a,5764,a)]. 85.19/85.06 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),B),C)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),B),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),C)). [resolve(5805,a,5775,a)]. 85.19/85.06 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),B),C)) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),B),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),C)). [resolve(5805,a,5776,a)]. 85.19/85.06 Derived: hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),C)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),C)). [resolve(5805,a,5801,a)]. 85.19/85.06 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),C)). [resolve(5805,a,5803,a)]. 85.19/85.06 5806 -comm_semiring_1(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(nat,nat,suc,C)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C)) # label(fact_1368_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) # label(axiom). [clausify(4775)]. 85.19/85.06 5807 ring(rat) # label(arity_Rat_Orat___Rings_Oring) # label(axiom). [assumption]. 85.19/85.06 5808 -ring(A) | hAPP(A,A,uminus_uminus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,uminus_uminus(A),B)),C) # label(fact_1907_minus__mult__left) # label(axiom). [clausify(78)]. 85.19/85.06 5809 -ring(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,uminus_uminus(A),B)),C) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(A,A,uminus_uminus(A),C)) # label(fact_1906_minus__mult__commute) # label(axiom). [clausify(680)]. 85.19/85.06 Derived: hAPP(rat,rat,uminus_uminus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,uminus_uminus(rat),A)),B). [resolve(5807,a,5808,a)]. 85.42/85.27 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,uminus_uminus(rat),A)),B) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),hAPP(rat,rat,uminus_uminus(rat),B)). [resolve(5807,a,5809,a)]. 85.42/85.27 5810 ring(real) # label(arity_RealDef_Oreal___Rings_Oring) # label(axiom). [assumption]. 85.42/85.27 Derived: hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,uminus_uminus(real),A)),B) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),hAPP(real,real,uminus_uminus(real),B)). [resolve(5810,a,5809,a)]. 85.42/85.27 5811 -ring(A) | hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),D) != hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),E),C)),F) | ti(A,D) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),E),B)),C)),F) # label(fact_753_eq__add__iff1) # label(axiom). [clausify(1417)]. 85.42/85.27 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),C) != hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),D),B)),E) | ti(rat,C) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),D),A)),B)),E). [resolve(5811,a,5807,a)]. 85.42/85.27 Derived: hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),C) != hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),D),B)),E) | ti(real,C) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),D),A)),B)),E). [resolve(5811,a,5810,a)]. 85.42/85.27 5812 -ring(A) | hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),E),C)),F) | ti(A,D) != hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),E),B)),C)),F) # label(fact_753_eq__add__iff1) # label(axiom). [clausify(1417)]. 85.42/85.27 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),C) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),D),B)),E) | ti(rat,C) != hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),D),A)),B)),E). [resolve(5812,a,5807,a)]. 85.42/85.27 Derived: hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),C) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),D),B)),E) | ti(real,C) != hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),D),A)),B)),E). [resolve(5812,a,5810,a)]. 85.42/85.27 5813 ring(complex) # label(arity_Complex_Ocomplex___Rings_Oring) # label(axiom). [assumption]. 85.42/85.27 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,uminus_uminus(complex),A)),B) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),hAPP(complex,complex,uminus_uminus(complex),B)). [resolve(5813,a,5809,a)]. 85.42/85.27 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)),C) != hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),D),B)),E) | ti(complex,C) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),D),A)),B)),E). [resolve(5813,a,5811,a)]. 85.49/85.35 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)),C) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),D),B)),E) | ti(complex,C) != hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),D),A)),B)),E). [resolve(5813,a,5812,a)]. 85.49/85.35 5814 -ring(A) | -number(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(int,A,number_number_of(A),B)),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(int,A,number_number_of(A),B)),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(int,A,number_number_of(A),B)),D)) # label(fact_756_right__diff__distrib__number__of) # label(axiom). [clausify(1758)]. 85.49/85.35 Derived: -number(rat) | hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(int,rat,number_number_of(rat),A)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),B),C)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(int,rat,number_number_of(rat),A)),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(int,rat,number_number_of(rat),A)),C)). [resolve(5814,a,5807,a)]. 85.49/85.35 5815 -ring(A) | hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),E)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),C),E))),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),D)),E)) # label(fact_755_mult__diff__mult) # label(axiom). [clausify(1933)]. 85.49/85.35 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),D)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),B),D))),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),C)),D)). [resolve(5815,a,5807,a)]. 85.49/85.35 Derived: hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),D)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),D))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),C)),D)). [resolve(5815,a,5810,a)]. 85.49/85.35 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),D)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),B),D))),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),A),C)),D)). [resolve(5815,a,5813,a)]. 85.49/85.35 5816 -number(A) | -ring(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C)),hAPP(int,A,number_number_of(A),D)) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(int,A,number_number_of(A),D))),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),hAPP(int,A,number_number_of(A),D))) # label(fact_757_left__diff__distrib__number__of) # label(axiom). [clausify(1975)]. 85.49/85.37 Derived: -number(rat) | hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B)),hAPP(int,rat,number_number_of(rat),C)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),hAPP(int,rat,number_number_of(rat),C))),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),hAPP(int,rat,number_number_of(rat),C))). [resolve(5816,b,5807,a)]. 85.49/85.37 5817 -ring(A) | hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),D) != hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),E),C)),F) | ti(A,F) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),E)),C)),D) # label(fact_754_eq__add__iff2) # label(axiom). [clausify(2021)]. 85.49/85.37 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),C) != hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),D),B)),E) | ti(rat,E) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),D)),B)),C). [resolve(5817,a,5807,a)]. 85.49/85.37 Derived: hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),C) != hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),D),B)),E) | ti(real,E) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),D)),B)),C). [resolve(5817,a,5810,a)]. 85.49/85.37 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)),C) != hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),D),B)),E) | ti(complex,E) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),A),D)),B)),C). [resolve(5817,a,5813,a)]. 85.49/85.37 5818 -ring(A) | hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),E),C)),F) | ti(A,F) != hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),E)),C)),D) # label(fact_754_eq__add__iff2) # label(axiom). [clausify(2021)]. 85.49/85.37 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),C) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),D),B)),E) | ti(rat,E) != hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),D)),B)),C). [resolve(5818,a,5807,a)]. 85.49/85.37 Derived: hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),C) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),D),B)),E) | ti(real,E) != hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),D)),B)),C). [resolve(5818,a,5810,a)]. 85.49/85.37 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B)),C) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),D),B)),E) | ti(complex,E) != hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),A),D)),B)),C). [resolve(5818,a,5813,a)]. 87.00/86.86 5819 -ring(A) | hAPP(A,A,uminus_uminus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(A,A,uminus_uminus(A),C)) # label(fact_1908_minus__mult__right) # label(axiom). [clausify(2478)]. 87.00/86.86 Derived: hAPP(rat,rat,uminus_uminus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),hAPP(rat,rat,uminus_uminus(rat),B)). [resolve(5819,a,5807,a)]. 87.00/86.86 5820 -ring(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,uminus_uminus(A),B)),hAPP(A,A,uminus_uminus(A),C)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C) # label(fact_1905_minus__mult__minus) # label(axiom). [clausify(2507)]. 87.00/86.86 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,uminus_uminus(rat),A)),hAPP(rat,rat,uminus_uminus(rat),B)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B). [resolve(5820,a,5807,a)]. 87.00/86.86 Derived: hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,uminus_uminus(real),A)),hAPP(real,real,uminus_uminus(real),B)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B). [resolve(5820,a,5810,a)]. 87.00/86.86 Derived: hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,uminus_uminus(complex),A)),hAPP(complex,complex,uminus_uminus(complex),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),B). [resolve(5820,a,5813,a)]. 87.00/86.86 5821 ring(int) # label(arity_Int_Oint___Rings_Oring) # label(axiom). [assumption]. 87.00/86.86 Derived: hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,uminus_uminus(int),A)),B). [resolve(5821,a,5808,a)]. 87.00/86.86 Derived: hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,uminus_uminus(int),A)),B) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),hAPP(int,int,uminus_uminus(int),B)). [resolve(5821,a,5809,a)]. 87.00/86.86 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),C) != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),D),B)),E) | ti(int,C) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),D),A)),B)),E). [resolve(5821,a,5811,a)]. 87.00/86.86 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),C) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),D),B)),E) | ti(int,C) != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),D),A)),B)),E). [resolve(5821,a,5812,a)]. 87.00/86.86 Derived: -number(int) | hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),A)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),C)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),A)),B)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),A)),C)). [resolve(5821,a,5814,a)]. 87.00/86.86 Derived: hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),D)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),D))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),C)),D)). [resolve(5821,a,5815,a)]. 89.68/89.49 Derived: -number(int) | hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B)),hAPP(int,int,number_number_of(int),C)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),hAPP(int,int,number_number_of(int),C))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),hAPP(int,int,number_number_of(int),C))). [resolve(5821,a,5816,b)]. 89.68/89.49 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),C) != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),D),B)),E) | ti(int,E) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),D)),B)),C). [resolve(5821,a,5817,a)]. 89.68/89.49 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),C) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),D),B)),E) | ti(int,E) != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),D)),B)),C). [resolve(5821,a,5818,a)]. 89.68/89.49 Derived: hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),hAPP(int,int,uminus_uminus(int),B)). [resolve(5821,a,5819,a)]. 89.68/89.49 Derived: hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,uminus_uminus(int),A)),hAPP(int,int,uminus_uminus(int),B)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B). [resolve(5821,a,5820,a)]. 89.68/89.49 5822 -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_161_of__nat__eq__iff) # label(axiom). [clausify(1029)]. 89.68/89.49 5823 semiring_char_0(quickcheck_code_int) # label(arity_Quickcheck__Narrowing_Ocode__int___Nat_Osemiring__char__0) # label(axiom). [assumption]. 89.68/89.49 5824 semiring_char_0(real) # label(arity_RealDef_Oreal___Nat_Osemiring__char__0) # label(axiom). [assumption]. 89.68/89.49 Derived: hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),A) != hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),B) | A = B. [resolve(5822,a,5823,a)]. 89.68/89.49 Derived: hAPP(nat,real,semiring_1_of_nat(real),A) != hAPP(nat,real,semiring_1_of_nat(real),B) | A = B. [resolve(5822,a,5824,a)]. 89.68/89.49 5825 -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_161_of__nat__eq__iff) # label(axiom). [clausify(1029)]. 89.68/89.49 Derived: hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),A) = hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),B) | A != B. [resolve(5825,a,5823,a)]. 89.68/89.49 Derived: hAPP(nat,real,semiring_1_of_nat(real),A) = hAPP(nat,real,semiring_1_of_nat(real),B) | A != B. [resolve(5825,a,5824,a)]. 89.68/89.49 5826 semiring_char_0(complex) # label(arity_Complex_Ocomplex___Nat_Osemiring__char__0) # label(axiom). [assumption]. 89.68/89.49 Derived: hAPP(nat,complex,semiring_1_of_nat(complex),A) != hAPP(nat,complex,semiring_1_of_nat(complex),B) | A = B. [resolve(5826,a,5822,a)]. 89.68/89.49 Derived: hAPP(nat,complex,semiring_1_of_nat(complex),A) = hAPP(nat,complex,semiring_1_of_nat(complex),B) | A != B. [resolve(5826,a,5825,a)]. 89.68/89.49 5827 -semiring_char_0(A) | zero_zero(A) != hAPP(nat,A,semiring_1_of_nat(A),hAPP(nat,nat,fact_fact(nat),B)) # label(fact_2541_of__nat__fact__not__zero) # label(axiom). [clausify(1937)]. 89.68/89.49 Derived: zero_zero(quickcheck_code_int) != hAPP(nat,quickcheck_code_int,semiring_1_of_nat(quickcheck_code_int),hAPP(nat,nat,fact_fact(nat),A)). [resolve(5827,a,5823,a)]. 89.68/89.49 Derived: zero_zero(real) != hAPP(nat,real,semiring_1_of_nat(real),hAPP(nat,nat,fact_fact(nat),A)). [resolve(5827,a,5824,a)]. 94.90/94.79 Derived: zero_zero(complex) != hAPP(nat,complex,semiring_1_of_nat(complex),hAPP(nat,nat,fact_fact(nat),A)). [resolve(5827,a,5826,a)]. 94.90/94.79 5828 semiring_char_0(int) # label(arity_Int_Oint___Nat_Osemiring__char__0) # label(axiom). [assumption]. 94.90/94.79 Derived: hAPP(nat,int,semiring_1_of_nat(int),A) != hAPP(nat,int,semiring_1_of_nat(int),B) | A = B. [resolve(5828,a,5822,a)]. 94.90/94.79 Derived: hAPP(nat,int,semiring_1_of_nat(int),A) = hAPP(nat,int,semiring_1_of_nat(int),B) | A != B. [resolve(5828,a,5825,a)]. 94.90/94.79 Derived: zero_zero(int) != hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,fact_fact(nat),A)). [resolve(5828,a,5827,a)]. 94.90/94.79 5829 semiring_char_0(code_code_numeral) # label(arity_Code__Numeral_Ocode__numeral___Nat_Osemiring__char__0) # label(axiom). [assumption]. 94.90/94.79 Derived: hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),A) != hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),B) | A = B. [resolve(5829,a,5822,a)]. 94.90/94.79 Derived: hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),A) = hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),B) | A != B. [resolve(5829,a,5825,a)]. 94.90/94.79 Derived: zero_zero(code_code_numeral) != hAPP(nat,code_code_numeral,semiring_1_of_nat(code_code_numeral),hAPP(nat,nat,fact_fact(nat),A)). [resolve(5829,a,5827,a)]. 94.90/94.79 5830 semiring_char_0(nat) # label(arity_Nat_Onat___Nat_Osemiring__char__0) # label(axiom). [assumption]. 94.90/94.79 Derived: hAPP(nat,nat,semiring_1_of_nat(nat),A) != hAPP(nat,nat,semiring_1_of_nat(nat),B) | A = B. [resolve(5830,a,5822,a)]. 94.90/94.79 Derived: hAPP(nat,nat,semiring_1_of_nat(nat),A) = hAPP(nat,nat,semiring_1_of_nat(nat),B) | A != B. [resolve(5830,a,5825,a)]. 94.90/94.79 Derived: zero_zero(nat) != hAPP(nat,nat,semiring_1_of_nat(nat),hAPP(nat,nat,fact_fact(nat),A)). [resolve(5830,a,5827,a)]. 94.90/94.79 5831 semiring_char_0(rat) # label(arity_Rat_Orat___Nat_Osemiring__char__0) # label(axiom). [assumption]. 94.90/94.79 Derived: hAPP(nat,rat,semiring_1_of_nat(rat),A) != hAPP(nat,rat,semiring_1_of_nat(rat),B) | A = B. [resolve(5831,a,5822,a)]. 94.90/94.79 Derived: hAPP(nat,rat,semiring_1_of_nat(rat),A) = hAPP(nat,rat,semiring_1_of_nat(rat),B) | A != B. [resolve(5831,a,5825,a)]. 94.90/94.79 Derived: zero_zero(rat) != hAPP(nat,rat,semiring_1_of_nat(rat),hAPP(nat,nat,fact_fact(nat),A)). [resolve(5831,a,5827,a)]. 94.90/94.79 5832 real_vector(real) # label(arity_RealDef_Oreal___RealVector_Oreal__vector) # label(axiom). [assumption]. 94.90/94.79 5833 -real_vector(A) | hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),D)),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),C),D)) # label(fact_4695_scaleR__left_Oadd) # label(axiom). [clausify(83)]. 94.90/94.79 5834 -real_vector(A) | zero_zero(real) = B | hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),C) != hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),D) | ti(A,C) = ti(A,D) # label(fact_4729_scaleR__left__imp__eq) # label(axiom). [clausify(228)]. 94.90/94.79 5835 -real_vector(A) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),hAPP(A,fun(B,A),hAPP(fun(B,fun(A,A)),fun(A,fun(B,A)),combc(B,A,A),hAPP(fun(B,real),fun(B,fun(A,A)),hAPP(fun(real,fun(A,A)),fun(fun(B,real),fun(B,fun(A,A))),combb(real,fun(A,A),B),scaleR_scaleR(A)),C)),D)),E) = hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),hAPP(fun(B,bool),real,hAPP(fun(B,real),fun(fun(B,bool),real),big_co1399186613setsum(B,real),C),E)),D) # label(fact_4755_scaleR__left_Osetsum) # label(axiom). [clausify(422)]. 94.90/94.79 5836 -real_vector(A) | hAPP(A,A,uminus_uminus(A),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),C)) = hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),hAPP(A,A,uminus_uminus(A),C)) # label(fact_4734_scaleR__minus__right) # label(axiom). [clausify(429)]. 94.90/94.79 5837 -real_vector(A) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),hAPP(fun(B,A),fun(B,A),hAPP(fun(A,A),fun(fun(B,A),fun(B,A)),combb(A,A,B),hAPP(real,fun(A,A),scaleR_scaleR(A),C)),D)),E) = hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),C),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),D),E)) # label(fact_4756_scaleR__right_Osetsum) # label(axiom). [clausify(708)]. 94.90/94.79 5838 -real_vector(A) | hAPP(A,A,uminus_uminus(A),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),C)) = hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),hAPP(real,real,uminus_uminus(real),B)),C) # label(fact_4740_scaleR__minus__left) # label(axiom). [clausify(975)]. 94.90/94.79 5839 -real_vector(A) | hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),D)),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),C),D)) # label(fact_4698_scaleR__left__diff__distrib) # label(axiom). [clausify(1237)]. 94.90/94.79 5840 -real_vector(A) | hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),C),D)) = hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),C),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),D)) # label(fact_4713_scaleR__left__commute) # label(axiom). [clausify(1403)]. 94.90/94.79 5841 -real_vector(A) | ti(A,B) = zero_zero(A) | hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),C),B) != hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),D),B) | C = D # label(fact_4733_scaleR__right__imp__eq) # label(axiom). [clausify(1672)]. 94.90/94.79 5842 -real_vector(A) | ti(A,B) = hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),one_one(real)),B) # label(fact_4703_real__vector_Oscale__one) # label(axiom). [clausify(1800)]. 94.90/94.79 5843 -real_vector(A) | hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),C)),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),D)) # label(fact_4700_scaleR__right_Oadd) # label(axiom). [clausify(2161)]. 94.90/94.79 5844 -real_vector(A) | hAPP(A,A,uminus_uminus(A),B) = hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),hAPP(int,real,number_number_of(real),min)),B) # label(fact_4690_scaleR__minus1__left) # label(axiom). [clausify(2723)]. 94.90/94.79 5845 -real_vector(A) | zero_zero(A) = hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),zero_zero(A)) # label(fact_4730_scaleR__right_Ozero) # label(axiom). [clausify(2947)]. 94.90/94.79 5846 -real_vector(A) | hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),D)),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),C),D)) # label(fact_4693_real__vector_Oscale__left__distrib) # label(axiom). [clausify(3313)]. 94.90/94.79 5847 -real_vector(A) | ti(A,B) != zero_zero(A) | zero_zero(A) = hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),C),B) # label(fact_4736_scaleR__eq__0__iff) # label(axiom). [clausify(3439)]. 94.90/94.79 5848 -real_vector(A) | zero_zero(real) != B | zero_zero(A) = hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),C) # label(fact_4736_scaleR__eq__0__iff) # label(axiom). [clausify(3439)]. 94.90/94.79 5849 -real_vector(A) | ti(A,B) = zero_zero(A) | zero_zero(real) = C | zero_zero(A) != hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),C),B) # label(fact_4736_scaleR__eq__0__iff) # label(axiom). [clausify(3439)]. 94.90/94.79 5850 -real_vector(A) | hAPP(A,A,uminus_uminus(A),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),C)) = hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),hAPP(real,real,uminus_uminus(real),B)),C) # label(fact_4739_scaleR__left_Ominus) # label(axiom). [clausify(3746)]. 94.90/94.79 5851 -real_vector(A) | hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),C)),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),D)) # label(fact_4701_scaleR__right_Odiff) # label(axiom). [clausify(3822)]. 94.90/94.79 5852 -real_vector(A) | zero_zero(A) = hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),zero_zero(real)),B) # label(fact_4737_scaleR__left_Ozero) # label(axiom). [clausify(3836)]. 94.90/94.79 5853 -real_vector(A) | hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),C) != hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),D) | zero_zero(real) = B | ti(A,C) = ti(A,D) # label(fact_4728_scaleR__cancel__left) # label(axiom). [clausify(4054)]. 94.90/94.80 5854 -real_vector(A) | hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),C) = hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),D) | zero_zero(real) != B # label(fact_4728_scaleR__cancel__left) # label(axiom). [clausify(4054)]. 94.90/94.80 5855 -real_vector(A) | hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),C) = hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),D) | ti(A,C) != ti(A,D) # label(fact_4728_scaleR__cancel__left) # label(axiom). [clausify(4054)]. 94.90/94.80 Derived: hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B)),C) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),C)),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),B),C)). [resolve(5832,a,5833,a)]. 94.90/94.80 Derived: zero_zero(real) = A | hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),B) != hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),C) | ti(real,B) = ti(real,C). [resolve(5832,a,5834,a)]. 94.90/94.80 Derived: hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),hAPP(real,fun(A,real),hAPP(fun(A,fun(real,real)),fun(real,fun(A,real)),combc(A,real,real),hAPP(fun(A,real),fun(A,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(A,real),fun(A,fun(real,real))),combb(real,fun(real,real),A),scaleR_scaleR(real)),B)),C)),D) = hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),B),D)),C). [resolve(5832,a,5835,a)]. 94.90/94.80 Derived: hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),B)) = hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),hAPP(real,real,uminus_uminus(real),B)). [resolve(5832,a,5836,a)]. 94.90/94.80 Derived: hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),hAPP(fun(A,real),fun(A,real),hAPP(fun(real,real),fun(fun(A,real),fun(A,real)),combb(real,real,A),hAPP(real,fun(real,real),scaleR_scaleR(real),B)),C)),D) = hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),B),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),C),D)). [resolve(5832,a,5837,a)]. 94.90/94.80 Derived: hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),B)) = hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),hAPP(real,real,uminus_uminus(real),A)),B). [resolve(5832,a,5838,a)]. 94.90/94.80 Derived: hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B)),C) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),C)),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),B),C)). [resolve(5832,a,5839,a)]. 94.90/94.80 Derived: hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),B),C)) = hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),B),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),C)). [resolve(5832,a,5840,a)]. 94.90/94.80 Derived: ti(real,A) = zero_zero(real) | hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),B),A) != hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),C),A) | B = C. [resolve(5832,a,5841,a)]. 94.90/94.80 Derived: ti(real,A) = hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),one_one(real)),A). [resolve(5832,a,5842,a)]. 94.90/94.80 Derived: hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),C)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),C)). [resolve(5832,a,5843,a)]. 94.90/94.80 Derived: hAPP(real,real,uminus_uminus(real),A) = hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),hAPP(int,real,number_number_of(real),min)),A). [resolve(5832,a,5844,a)]. 95.20/95.02 Derived: zero_zero(real) = hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),zero_zero(real)). [resolve(5832,a,5845,a)]. 95.20/95.02 Derived: ti(real,A) != zero_zero(real) | zero_zero(real) = hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),B),A). [resolve(5832,a,5847,a)]. 95.20/95.02 Derived: zero_zero(real) != A | zero_zero(real) = hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),B). [resolve(5832,a,5848,a)]. 95.20/95.02 Derived: ti(real,A) = zero_zero(real) | zero_zero(real) = B | zero_zero(real) != hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),B),A). [resolve(5832,a,5849,a)]. 95.20/95.02 Derived: hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),C)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),C)). [resolve(5832,a,5851,a)]. 95.20/95.02 Derived: zero_zero(real) = hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),zero_zero(real)),A). [resolve(5832,a,5852,a)]. 95.20/95.02 Derived: hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),B) = hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),C) | zero_zero(real) != A. [resolve(5832,a,5854,a)]. 95.20/95.02 Derived: hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),B) = hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),C) | ti(real,B) != ti(real,C). [resolve(5832,a,5855,a)]. 95.20/95.02 5856 -real_vector(A) | hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),C) != hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),D),C) | B = D | ti(A,C) = zero_zero(A) # label(fact_4732_scaleR__cancel__right) # label(axiom). [clausify(4219)]. 95.20/95.02 5857 -real_vector(A) | hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),C) = hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),D),C) | B != D # label(fact_4732_scaleR__cancel__right) # label(axiom). [clausify(4219)]. 95.20/95.02 Derived: hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),B) = hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),C),B) | A != C. [resolve(5857,a,5832,a)]. 95.20/95.02 5858 -real_vector(A) | hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),C) = hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),D),C) | ti(A,C) != zero_zero(A) # label(fact_4732_scaleR__cancel__right) # label(axiom). [clausify(4219)]. 95.20/95.02 Derived: hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),B) = hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),C),B) | ti(real,B) != zero_zero(real). [resolve(5858,a,5832,a)]. 95.20/95.02 5859 real_vector(complex) # label(arity_Complex_Ocomplex___RealVector_Oreal__vector) # label(axiom). [assumption]. 95.20/95.02 Derived: hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B)),C) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),C)),hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),B),C)). [resolve(5859,a,5833,a)]. 95.20/95.02 Derived: zero_zero(real) = A | hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),B) != hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),C) | ti(complex,B) = ti(complex,C). [resolve(5859,a,5834,a)]. 95.20/95.02 Derived: hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co1399186613setsum(A,complex),hAPP(complex,fun(A,complex),hAPP(fun(A,fun(complex,complex)),fun(complex,fun(A,complex)),combc(A,complex,complex),hAPP(fun(A,real),fun(A,fun(complex,complex)),hAPP(fun(real,fun(complex,complex)),fun(fun(A,real),fun(A,fun(complex,complex))),combb(real,fun(complex,complex),A),scaleR_scaleR(complex)),B)),C)),D) = hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),B),D)),C). [resolve(5859,a,5835,a)]. 95.20/95.02 Derived: hAPP(complex,complex,uminus_uminus(complex),hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),B)) = hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),hAPP(complex,complex,uminus_uminus(complex),B)). [resolve(5859,a,5836,a)]. 95.22/95.03 Derived: hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co1399186613setsum(A,complex),hAPP(fun(A,complex),fun(A,complex),hAPP(fun(complex,complex),fun(fun(A,complex),fun(A,complex)),combb(complex,complex,A),hAPP(real,fun(complex,complex),scaleR_scaleR(complex),B)),C)),D) = hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),B),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co1399186613setsum(A,complex),C),D)). [resolve(5859,a,5837,a)]. 95.22/95.03 Derived: hAPP(complex,complex,uminus_uminus(complex),hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),B)) = hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),hAPP(real,real,uminus_uminus(real),A)),B). [resolve(5859,a,5838,a)]. 95.22/95.03 Derived: hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B)),C) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),C)),hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),B),C)). [resolve(5859,a,5839,a)]. 95.22/95.03 Derived: hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),B),C)) = hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),B),hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),C)). [resolve(5859,a,5840,a)]. 95.22/95.03 Derived: ti(complex,A) = zero_zero(complex) | hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),B),A) != hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),C),A) | B = C. [resolve(5859,a,5841,a)]. 95.22/95.03 Derived: ti(complex,A) = hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),one_one(real)),A). [resolve(5859,a,5842,a)]. 95.22/95.03 Derived: hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),B),C)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),B)),hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),C)). [resolve(5859,a,5843,a)]. 95.22/95.03 Derived: hAPP(complex,complex,uminus_uminus(complex),A) = hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),hAPP(int,real,number_number_of(real),min)),A). [resolve(5859,a,5844,a)]. 95.22/95.03 Derived: zero_zero(complex) = hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),zero_zero(complex)). [resolve(5859,a,5845,a)]. 95.22/95.03 Derived: ti(complex,A) != zero_zero(complex) | zero_zero(complex) = hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),B),A). [resolve(5859,a,5847,a)]. 95.22/95.03 Derived: zero_zero(real) != A | zero_zero(complex) = hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),B). [resolve(5859,a,5848,a)]. 95.22/95.03 Derived: ti(complex,A) = zero_zero(complex) | zero_zero(real) = B | zero_zero(complex) != hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),B),A). [resolve(5859,a,5849,a)]. 95.22/95.03 Derived: hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),B),C)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),B)),hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),C)). [resolve(5859,a,5851,a)]. 95.22/95.03 Derived: zero_zero(complex) = hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),zero_zero(real)),A). [resolve(5859,a,5852,a)]. 98.02/97.83 Derived: hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),B) = hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),C) | zero_zero(real) != A. [resolve(5859,a,5854,a)]. 98.02/97.83 Derived: hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),B) = hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),C) | ti(complex,B) != ti(complex,C). [resolve(5859,a,5855,a)]. 98.02/97.83 Derived: hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),B) = hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),C),B) | A != C. [resolve(5859,a,5857,a)]. 98.02/97.83 Derived: hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),B) = hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),C),B) | ti(complex,B) != zero_zero(complex). [resolve(5859,a,5858,a)]. 98.02/97.83 5860 -real_vector(A) | hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),C)),D) = hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),C),D)) # label(fact_4716_real__vector_Oscale__scale) # label(axiom). [clausify(4562)]. 98.02/97.83 Derived: hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),C) = hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),B),C)). [resolve(5860,a,5832,a)]. 98.02/97.83 Derived: hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),C) = hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),B),C)). [resolve(5860,a,5859,a)]. 98.02/97.83 5861 -real_vector(A) | ti(fun(real,fun(A,A)),scaleR_scaleR(A)) = scaleR_scaleR(A) # label(tsy_c_RealVector_OscaleR__class_OscaleR_1_res) # label(axiom). [clausify(4571)]. 98.02/97.83 Derived: ti(fun(real,fun(real,real)),scaleR_scaleR(real)) = scaleR_scaleR(real). [resolve(5861,a,5832,a)]. 98.02/97.83 Derived: ti(fun(real,fun(complex,complex)),scaleR_scaleR(complex)) = scaleR_scaleR(complex). [resolve(5861,a,5859,a)]. 98.02/97.83 5862 -real_vector(A) | hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),D)),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),C),D)) # label(fact_4696_scaleR__left_Odiff) # label(axiom). [clausify(4723)]. 98.02/97.83 5863 ring_div(int) # label(arity_Int_Oint___Divides_Oring__div) # label(axiom). [assumption]. 98.02/97.83 5864 -ring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),C) != hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),D),C) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,uminus_uminus(A),B)),C) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,uminus_uminus(A),D)),C) # label(fact_1945_mod__minus__cong) # label(axiom). [clausify(88)]. 98.02/97.83 5865 -ring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),C)),D)),C) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),D)),C) # label(fact_1300_mod__diff__left__eq) # label(axiom). [clausify(658)]. 98.02/97.83 Derived: hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A),B) != hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),C),B) | hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),A)),B) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),C)),B). [resolve(5863,a,5864,a)]. 98.02/97.83 5866 -ring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | hAPP(A,A,uminus_uminus(A),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),C),B)) = hAPP(A,A,hAPP(A,fun(A,A),div_div(A),hAPP(A,A,uminus_uminus(A),C)),B) # label(fact_1985_dvd__neg__div) # label(axiom). [clausify(1799)]. 98.02/97.83 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),A)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,uminus_uminus(int),B)),A). [resolve(5866,a,5863,a)]. 101.89/101.71 5867 -ring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | hAPP(A,A,uminus_uminus(A),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),C),B)) = hAPP(A,A,hAPP(A,fun(A,A),div_div(A),C),hAPP(A,A,uminus_uminus(A),B)) # label(fact_1986_dvd__div__neg) # label(axiom). [clausify(1959)]. 101.89/101.71 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),A)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),hAPP(int,int,uminus_uminus(int),A)). [resolve(5867,a,5863,a)]. 101.89/101.71 5868 -ring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),C) != hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),D),C) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),E),C) != hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),F),C) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),E)),C) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),D),F)),C) # label(fact_1298_mod__diff__cong) # label(axiom). [clausify(3025)]. 101.89/101.71 Derived: hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A),B) != hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),C),B) | hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),D),B) != hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),E),B) | hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),D)),B) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),C),E)),B). [resolve(5868,a,5863,a)]. 101.89/101.71 5869 -ring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),D),C))),C) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),D)),C) # label(fact_1299_mod__diff__eq) # label(axiom). [clausify(3214)]. 101.89/101.71 Derived: hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),C),B))),B) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),C)),B). [resolve(5869,a,5863,a)]. 101.89/101.71 5870 -ring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),C),D))),D) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C)),D) # label(fact_1301_mod__diff__right__eq) # label(axiom). [clausify(3911)]. 101.89/101.71 Derived: hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),C))),C) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B)),C). [resolve(5870,a,5863,a)]. 101.89/101.71 5871 -ring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,uminus_uminus(A),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),C))),C) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,uminus_uminus(A),B)),C) # label(fact_1944_mod__minus__eq) # label(axiom). [clausify(4812)]. 101.89/101.71 Derived: hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A),B))),B) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),A)),B). [resolve(5871,a,5863,a)]. 101.89/101.71 5872 linorder(int) # label(arity_Int_Oint___Orderings_Olinorder) # label(axiom). [assumption]. 101.89/101.71 5873 -linorder(A) | hAPP(A,fun(A,bool),ord_atMost(A),B) != hAPP(A,fun(A,bool),ord_atMost(A),C) | ti(A,B) = ti(A,C) # label(fact_4965_atMost__eq__iff) # label(axiom). [clausify(94)]. 101.89/101.71 5874 -linorder(A) | hAPP(A,fun(A,bool),ord_atMost(A),B) = hAPP(A,fun(A,bool),ord_atMost(A),C) | ti(A,B) != ti(A,C) # label(fact_4965_atMost__eq__iff) # label(axiom). [clausify(94)]. 101.89/101.71 5875 -linorder(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),B),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),C),D))),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),B),D)))) # label(fact_4980_min__max_Odistrib__sup__le) # label(axiom). [clausify(147)]. 101.89/101.76 5876 -linorder(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),C),B)) # label(fact_566_leI) # label(axiom). [clausify(187)]. 101.89/101.76 5877 -linorder(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | ti(A,C) != ti(A,B) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_564_linorder__antisym__conv1) # label(axiom). [clausify(205)]. 101.89/101.76 5878 -linorder(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | ti(A,C) = ti(A,B) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_564_linorder__antisym__conv1) # label(axiom). [clausify(205)]. 101.89/101.76 Derived: hAPP(int,fun(int,bool),ord_atMost(int),A) != hAPP(int,fun(int,bool),ord_atMost(int),B) | ti(int,A) = ti(int,B). [resolve(5872,a,5873,a)]. 101.89/101.76 Derived: hAPP(int,fun(int,bool),ord_atMost(int),A) = hAPP(int,fun(int,bool),ord_atMost(int),B) | ti(int,A) != ti(int,B). [resolve(5872,a,5874,a)]. 101.89/101.76 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),ord_max(int),A),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),B),C))),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),hAPP(int,int,hAPP(int,fun(int,int),ord_max(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),ord_max(int),A),C)))). [resolve(5872,a,5875,a)]. 101.89/101.76 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),B),A)). [resolve(5872,a,5876,a)]. 101.89/101.76 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | ti(int,B) != ti(int,A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(5872,a,5877,a)]. 101.89/101.76 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | ti(int,B) = ti(int,A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(5872,a,5878,a)]. 101.89/101.76 5879 linorder(rat) # label(arity_Rat_Orat___Orderings_Olinorder) # label(axiom). [assumption]. 101.89/101.76 Derived: hAPP(rat,fun(rat,bool),ord_atMost(rat),A) != hAPP(rat,fun(rat,bool),ord_atMost(rat),B) | ti(rat,A) = ti(rat,B). [resolve(5879,a,5873,a)]. 101.89/101.76 Derived: hAPP(rat,fun(rat,bool),ord_atMost(rat),A) = hAPP(rat,fun(rat,bool),ord_atMost(rat),B) | ti(rat,A) != ti(rat,B). [resolve(5879,a,5874,a)]. 101.89/101.76 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_max(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),B),C))),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_max(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_max(rat),A),C)))). [resolve(5879,a,5875,a)]. 101.89/101.76 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),A)). [resolve(5879,a,5876,a)]. 101.89/101.76 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | ti(rat,B) != ti(rat,A) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)). [resolve(5879,a,5877,a)]. 101.89/101.76 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | ti(rat,B) = ti(rat,A) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)). [resolve(5879,a,5878,a)]. 101.89/101.76 5880 -linorder(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),C),B)) # label(fact_565_not__leE) # label(axiom). [clausify(369)]. 101.89/101.76 5881 -linorder(A) | hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),C) # label(fact_4834_min__max_Oinf__left__idem) # label(axiom). [clausify(384)]. 101.89/101.76 Derived: hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),B)) = hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),B). [resolve(5881,a,5872,a)]. 101.89/101.76 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),B)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),B). [resolve(5881,a,5879,a)]. 101.99/101.82 5882 -linorder(A) | hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),B),D)) = hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),B),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),C),D)) # label(fact_4976_min__max_Osup__inf__distrib1) # label(axiom). [clausify(415)]. 101.99/101.82 Derived: hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),hAPP(int,int,hAPP(int,fun(int,int),ord_max(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),ord_max(int),A),C)) = hAPP(int,int,hAPP(int,fun(int,int),ord_max(int),A),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),B),C)). [resolve(5882,a,5872,a)]. 101.99/101.82 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_max(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_max(rat),A),C)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_max(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),B),C)). [resolve(5882,a,5879,a)]. 101.99/101.82 5883 -linorder(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),C),B)) # label(fact_571_linorder__not__le) # label(axiom). [clausify(433)]. 101.99/101.82 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),B),A)). [resolve(5883,a,5872,a)]. 101.99/101.82 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),A)). [resolve(5883,a,5879,a)]. 101.99/101.82 5884 -linorder(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),C),B)) # label(fact_571_linorder__not__le) # label(axiom). [clausify(433)]. 101.99/101.82 5885 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),hAPP(A,fun(A,bool),hAPP(A,fun(A,fun(A,bool)),ord_atLeastLessThan(A),B),D)),hAPP(A,fun(A,bool),hAPP(A,fun(A,fun(A,bool)),ord_atLeastLessThan(A),B),C)) = hAPP(A,fun(A,bool),hAPP(A,fun(A,fun(A,bool)),ord_atLeastLessThan(A),C),D) # label(fact_4220_ivl__diff) # label(axiom). [clausify(492)]. 101.99/101.82 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,bool),fun(fun(int,bool),fun(int,bool)),minus_minus(fun(int,bool)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),A),C)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),A),B)) = hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),B),C). [resolve(5885,a,5872,a)]. 101.99/101.82 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | hAPP(fun(rat,bool),fun(rat,bool),hAPP(fun(rat,bool),fun(fun(rat,bool),fun(rat,bool)),minus_minus(fun(rat,bool)),hAPP(rat,fun(rat,bool),hAPP(rat,fun(rat,fun(rat,bool)),ord_atLeastLessThan(rat),A),C)),hAPP(rat,fun(rat,bool),hAPP(rat,fun(rat,fun(rat,bool)),ord_atLeastLessThan(rat),A),B)) = hAPP(rat,fun(rat,bool),hAPP(rat,fun(rat,fun(rat,bool)),ord_atLeastLessThan(rat),B),C). [resolve(5885,a,5879,a)]. 101.99/101.82 5886 -number(A) | -linorder(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(int,A,number_number_of(A),B)),hAPP(int,A,number_number_of(A),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(int,A,number_number_of(A),C)),hAPP(int,A,number_number_of(A),B))) # label(fact_250_le__number__of__eq__not__less) # label(axiom). [clausify(628)]. 101.99/101.82 5887 -number(A) | -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(int,A,number_number_of(A),B)),hAPP(int,A,number_number_of(A),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(int,A,number_number_of(A),C)),hAPP(int,A,number_number_of(A),B))) # label(fact_250_le__number__of__eq__not__less) # label(axiom). [clausify(628)]. 101.99/101.82 5888 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less(fun(A,bool)),hAPP(A,fun(A,bool),ord_lessThan(A),B)),hAPP(A,fun(A,bool),ord_lessThan(A),C))) # label(fact_4347_lessThan__strict__subset__iff) # label(axiom). [clausify(731)]. 102.01/101.91 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,bool),fun(fun(int,bool),bool),ord_less(fun(int,bool)),hAPP(int,fun(int,bool),ord_lessThan(int),A)),hAPP(int,fun(int,bool),ord_lessThan(int),B))). [resolve(5888,a,5872,a)]. 102.01/101.91 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | hBOOL(hAPP(fun(rat,bool),bool,hAPP(fun(rat,bool),fun(fun(rat,bool),bool),ord_less(fun(rat,bool)),hAPP(rat,fun(rat,bool),ord_lessThan(rat),A)),hAPP(rat,fun(rat,bool),ord_lessThan(rat),B))). [resolve(5888,a,5879,a)]. 102.01/101.91 5889 -linorder(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less(fun(A,bool)),hAPP(A,fun(A,bool),ord_lessThan(A),B)),hAPP(A,fun(A,bool),ord_lessThan(A),C))) # label(fact_4347_lessThan__strict__subset__iff) # label(axiom). [clausify(731)]. 102.01/101.91 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,bool),fun(fun(int,bool),bool),ord_less(fun(int,bool)),hAPP(int,fun(int,bool),ord_lessThan(int),A)),hAPP(int,fun(int,bool),ord_lessThan(int),B))). [resolve(5889,a,5872,a)]. 102.01/101.91 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | -hBOOL(hAPP(fun(rat,bool),bool,hAPP(fun(rat,bool),fun(fun(rat,bool),bool),ord_less(fun(rat,bool)),hAPP(rat,fun(rat,bool),ord_lessThan(rat),A)),hAPP(rat,fun(rat,bool),ord_lessThan(rat),B))). [resolve(5889,a,5879,a)]. 102.01/101.91 5890 -linorder(A) | hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),C),D)) # label(fact_4837_min__max_Oinf_Oassoc) # label(axiom). [clausify(776)]. 102.01/101.91 Derived: hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),B)),C) = hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),B),C)). [resolve(5890,a,5872,a)]. 102.01/101.91 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),B)),C) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),B),C)). [resolve(5890,a,5879,a)]. 102.01/101.91 5891 -linorder(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | ti(A,C) = ti(A,B) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),B)) # label(fact_533_linorder__less__linear) # label(axiom). [clausify(1092)]. 102.01/101.91 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | ti(rat,B) = ti(rat,A) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),A)). [resolve(5891,a,5879,a)]. 102.01/101.91 5892 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),B),C)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),D)) # label(fact_4983_max__less__iff__conj) # label(axiom). [clausify(1173)]. 102.01/101.91 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),ord_max(int),A),B)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),C)). [resolve(5892,a,5872,a)]. 102.01/101.91 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_max(rat),A),B)),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),C)). [resolve(5892,a,5879,a)]. 102.01/101.91 5893 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),B),C)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) # label(fact_4983_max__less__iff__conj) # label(axiom). [clausify(1173)]. 102.01/101.91 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),ord_max(int),A),B)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)). [resolve(5893,a,5872,a)]. 102.01/101.91 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_max(rat),A),B)),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),C)). [resolve(5893,a,5879,a)]. 102.11/101.97 5894 -linorder(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),B),C)),D)) | -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),D)) # label(fact_4983_max__less__iff__conj) # label(axiom). [clausify(1173)]. 102.11/101.97 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),ord_max(int),A),B)),C)) | -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),C)). [resolve(5894,a,5872,a)]. 102.11/101.97 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_max(rat),A),B)),C)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),C)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),C)). [resolve(5894,a,5879,a)]. 102.11/101.97 5895 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),C),E))) # label(fact_4853_min__max_Oinf__mono) # label(axiom). [clausify(1256)]. 102.11/101.97 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),D)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),C)),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),B),D))). [resolve(5895,a,5872,a)]. 102.11/101.97 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),C),D)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),C)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),B),D))). [resolve(5895,a,5879,a)]. 102.11/101.97 5896 -linorder(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)) # label(fact_534_not__less__iff__gr__or__eq) # label(axiom). [clausify(1268)]. 102.11/101.97 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)). [resolve(5896,a,5872,a)]. 102.11/101.97 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),A)). [resolve(5896,a,5879,a)]. 102.11/101.97 5897 -linorder(A) | ti(A,B) != ti(A,C) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),B)) # label(fact_534_not__less__iff__gr__or__eq) # label(axiom). [clausify(1268)]. 102.11/101.97 Derived: ti(int,A) != ti(int,B) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),A)). [resolve(5897,a,5872,a)]. 102.11/101.97 Derived: ti(rat,A) != ti(rat,B) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),A)). [resolve(5897,a,5879,a)]. 102.11/101.97 5898 -linorder(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | ti(A,B) = ti(A,C) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),B)) # label(fact_534_not__less__iff__gr__or__eq) # label(axiom). [clausify(1268)]. 102.11/101.97 5899 -linorder(A) | ti(A,B) != ti(A,C) | hAPP(A,fun(A,bool),ord_lessThan(A),B) = hAPP(A,fun(A,bool),ord_lessThan(A),C) # label(fact_4343_lessThan__eq__iff) # label(axiom). [clausify(1377)]. 102.11/101.97 Derived: ti(int,A) != ti(int,B) | hAPP(int,fun(int,bool),ord_lessThan(int),A) = hAPP(int,fun(int,bool),ord_lessThan(int),B). [resolve(5899,a,5872,a)]. 102.11/101.97 Derived: ti(rat,A) != ti(rat,B) | hAPP(rat,fun(rat,bool),ord_lessThan(rat),A) = hAPP(rat,fun(rat,bool),ord_lessThan(rat),B). [resolve(5899,a,5879,a)]. 102.11/101.97 5900 -linorder(A) | ti(A,B) = ti(A,C) | hAPP(A,fun(A,bool),ord_lessThan(A),B) != hAPP(A,fun(A,bool),ord_lessThan(A),C) # label(fact_4343_lessThan__eq__iff) # label(axiom). [clausify(1377)]. 102.11/101.97 Derived: ti(int,A) = ti(int,B) | hAPP(int,fun(int,bool),ord_lessThan(int),A) != hAPP(int,fun(int,bool),ord_lessThan(int),B). [resolve(5900,a,5872,a)]. 102.20/102.04 Derived: ti(rat,A) = ti(rat,B) | hAPP(rat,fun(rat,bool),ord_lessThan(rat),A) != hAPP(rat,fun(rat,bool),ord_lessThan(rat),B). [resolve(5900,a,5879,a)]. 102.20/102.04 5901 -linorder(A) | hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),C),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),D)) # label(fact_4835_min__max_Oinf_Oleft__commute) # label(axiom). [clausify(1383)]. 102.20/102.04 Derived: hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),B),C)) = hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),B),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),C)). [resolve(5901,a,5872,a)]. 102.20/102.04 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),B),C)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),C)). [resolve(5901,a,5879,a)]. 102.20/102.04 5902 -linorder(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)) | ti(A,B) = ti(A,C) # label(fact_532_linorder__antisym__conv3) # label(axiom). [clausify(1474)]. 102.20/102.04 5903 -linorder(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)) | ti(A,B) != ti(A,C) # label(fact_532_linorder__antisym__conv3) # label(axiom). [clausify(1474)]. 102.20/102.04 5904 -linorder(A) | ti(A,B) = ti(A,C) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) # label(fact_531_linorder__neqE) # label(axiom). [clausify(1525)]. 102.20/102.04 5905 -linorder(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),C)),C)) # label(fact_4844_min__max_Oinf__le2) # label(axiom). [clausify(1569)]. 102.20/102.04 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),B)),B)). [resolve(5905,a,5872,a)]. 102.20/102.04 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),B)),B)). [resolve(5905,a,5879,a)]. 102.20/102.04 5906 linorder(code_code_numeral) # label(arity_Code__Numeral_Ocode__numeral___Orderings_Olinorder) # label(axiom). [assumption]. 102.20/102.04 Derived: hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_atMost(code_code_numeral),A) != hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_atMost(code_code_numeral),B) | ti(code_code_numeral,A) = ti(code_code_numeral,B). [resolve(5906,a,5873,a)]. 102.20/102.04 Derived: hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_atMost(code_code_numeral),A) = hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_atMost(code_code_numeral),B) | ti(code_code_numeral,A) != ti(code_code_numeral,B). [resolve(5906,a,5874,a)]. 102.20/102.04 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_max(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),B),C))),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_max(code_code_numeral),A),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_max(code_code_numeral),A),C)))). [resolve(5906,a,5875,a)]. 102.20/102.04 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),B),A)). [resolve(5906,a,5876,a)]. 102.20/102.04 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),B)) | ti(code_code_numeral,B) != ti(code_code_numeral,A) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),B)). [resolve(5906,a,5877,a)]. 102.20/102.06 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),B)) | ti(code_code_numeral,B) = ti(code_code_numeral,A) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),B)). [resolve(5906,a,5878,a)]. 102.20/102.06 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),B)) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),B). [resolve(5906,a,5881,a)]. 102.20/102.06 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_max(code_code_numeral),A),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_max(code_code_numeral),A),C)) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_max(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),B),C)). [resolve(5906,a,5882,a)]. 102.20/102.06 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),B),A)). [resolve(5906,a,5883,a)]. 102.20/102.06 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),B)) | hAPP(fun(code_code_numeral,bool),fun(code_code_numeral,bool),hAPP(fun(code_code_numeral,bool),fun(fun(code_code_numeral,bool),fun(code_code_numeral,bool)),minus_minus(fun(code_code_numeral,bool)),hAPP(code_code_numeral,fun(code_code_numeral,bool),hAPP(code_code_numeral,fun(code_code_numeral,fun(code_code_numeral,bool)),ord_atLeastLessThan(code_code_numeral),A),C)),hAPP(code_code_numeral,fun(code_code_numeral,bool),hAPP(code_code_numeral,fun(code_code_numeral,fun(code_code_numeral,bool)),ord_atLeastLessThan(code_code_numeral),A),B)) = hAPP(code_code_numeral,fun(code_code_numeral,bool),hAPP(code_code_numeral,fun(code_code_numeral,fun(code_code_numeral,bool)),ord_atLeastLessThan(code_code_numeral),B),C). [resolve(5906,a,5885,a)]. 102.20/102.06 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),B)) | hBOOL(hAPP(fun(code_code_numeral,bool),bool,hAPP(fun(code_code_numeral,bool),fun(fun(code_code_numeral,bool),bool),ord_less(fun(code_code_numeral,bool)),hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_lessThan(code_code_numeral),A)),hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_lessThan(code_code_numeral),B))). [resolve(5906,a,5888,a)]. 102.20/102.06 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),B)) | -hBOOL(hAPP(fun(code_code_numeral,bool),bool,hAPP(fun(code_code_numeral,bool),fun(fun(code_code_numeral,bool),bool),ord_less(fun(code_code_numeral,bool)),hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_lessThan(code_code_numeral),A)),hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_lessThan(code_code_numeral),B))). [resolve(5906,a,5889,a)]. 102.20/102.06 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),B)),C) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),B),C)). [resolve(5906,a,5890,a)]. 102.20/102.07 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),B)) | ti(code_code_numeral,B) = ti(code_code_numeral,A) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),B),A)). [resolve(5906,a,5891,a)]. 102.20/102.07 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_max(code_code_numeral),A),B)),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),C)). [resolve(5906,a,5892,a)]. 102.20/102.07 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_max(code_code_numeral),A),B)),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),B),C)). [resolve(5906,a,5893,a)]. 102.20/102.07 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_max(code_code_numeral),A),B)),C)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),C)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),B),C)). [resolve(5906,a,5894,a)]. 102.20/102.07 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),C),D)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),C)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),B),D))). [resolve(5906,a,5895,a)]. 102.20/102.07 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),B),A)). [resolve(5906,a,5896,a)]. 102.20/102.07 Derived: ti(code_code_numeral,A) != ti(code_code_numeral,B) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),B),A)). [resolve(5906,a,5897,a)]. 102.20/102.07 Derived: ti(code_code_numeral,A) != ti(code_code_numeral,B) | hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_lessThan(code_code_numeral),A) = hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_lessThan(code_code_numeral),B). [resolve(5906,a,5899,a)]. 102.20/102.07 Derived: ti(code_code_numeral,A) = ti(code_code_numeral,B) | hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_lessThan(code_code_numeral),A) != hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_lessThan(code_code_numeral),B). [resolve(5906,a,5900,a)]. 102.20/102.07 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),B),C)) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),B),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),C)). [resolve(5906,a,5901,a)]. 102.28/102.09 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),B)),B)). [resolve(5906,a,5905,a)]. 102.28/102.09 5907 -linorder(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),B),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),C),D))) # label(fact_4851_min__max_Ole__infI) # label(axiom). [clausify(1594)]. 102.28/102.09 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),B),C))). [resolve(5907,a,5872,a)]. 102.28/102.09 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),B),C))). [resolve(5907,a,5879,a)]. 102.28/102.09 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),B),C))). [resolve(5907,a,5906,a)]. 102.28/102.09 5908 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),hAPP(A,fun(A,bool),ord_lessThan(A),B)),hAPP(A,fun(A,bool),ord_lessThan(A),C))) # label(fact_4346_lessThan__subset__iff) # label(axiom). [clausify(1610)]. 102.28/102.09 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,bool),fun(fun(int,bool),bool),ord_less_eq(fun(int,bool)),hAPP(int,fun(int,bool),ord_lessThan(int),A)),hAPP(int,fun(int,bool),ord_lessThan(int),B))). [resolve(5908,a,5872,a)]. 102.28/102.09 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | hBOOL(hAPP(fun(rat,bool),bool,hAPP(fun(rat,bool),fun(fun(rat,bool),bool),ord_less_eq(fun(rat,bool)),hAPP(rat,fun(rat,bool),ord_lessThan(rat),A)),hAPP(rat,fun(rat,bool),ord_lessThan(rat),B))). [resolve(5908,a,5879,a)]. 102.28/102.09 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),B)) | hBOOL(hAPP(fun(code_code_numeral,bool),bool,hAPP(fun(code_code_numeral,bool),fun(fun(code_code_numeral,bool),bool),ord_less_eq(fun(code_code_numeral,bool)),hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_lessThan(code_code_numeral),A)),hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_lessThan(code_code_numeral),B))). [resolve(5908,a,5906,a)]. 102.28/102.09 5909 -linorder(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),hAPP(A,fun(A,bool),ord_lessThan(A),B)),hAPP(A,fun(A,bool),ord_lessThan(A),C))) # label(fact_4346_lessThan__subset__iff) # label(axiom). [clausify(1610)]. 102.28/102.09 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,bool),fun(fun(int,bool),bool),ord_less_eq(fun(int,bool)),hAPP(int,fun(int,bool),ord_lessThan(int),A)),hAPP(int,fun(int,bool),ord_lessThan(int),B))). [resolve(5909,a,5872,a)]. 102.28/102.09 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | -hBOOL(hAPP(fun(rat,bool),bool,hAPP(fun(rat,bool),fun(fun(rat,bool),bool),ord_less_eq(fun(rat,bool)),hAPP(rat,fun(rat,bool),ord_lessThan(rat),A)),hAPP(rat,fun(rat,bool),ord_lessThan(rat),B))). [resolve(5909,a,5879,a)]. 102.41/102.24 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),B)) | -hBOOL(hAPP(fun(code_code_numeral,bool),bool,hAPP(fun(code_code_numeral,bool),fun(fun(code_code_numeral,bool),bool),ord_less_eq(fun(code_code_numeral,bool)),hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_lessThan(code_code_numeral),A)),hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_lessThan(code_code_numeral),B))). [resolve(5909,a,5906,a)]. 102.41/102.24 5910 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),D),B)),C)) # label(fact_4825_min__max_Oless__infI2) # label(axiom). [clausify(1695)]. 102.41/102.24 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),C),A)),B)). [resolve(5910,a,5872,a)]. 102.41/102.24 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),C),A)),B)). [resolve(5910,a,5879,a)]. 102.41/102.24 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),C),A)),B)). [resolve(5910,a,5906,a)]. 102.41/102.24 5911 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),C) # label(fact_4845_min__max_Ole__iff__inf) # label(axiom). [clausify(1818)]. 102.41/102.24 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | ti(int,A) = hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),B). [resolve(5911,a,5872,a)]. 102.41/102.24 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | ti(rat,A) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),B). [resolve(5911,a,5879,a)]. 102.41/102.24 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),B)) | ti(code_code_numeral,A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),B). [resolve(5911,a,5906,a)]. 102.41/102.24 5912 -linorder(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | ti(A,B) != hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),C) # label(fact_4845_min__max_Ole__iff__inf) # label(axiom). [clausify(1818)]. 102.41/102.24 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | ti(int,A) != hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),B). [resolve(5912,a,5872,a)]. 102.41/102.24 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | ti(rat,A) != hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),B). [resolve(5912,a,5879,a)]. 102.41/102.24 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),B)) | ti(code_code_numeral,A) != hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),B). [resolve(5912,a,5906,a)]. 102.41/102.24 5913 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),C),D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_4854_min__max_Ole__infE) # label(axiom). [clausify(2082)]. 102.41/102.24 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),B),C))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(5913,a,5872,a)]. 102.41/102.24 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),B),C))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)). [resolve(5913,a,5879,a)]. 102.41/102.26 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),B),C))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),B)). [resolve(5913,a,5906,a)]. 102.41/102.26 5914 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),C),D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),D)) # label(fact_4854_min__max_Ole__infE) # label(axiom). [clausify(2082)]. 102.41/102.26 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),B),C))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),C)). [resolve(5914,a,5872,a)]. 102.41/102.26 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),B),C))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),C)). [resolve(5914,a,5879,a)]. 102.41/102.26 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),B),C))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),C)). [resolve(5914,a,5906,a)]. 102.41/102.26 5915 -linorder(A) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),hAPP(A,fun(A,bool),hAPP(A,fun(A,fun(A,bool)),ord_atLeastLessThan(A),B),C)),hAPP(A,fun(A,bool),hAPP(A,fun(A,fun(A,bool)),ord_atLeastLessThan(A),D),E))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),E)) # label(fact_4230_atLeastLessThan__subset__iff) # label(axiom). [clausify(2127)]. 102.41/102.26 Derived: -hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,bool),fun(fun(int,bool),bool),ord_less_eq(fun(int,bool)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),A),B)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),C),D))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),D)). [resolve(5915,a,5872,a)]. 102.41/102.26 Derived: -hBOOL(hAPP(fun(rat,bool),bool,hAPP(fun(rat,bool),fun(fun(rat,bool),bool),ord_less_eq(fun(rat,bool)),hAPP(rat,fun(rat,bool),hAPP(rat,fun(rat,fun(rat,bool)),ord_atLeastLessThan(rat),A),B)),hAPP(rat,fun(rat,bool),hAPP(rat,fun(rat,fun(rat,bool)),ord_atLeastLessThan(rat),C),D))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),D)). [resolve(5915,a,5879,a)]. 102.41/102.26 Derived: -hBOOL(hAPP(fun(code_code_numeral,bool),bool,hAPP(fun(code_code_numeral,bool),fun(fun(code_code_numeral,bool),bool),ord_less_eq(fun(code_code_numeral,bool)),hAPP(code_code_numeral,fun(code_code_numeral,bool),hAPP(code_code_numeral,fun(code_code_numeral,fun(code_code_numeral,bool)),ord_atLeastLessThan(code_code_numeral),A),B)),hAPP(code_code_numeral,fun(code_code_numeral,bool),hAPP(code_code_numeral,fun(code_code_numeral,fun(code_code_numeral,bool)),ord_atLeastLessThan(code_code_numeral),C),D))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),B),A)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),B),D)). [resolve(5915,a,5906,a)]. 102.41/102.26 5916 -linorder(A) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),hAPP(A,fun(A,bool),hAPP(A,fun(A,fun(A,bool)),ord_atLeastLessThan(A),B),C)),hAPP(A,fun(A,bool),hAPP(A,fun(A,fun(A,bool)),ord_atLeastLessThan(A),D),E))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),B)) # label(fact_4230_atLeastLessThan__subset__iff) # label(axiom). [clausify(2127)]. 102.41/102.28 Derived: -hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,bool),fun(fun(int,bool),bool),ord_less_eq(fun(int,bool)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),A),B)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),C),D))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),A)). [resolve(5916,a,5872,a)]. 102.41/102.28 Derived: -hBOOL(hAPP(fun(rat,bool),bool,hAPP(fun(rat,bool),fun(fun(rat,bool),bool),ord_less_eq(fun(rat,bool)),hAPP(rat,fun(rat,bool),hAPP(rat,fun(rat,fun(rat,bool)),ord_atLeastLessThan(rat),A),B)),hAPP(rat,fun(rat,bool),hAPP(rat,fun(rat,fun(rat,bool)),ord_atLeastLessThan(rat),C),D))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),C),A)). [resolve(5916,a,5879,a)]. 102.41/102.28 Derived: -hBOOL(hAPP(fun(code_code_numeral,bool),bool,hAPP(fun(code_code_numeral,bool),fun(fun(code_code_numeral,bool),bool),ord_less_eq(fun(code_code_numeral,bool)),hAPP(code_code_numeral,fun(code_code_numeral,bool),hAPP(code_code_numeral,fun(code_code_numeral,fun(code_code_numeral,bool)),ord_atLeastLessThan(code_code_numeral),A),B)),hAPP(code_code_numeral,fun(code_code_numeral,bool),hAPP(code_code_numeral,fun(code_code_numeral,fun(code_code_numeral,bool)),ord_atLeastLessThan(code_code_numeral),C),D))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),B),A)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),C),A)). [resolve(5916,a,5906,a)]. 102.41/102.28 5917 linorder(real) # label(arity_RealDef_Oreal___Orderings_Olinorder) # label(axiom). [assumption]. 102.41/102.28 Derived: hAPP(real,fun(real,bool),ord_atMost(real),A) != hAPP(real,fun(real,bool),ord_atMost(real),B) | ti(real,A) = ti(real,B). [resolve(5917,a,5873,a)]. 102.41/102.28 Derived: hAPP(real,fun(real,bool),ord_atMost(real),A) = hAPP(real,fun(real,bool),ord_atMost(real),B) | ti(real,A) != ti(real,B). [resolve(5917,a,5874,a)]. 102.41/102.28 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),ord_max(real),A),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),B),C))),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),hAPP(real,real,hAPP(real,fun(real,real),ord_max(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),ord_max(real),A),C)))). [resolve(5917,a,5875,a)]. 102.41/102.28 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),B),A)). [resolve(5917,a,5876,a)]. 102.41/102.28 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | ti(real,B) != ti(real,A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)). [resolve(5917,a,5877,a)]. 102.41/102.28 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | ti(real,B) = ti(real,A) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)). [resolve(5917,a,5878,a)]. 102.41/102.28 Derived: hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),B)) = hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),B). [resolve(5917,a,5881,a)]. 102.41/102.28 Derived: hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),hAPP(real,real,hAPP(real,fun(real,real),ord_max(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),ord_max(real),A),C)) = hAPP(real,real,hAPP(real,fun(real,real),ord_max(real),A),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),B),C)). [resolve(5917,a,5882,a)]. 102.41/102.28 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),B),A)). [resolve(5917,a,5883,a)]. 102.41/102.31 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,bool),fun(fun(real,bool),fun(real,bool)),minus_minus(fun(real,bool)),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastLessThan(real),A),C)),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastLessThan(real),A),B)) = hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastLessThan(real),B),C). [resolve(5917,a,5885,a)]. 102.41/102.31 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | hBOOL(hAPP(fun(real,bool),bool,hAPP(fun(real,bool),fun(fun(real,bool),bool),ord_less(fun(real,bool)),hAPP(real,fun(real,bool),ord_lessThan(real),A)),hAPP(real,fun(real,bool),ord_lessThan(real),B))). [resolve(5917,a,5888,a)]. 102.41/102.31 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | -hBOOL(hAPP(fun(real,bool),bool,hAPP(fun(real,bool),fun(fun(real,bool),bool),ord_less(fun(real,bool)),hAPP(real,fun(real,bool),ord_lessThan(real),A)),hAPP(real,fun(real,bool),ord_lessThan(real),B))). [resolve(5917,a,5889,a)]. 102.41/102.31 Derived: hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),B)),C) = hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),B),C)). [resolve(5917,a,5890,a)]. 102.41/102.31 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | ti(real,B) = ti(real,A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),A)). [resolve(5917,a,5891,a)]. 102.41/102.31 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),ord_max(real),A),B)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),C)). [resolve(5917,a,5892,a)]. 102.41/102.31 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),ord_max(real),A),B)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),C)). [resolve(5917,a,5893,a)]. 102.41/102.31 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),ord_max(real),A),B)),C)) | -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),C)). [resolve(5917,a,5894,a)]. 102.41/102.31 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),C)),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),B),D))). [resolve(5917,a,5895,a)]. 102.41/102.31 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)). [resolve(5917,a,5896,a)]. 102.41/102.31 Derived: ti(real,A) != ti(real,B) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),A)). [resolve(5917,a,5897,a)]. 102.41/102.31 Derived: ti(real,A) != ti(real,B) | hAPP(real,fun(real,bool),ord_lessThan(real),A) = hAPP(real,fun(real,bool),ord_lessThan(real),B). [resolve(5917,a,5899,a)]. 102.41/102.31 Derived: ti(real,A) = ti(real,B) | hAPP(real,fun(real,bool),ord_lessThan(real),A) != hAPP(real,fun(real,bool),ord_lessThan(real),B). [resolve(5917,a,5900,a)]. 102.41/102.31 Derived: hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),B),C)) = hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),B),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),C)). [resolve(5917,a,5901,a)]. 102.41/102.31 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),B)),B)). [resolve(5917,a,5905,a)]. 102.41/102.31 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),B),C))). [resolve(5917,a,5907,a)]. 102.50/102.33 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | hBOOL(hAPP(fun(real,bool),bool,hAPP(fun(real,bool),fun(fun(real,bool),bool),ord_less_eq(fun(real,bool)),hAPP(real,fun(real,bool),ord_lessThan(real),A)),hAPP(real,fun(real,bool),ord_lessThan(real),B))). [resolve(5917,a,5908,a)]. 102.50/102.33 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(fun(real,bool),bool,hAPP(fun(real,bool),fun(fun(real,bool),bool),ord_less_eq(fun(real,bool)),hAPP(real,fun(real,bool),ord_lessThan(real),A)),hAPP(real,fun(real,bool),ord_lessThan(real),B))). [resolve(5917,a,5909,a)]. 102.50/102.33 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),C),A)),B)). [resolve(5917,a,5910,a)]. 102.50/102.33 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | ti(real,A) = hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),B). [resolve(5917,a,5911,a)]. 102.50/102.33 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | ti(real,A) != hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),B). [resolve(5917,a,5912,a)]. 102.50/102.33 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),B),C))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)). [resolve(5917,a,5913,a)]. 102.50/102.33 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),B),C))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),C)). [resolve(5917,a,5914,a)]. 102.50/102.33 Derived: -hBOOL(hAPP(fun(real,bool),bool,hAPP(fun(real,bool),fun(fun(real,bool),bool),ord_less_eq(fun(real,bool)),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastLessThan(real),A),B)),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastLessThan(real),C),D))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),D)). [resolve(5917,a,5915,a)]. 102.50/102.33 Derived: -hBOOL(hAPP(fun(real,bool),bool,hAPP(fun(real,bool),fun(fun(real,bool),bool),ord_less_eq(fun(real,bool)),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastLessThan(real),A),B)),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastLessThan(real),C),D))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),A)). [resolve(5917,a,5916,a)]. 102.50/102.33 5918 -linorder(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),C)),B)) # label(fact_4843_min__max_Oinf__le1) # label(axiom). [clausify(2152)]. 102.50/102.33 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),B)),A)). [resolve(5918,a,5872,a)]. 102.50/102.33 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),B)),A)). [resolve(5918,a,5879,a)]. 102.50/102.33 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),B)),A)). [resolve(5918,a,5906,a)]. 102.50/102.33 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),B)),A)). [resolve(5918,a,5917,a)]. 102.50/102.33 5919 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),D)),C)) # label(fact_4826_min__max_Oless__infI1) # label(axiom). [clausify(2153)]. 102.50/102.33 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),C)),B)). [resolve(5919,a,5872,a)]. 102.59/102.40 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),C)),B)). [resolve(5919,a,5879,a)]. 102.59/102.40 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),C)),B)). [resolve(5919,a,5906,a)]. 102.59/102.40 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),C)),B)). [resolve(5919,a,5917,a)]. 102.59/102.40 5920 -linorder(A) | hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),D),C)) = hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),D)),C) # label(fact_4978_min__max_Osup__inf__distrib2) # label(axiom). [clausify(2226)]. 102.59/102.40 Derived: hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),hAPP(int,int,hAPP(int,fun(int,int),ord_max(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),ord_max(int),C),B)) = hAPP(int,int,hAPP(int,fun(int,int),ord_max(int),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),C)),B). [resolve(5920,a,5872,a)]. 102.59/102.40 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_max(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_max(rat),C),B)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_max(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),C)),B). [resolve(5920,a,5879,a)]. 102.59/102.40 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_max(code_code_numeral),A),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_max(code_code_numeral),C),B)) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_max(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),C)),B). [resolve(5920,a,5906,a)]. 102.59/102.40 Derived: hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),hAPP(real,real,hAPP(real,fun(real,real),ord_max(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),ord_max(real),C),B)) = hAPP(real,real,hAPP(real,fun(real,real),ord_max(real),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),C)),B). [resolve(5920,a,5917,a)]. 102.59/102.40 5921 -linorder(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),B),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),C),D))) # label(fact_4852_min__max_Oinf__greatest) # label(axiom). [clausify(2285)]. 102.59/102.40 5922 -linorder(A) | hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),D)) # label(fact_4977_min__max_Oinf__sup__distrib1) # label(axiom). [clausify(2293)]. 102.59/102.40 Derived: hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),hAPP(int,int,hAPP(int,fun(int,int),ord_max(int),B),C)) = hAPP(int,int,hAPP(int,fun(int,int),ord_max(int),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),C)). [resolve(5922,a,5872,a)]. 102.59/102.40 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_max(rat),B),C)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_max(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),C)). [resolve(5922,a,5879,a)]. 102.59/102.43 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_max(code_code_numeral),B),C)) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_max(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),C)). [resolve(5922,a,5906,a)]. 102.59/102.43 Derived: hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),hAPP(real,real,hAPP(real,fun(real,real),ord_max(real),B),C)) = hAPP(real,real,hAPP(real,fun(real,real),ord_max(real),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),C)). [resolve(5922,a,5917,a)]. 102.59/102.43 5923 -linorder(A) | hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),C) = hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),C),B) # label(fact_4831_min__max_Oinf_Ocommute) # label(axiom). [clausify(2298)]. 102.59/102.43 Derived: hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),B) = hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),B),A). [resolve(5923,a,5872,a)]. 102.59/102.43 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),B) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),B),A). [resolve(5923,a,5879,a)]. 102.59/102.43 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),B) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),B),A). [resolve(5923,a,5906,a)]. 102.59/102.43 Derived: hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),B) = hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),B),A). [resolve(5923,a,5917,a)]. 102.59/102.43 5924 -linorder(A) | hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),C),D)) # label(fact_4979_min__max_Oinf__sup__distrib2) # label(axiom). [clausify(2329)]. 102.59/102.43 Derived: hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),hAPP(int,int,hAPP(int,fun(int,int),ord_max(int),A),B)),C) = hAPP(int,int,hAPP(int,fun(int,int),ord_max(int),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),C)),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),B),C)). [resolve(5924,a,5872,a)]. 102.59/102.43 Derived: hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_max(rat),A),B)),C) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_max(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),C)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),B),C)). [resolve(5924,a,5879,a)]. 102.59/102.43 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_max(code_code_numeral),A),B)),C) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_max(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),C)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),B),C)). [resolve(5924,a,5906,a)]. 102.59/102.43 Derived: hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),hAPP(real,real,hAPP(real,fun(real,real),ord_max(real),A),B)),C) = hAPP(real,real,hAPP(real,fun(real,real),ord_max(real),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),C)),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),B),C)). [resolve(5924,a,5917,a)]. 102.59/102.43 5925 -linorder(A) | hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),C),D)) # label(fact_4838_min__max_Oinf__assoc) # label(axiom). [clausify(2335)]. 102.74/102.53 5926 -linorder(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),C),B)) # label(fact_499_linorder__le__cases) # label(axiom). [clausify(2415)]. 102.74/102.53 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),A)). [resolve(5926,a,5872,a)]. 102.74/102.53 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),A)). [resolve(5926,a,5879,a)]. 102.74/102.53 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),B),A)). [resolve(5926,a,5906,a)]. 102.74/102.53 5927 -linorder(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),B) # label(fact_4830_min__max_Oinf_Oidem) # label(axiom). [clausify(2481)]. 102.74/102.53 Derived: ti(int,A) = hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),A). [resolve(5927,a,5872,a)]. 102.74/102.53 Derived: ti(rat,A) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),A). [resolve(5927,a,5879,a)]. 102.74/102.53 Derived: ti(code_code_numeral,A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),A). [resolve(5927,a,5906,a)]. 102.74/102.53 Derived: ti(real,A) = hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),A). [resolve(5927,a,5917,a)]. 102.74/102.53 5928 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),D),B)),C)) # label(fact_4855_min__le__iff__disj) # label(axiom). [clausify(2526)]. 102.74/102.53 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),C),A)),B)). [resolve(5928,a,5872,a)]. 102.74/102.53 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),C),A)),B)). [resolve(5928,a,5879,a)]. 102.74/102.53 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),C),A)),B)). [resolve(5928,a,5906,a)]. 102.74/102.53 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),C),A)),B)). [resolve(5928,a,5917,a)]. 102.74/102.53 5929 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),D)),C)) # label(fact_4855_min__le__iff__disj) # label(axiom). [clausify(2526)]. 102.74/102.53 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),C)),B)). [resolve(5929,a,5872,a)]. 102.74/102.53 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),C)),B)). [resolve(5929,a,5879,a)]. 102.74/102.53 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),C)),B)). [resolve(5929,a,5906,a)]. 102.75/102.57 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),C)),B)). [resolve(5929,a,5917,a)]. 102.75/102.57 5930 -linorder(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),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),D),B)),C)) # label(fact_4855_min__le__iff__disj) # label(axiom). [clausify(2526)]. 102.75/102.57 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),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),C),A)),B)). [resolve(5930,a,5872,a)]. 102.75/102.57 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),C),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),C),A)),B)). [resolve(5930,a,5879,a)]. 102.75/102.57 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),C),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),C),A)),B)). [resolve(5930,a,5906,a)]. 102.75/102.57 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),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),C),A)),B)). [resolve(5930,a,5917,a)]. 102.75/102.57 5931 -linorder(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),D))),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),C),D)))) # label(fact_4981_min__max_Odistrib__inf__le) # label(axiom). [clausify(2586)]. 102.75/102.57 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),ord_max(int),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),C))),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),hAPP(int,int,hAPP(int,fun(int,int),ord_max(int),B),C)))). [resolve(5931,a,5872,a)]. 102.75/102.57 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_max(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),C))),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_max(rat),B),C)))). [resolve(5931,a,5879,a)]. 102.75/102.57 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_max(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),C))),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_max(code_code_numeral),B),C)))). [resolve(5931,a,5906,a)]. 102.75/102.57 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),ord_max(real),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),C))),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),hAPP(real,real,hAPP(real,fun(real,real),ord_max(real),B),C)))). [resolve(5931,a,5917,a)]. 102.88/102.69 5932 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),D),B)),C)) # label(fact_4848_min__max_Ole__infI2) # label(axiom). [clausify(2588)]. 102.88/102.69 5933 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),C),D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_4846_min__max_Ole__inf__iff) # label(axiom). [clausify(2598)]. 102.88/102.69 5934 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),C),D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),D)) # label(fact_4846_min__max_Ole__inf__iff) # label(axiom). [clausify(2598)]. 102.88/102.69 5935 -linorder(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),C),D))) | -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),B),D)) # label(fact_4846_min__max_Ole__inf__iff) # label(axiom). [clausify(2598)]. 102.88/102.69 5936 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | ti(A,B) != ti(A,C) # label(fact_535_linorder__neq__iff) # label(axiom). [clausify(2647)]. 102.88/102.69 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | ti(int,A) != ti(int,B). [resolve(5936,a,5872,a)]. 102.88/102.69 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | ti(rat,A) != ti(rat,B). [resolve(5936,a,5879,a)]. 102.88/102.69 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),B)) | ti(code_code_numeral,A) != ti(code_code_numeral,B). [resolve(5936,a,5906,a)]. 102.88/102.69 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | ti(real,A) != ti(real,B). [resolve(5936,a,5917,a)]. 102.88/102.69 5937 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | ti(A,C) != ti(A,B) # label(fact_535_linorder__neq__iff) # label(axiom). [clausify(2647)]. 102.88/102.69 5938 -linorder(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)) | ti(A,B) = ti(A,C) # label(fact_535_linorder__neq__iff) # label(axiom). [clausify(2647)]. 102.88/102.69 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)) | ti(int,A) = ti(int,B). [resolve(5938,a,5872,a)]. 102.88/102.69 5939 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),f463(A,B)),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),B)) # label(fact_2857_pinf_I6_J) # label(axiom). [clausify(2754)]. 102.88/102.69 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),f463(int,A)),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),A)). [resolve(5939,a,5872,a)]. 102.88/102.69 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),f463(rat,A)),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),A)). [resolve(5939,a,5879,a)]. 102.88/102.69 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),f463(code_code_numeral,A)),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),B),A)). [resolve(5939,a,5906,a)]. 102.88/102.69 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),f463(real,A)),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),A)). [resolve(5939,a,5917,a)]. 102.88/102.69 5940 -plus(A) | -dvd(A) | -linorder(A) | ti(fun(A,fun(A,A)),plus_plus(A)) = plus_plus(A) # label(tsy_c_Groups_Oplus__class_Oplus_0_res) # label(axiom). [clausify(2833)]. 102.88/102.69 Derived: -plus(int) | -dvd(int) | ti(fun(int,fun(int,int)),plus_plus(int)) = plus_plus(int). [resolve(5940,c,5872,a)]. 103.35/103.15 Derived: -plus(rat) | -dvd(rat) | ti(fun(rat,fun(rat,rat)),plus_plus(rat)) = plus_plus(rat). [resolve(5940,c,5879,a)]. 103.35/103.15 Derived: -plus(code_code_numeral) | -dvd(code_code_numeral) | ti(fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)),plus_plus(code_code_numeral)) = plus_plus(code_code_numeral). [resolve(5940,c,5906,a)]. 103.35/103.15 Derived: -plus(real) | -dvd(real) | ti(fun(real,fun(real,real)),plus_plus(real)) = plus_plus(real). [resolve(5940,c,5917,a)]. 103.35/103.15 5941 -linorder(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),C),B)) # label(fact_561_leD) # label(axiom). [clausify(2929)]. 103.35/103.15 5942 -linorder(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),C),B)) # label(fact_513_linorder__linear) # label(axiom). [clausify(3055)]. 103.35/103.15 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),A)). [resolve(5942,a,5917,a)]. 103.35/103.15 5943 -linorder(A) | hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),C)) = hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),C) # label(fact_4833_min__max_Oinf_Oleft__idem) # label(axiom). [clausify(3106)]. 103.35/103.15 5944 -linorder(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),B),C)) # label(fact_4975_min__max_Oinf__sup__absorb) # label(axiom). [clausify(3168)]. 103.35/103.15 Derived: ti(int,A) = hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),hAPP(int,int,hAPP(int,fun(int,int),ord_max(int),A),B)). [resolve(5944,a,5872,a)]. 103.35/103.15 Derived: ti(rat,A) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_max(rat),A),B)). [resolve(5944,a,5879,a)]. 103.35/103.15 Derived: ti(code_code_numeral,A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_max(code_code_numeral),A),B)). [resolve(5944,a,5906,a)]. 103.35/103.15 Derived: ti(real,A) = hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),hAPP(real,real,hAPP(real,fun(real,real),ord_max(real),A),B)). [resolve(5944,a,5917,a)]. 103.35/103.15 5945 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),f550(A,B)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_2852_pinf_I8_J) # label(axiom). [clausify(3276)]. 103.35/103.15 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),f550(int,A)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(5945,a,5872,a)]. 103.35/103.15 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),f550(rat,A)),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)). [resolve(5945,a,5879,a)]. 103.35/103.15 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),f550(code_code_numeral,A)),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),B)). [resolve(5945,a,5906,a)]. 103.35/103.15 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),f550(real,A)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)). [resolve(5945,a,5917,a)]. 103.35/103.15 5946 -linorder(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | ti(A,C) = ti(A,B) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),B)) # label(fact_515_linorder__cases) # label(axiom). [clausify(3332)]. 103.35/103.15 5947 -linorder(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),C),B)) # label(fact_572_linorder__not__less) # label(axiom). [clausify(3408)]. 103.35/103.15 5948 -linorder(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),C),B)) # label(fact_572_linorder__not__less) # label(axiom). [clausify(3408)]. 103.35/103.15 5949 -linorder(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),B),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),C)) # label(fact_4974_min__max_Osup__inf__absorb) # label(axiom). [clausify(3447)]. 103.51/103.33 Derived: ti(int,A) = hAPP(int,int,hAPP(int,fun(int,int),ord_max(int),A),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),B)). [resolve(5949,a,5872,a)]. 103.51/103.33 Derived: ti(rat,A) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_max(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),B)). [resolve(5949,a,5879,a)]. 103.51/103.33 Derived: ti(code_code_numeral,A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_max(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),B)). [resolve(5949,a,5906,a)]. 103.51/103.33 Derived: ti(real,A) = hAPP(real,real,hAPP(real,fun(real,real),ord_max(real),A),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),B)). [resolve(5949,a,5917,a)]. 103.51/103.33 5950 -linorder(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),B),hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),C),D))) # label(fact_4986_min__max_Oless__supI1) # label(axiom). [clausify(3491)]. 103.51/103.33 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(int,int,hAPP(int,fun(int,int),ord_max(int),B),C))). [resolve(5950,a,5872,a)]. 103.51/103.33 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_max(rat),B),C))). [resolve(5950,a,5879,a)]. 103.51/103.33 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_max(code_code_numeral),B),C))). [resolve(5950,a,5906,a)]. 103.51/103.33 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(real,real,hAPP(real,fun(real,real),ord_max(real),B),C))). [resolve(5950,a,5917,a)]. 103.51/103.33 5951 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),C) # label(fact_4849_min__max_Oinf__absorb1) # label(axiom). [clausify(3520)]. 103.51/103.33 5952 -linorder(A) | ord_min(A) = hAPP(fun(A,fun(A,bool)),fun(A,fun(A,A)),max(A),hAPP(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),ord_less_eq(A))) # label(fact_4859_dual__max) # label(axiom). [clausify(3544)]. 103.51/103.33 Derived: ord_min(int) = hAPP(fun(int,fun(int,bool)),fun(int,fun(int,int)),max(int),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less_eq(int))). [resolve(5952,a,5872,a)]. 103.51/103.33 Derived: ord_min(rat) = hAPP(fun(rat,fun(rat,bool)),fun(rat,fun(rat,rat)),max(rat),hAPP(fun(rat,fun(rat,bool)),fun(rat,fun(rat,bool)),combc(rat,rat,bool),ord_less_eq(rat))). [resolve(5952,a,5879,a)]. 103.51/103.33 Derived: ord_min(code_code_numeral) = hAPP(fun(code_code_numeral,fun(code_code_numeral,bool)),fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)),max(code_code_numeral),hAPP(fun(code_code_numeral,fun(code_code_numeral,bool)),fun(code_code_numeral,fun(code_code_numeral,bool)),combc(code_code_numeral,code_code_numeral,bool),ord_less_eq(code_code_numeral))). [resolve(5952,a,5906,a)]. 103.51/103.33 Derived: ord_min(real) = hAPP(fun(real,fun(real,bool)),fun(real,fun(real,real)),max(real),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),ord_less_eq(real))). [resolve(5952,a,5917,a)]. 103.51/103.33 5953 -linorder(A) | hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),C),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),D)) # label(fact_4836_min__max_Oinf__left__commute) # label(axiom). [clausify(3600)]. 103.51/103.33 5954 -linorder(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),B),hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),D),C))) # label(fact_4985_min__max_Oless__supI2) # label(axiom). [clausify(3658)]. 103.65/103.48 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),A),hAPP(int,int,hAPP(int,fun(int,int),ord_max(int),C),B))). [resolve(5954,a,5872,a)]. 103.65/103.48 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_max(rat),C),B))). [resolve(5954,a,5879,a)]. 103.65/103.48 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_max(code_code_numeral),C),B))). [resolve(5954,a,5906,a)]. 103.65/103.48 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),A),hAPP(real,real,hAPP(real,fun(real,real),ord_max(real),C),B))). [resolve(5954,a,5917,a)]. 103.65/103.48 5955 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),C),D))) | 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),B),C)) # label(fact_4984_less__max__iff__disj) # label(axiom). [clausify(3831)]. 103.65/103.48 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(int,int,hAPP(int,fun(int,int),ord_max(int),B),C))) | 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),A),B)). [resolve(5955,a,5872,a)]. 103.65/103.48 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_max(rat),B),C))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)). [resolve(5955,a,5879,a)]. 103.65/103.48 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_max(code_code_numeral),B),C))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),B)). [resolve(5955,a,5906,a)]. 103.65/103.48 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(real,real,hAPP(real,fun(real,real),ord_max(real),B),C))) | 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),A),B)). [resolve(5955,a,5917,a)]. 103.65/103.48 5956 -linorder(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),C),D))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),D)) # label(fact_4984_less__max__iff__disj) # label(axiom). [clausify(3831)]. 103.65/103.48 5957 -linorder(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(A,A,hAPP(A,fun(A,A),ord_max(A),C),D))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) # label(fact_4984_less__max__iff__disj) # label(axiom). [clausify(3831)]. 103.65/103.48 5958 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),f651(A,C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_2853_minf_I6_J) # label(axiom). [clausify(3847)]. 103.65/103.48 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),f651(int,B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(5958,a,5872,a)]. 103.65/103.48 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),f651(rat,B))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)). [resolve(5958,a,5879,a)]. 103.65/103.48 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),f651(code_code_numeral,B))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),B)). [resolve(5958,a,5906,a)]. 103.95/103.77 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),f651(real,B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)). [resolve(5958,a,5917,a)]. 103.95/103.77 5959 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),D)),C)) # label(fact_4847_min__max_Ole__infI1) # label(axiom). [clausify(4009)]. 103.95/103.77 5960 linorder(nat) # label(arity_Nat_Onat___Orderings_Olinorder) # label(axiom). [assumption]. 103.95/103.77 Derived: hAPP(nat,fun(nat,bool),ord_atMost(nat),A) != hAPP(nat,fun(nat,bool),ord_atMost(nat),B) | ti(nat,A) = ti(nat,B). [resolve(5960,a,5873,a)]. 103.95/103.77 Derived: hAPP(nat,fun(nat,bool),ord_atMost(nat),A) = hAPP(nat,fun(nat,bool),ord_atMost(nat),B) | ti(nat,A) != ti(nat,B). [resolve(5960,a,5874,a)]. 103.95/103.77 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),B),C))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),A),C)))). [resolve(5960,a,5875,a)]. 103.95/103.77 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),B),A)). [resolve(5960,a,5876,a)]. 103.95/103.77 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | ti(nat,B) != ti(nat,A) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)). [resolve(5960,a,5877,a)]. 103.95/103.77 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | ti(nat,B) = ti(nat,A) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)). [resolve(5960,a,5878,a)]. 103.95/103.77 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),B)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),B). [resolve(5960,a,5881,a)]. 103.95/103.77 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),A),C)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),B),C)). [resolve(5960,a,5882,a)]. 103.95/103.77 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),B),A)). [resolve(5960,a,5883,a)]. 103.95/103.77 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,bool),fun(fun(nat,bool),fun(nat,bool)),minus_minus(fun(nat,bool)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),A),C)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),A),B)) = hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),B),C). [resolve(5960,a,5885,a)]. 103.95/103.77 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less(fun(nat,bool)),hAPP(nat,fun(nat,bool),ord_lessThan(nat),A)),hAPP(nat,fun(nat,bool),ord_lessThan(nat),B))). [resolve(5960,a,5888,a)]. 103.95/103.77 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less(fun(nat,bool)),hAPP(nat,fun(nat,bool),ord_lessThan(nat),A)),hAPP(nat,fun(nat,bool),ord_lessThan(nat),B))). [resolve(5960,a,5889,a)]. 103.95/103.77 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),B)),C) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),B),C)). [resolve(5960,a,5890,a)]. 103.95/103.77 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | ti(nat,B) = ti(nat,A) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),A)). [resolve(5960,a,5891,a)]. 103.95/103.79 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),A),B)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),C)). [resolve(5960,a,5892,a)]. 103.95/103.79 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),A),B)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)). [resolve(5960,a,5893,a)]. 103.95/103.79 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),A),B)),C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)). [resolve(5960,a,5894,a)]. 103.95/103.79 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),B),D))). [resolve(5960,a,5895,a)]. 103.95/103.79 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),B),A)). [resolve(5960,a,5896,a)]. 103.95/103.79 Derived: ti(nat,A) != ti(nat,B) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),A)). [resolve(5960,a,5897,a)]. 103.95/103.79 Derived: ti(nat,A) != ti(nat,B) | hAPP(nat,fun(nat,bool),ord_lessThan(nat),A) = hAPP(nat,fun(nat,bool),ord_lessThan(nat),B). [resolve(5960,a,5899,a)]. 103.95/103.79 Derived: ti(nat,A) = ti(nat,B) | hAPP(nat,fun(nat,bool),ord_lessThan(nat),A) != hAPP(nat,fun(nat,bool),ord_lessThan(nat),B). [resolve(5960,a,5900,a)]. 103.95/103.79 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),B),C)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),C)). [resolve(5960,a,5901,a)]. 103.95/103.79 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),B)),B)). [resolve(5960,a,5905,a)]. 103.95/103.79 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),A),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),B),C))). [resolve(5960,a,5907,a)]. 103.95/103.79 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less_eq(fun(nat,bool)),hAPP(nat,fun(nat,bool),ord_lessThan(nat),A)),hAPP(nat,fun(nat,bool),ord_lessThan(nat),B))). [resolve(5960,a,5908,a)]. 103.95/103.79 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less_eq(fun(nat,bool)),hAPP(nat,fun(nat,bool),ord_lessThan(nat),A)),hAPP(nat,fun(nat,bool),ord_lessThan(nat),B))). [resolve(5960,a,5909,a)]. 103.95/103.79 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,hAPP(nat,fun(nat,nat),ord_min(nat),C),A)),B)). [resolve(5960,a,5910,a)]. 103.95/103.79 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | ti(nat,A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),B). [resolve(5960,a,5911,a)]. 103.95/103.79 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | ti(nat,A) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),B). [resolve(5960,a,5912,a)]. 103.95/103.79 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),B),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)). [resolve(5960,a,5913,a)]. 103.95/103.79 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),B),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),C)). [resolve(5960,a,5914,a)]. 103.95/103.79 Derived: -hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less_eq(fun(nat,bool)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),A),B)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),C),D))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),D)). [resolve(5960,a,5915,a)]. 103.95/103.81 Derived: -hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less_eq(fun(nat,bool)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),A),B)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),C),D))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),A)). [resolve(5960,a,5916,a)]. 103.95/103.81 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),B)),A)). [resolve(5960,a,5918,a)]. 103.95/103.81 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,hAPP(nat,fun(nat,nat),ord_min(nat),A),C)),B)). [resolve(5960,a,5919,a)]. 103.95/103.81 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),C),B)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),C)),B). [resolve(5960,a,5920,a)]. 103.95/103.81 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),B),C)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),C)). [resolve(5960,a,5922,a)]. 103.95/103.81 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),B),A). [resolve(5960,a,5923,a)]. 103.95/103.81 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),A),B)),C) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),B),C)). [resolve(5960,a,5924,a)]. 103.95/103.81 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),B),A)). [resolve(5960,a,5926,a)]. 103.95/103.81 Derived: ti(nat,A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),A). [resolve(5960,a,5927,a)]. 103.95/103.81 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,hAPP(nat,fun(nat,nat),ord_min(nat),C),A)),B)). [resolve(5960,a,5928,a)]. 103.95/103.81 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,hAPP(nat,fun(nat,nat),ord_min(nat),A),C)),B)). [resolve(5960,a,5929,a)]. 103.95/103.81 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),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),C),A)),B)). [resolve(5960,a,5930,a)]. 103.95/103.81 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),C))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),B),C)))). [resolve(5960,a,5931,a)]. 103.95/103.81 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | ti(nat,A) != ti(nat,B). [resolve(5960,a,5936,a)]. 103.95/103.81 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),f463(nat,A)),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),A)). [resolve(5960,a,5939,a)]. 103.95/103.81 Derived: -plus(nat) | -dvd(nat) | ti(fun(nat,fun(nat,nat)),plus_plus(nat)) = plus_plus(nat). [resolve(5960,a,5940,c)]. 104.09/103.90 Derived: ti(nat,A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),A),B)). [resolve(5960,a,5944,a)]. 104.09/103.90 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),f550(nat,A)),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)). [resolve(5960,a,5945,a)]. 104.09/103.90 Derived: ti(nat,A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),B)). [resolve(5960,a,5949,a)]. 104.09/103.90 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),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),B),C))). [resolve(5960,a,5950,a)]. 104.09/103.90 Derived: ord_min(nat) = hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,nat)),max(nat),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less_eq(nat))). [resolve(5960,a,5952,a)]. 104.09/103.90 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),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),C),B))). [resolve(5960,a,5954,a)]. 104.09/103.90 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),B),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)). [resolve(5960,a,5955,a)]. 104.09/103.90 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),f651(nat,B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)). [resolve(5960,a,5958,a)]. 104.09/103.90 5961 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),C),B) # label(fact_4850_min__max_Oinf__absorb2) # label(axiom). [clausify(4253)]. 104.09/103.90 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | ti(int,A) = hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),B),A). [resolve(5961,a,5872,a)]. 104.09/103.90 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | ti(rat,A) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),B),A). [resolve(5961,a,5879,a)]. 104.09/103.90 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),B)) | ti(code_code_numeral,A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),B),A). [resolve(5961,a,5906,a)]. 104.09/103.90 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | ti(real,A) = hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),B),A). [resolve(5961,a,5917,a)]. 104.09/103.90 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | ti(nat,A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),B),A). [resolve(5961,a,5960,a)]. 104.09/103.90 5962 -linorder(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),B) # label(fact_4842_Min_Oidem) # label(axiom). [clausify(4288)]. 104.09/103.90 5963 -linorder(A) | hAPP(A,fun(A,bool),hAPP(A,fun(A,fun(A,bool)),ord_atLeastLessThan(A),B),C) != hAPP(A,fun(A,bool),hAPP(A,fun(A,fun(A,bool)),ord_atLeastLessThan(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)) | ti(A,C) = ti(A,E) # label(fact_4222_atLeastLessThan__inj_I2_J) # label(axiom). [clausify(4292)]. 104.09/103.90 Derived: hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),A),B) != hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(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)) | ti(int,B) = ti(int,D). [resolve(5963,a,5872,a)]. 104.09/103.90 Derived: hAPP(rat,fun(rat,bool),hAPP(rat,fun(rat,fun(rat,bool)),ord_atLeastLessThan(rat),A),B) != hAPP(rat,fun(rat,bool),hAPP(rat,fun(rat,fun(rat,bool)),ord_atLeastLessThan(rat),C),D) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),C),D)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | ti(rat,B) = ti(rat,D). [resolve(5963,a,5879,a)]. 104.25/104.06 Derived: hAPP(code_code_numeral,fun(code_code_numeral,bool),hAPP(code_code_numeral,fun(code_code_numeral,fun(code_code_numeral,bool)),ord_atLeastLessThan(code_code_numeral),A),B) != hAPP(code_code_numeral,fun(code_code_numeral,bool),hAPP(code_code_numeral,fun(code_code_numeral,fun(code_code_numeral,bool)),ord_atLeastLessThan(code_code_numeral),C),D) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),C),D)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),B)) | ti(code_code_numeral,B) = ti(code_code_numeral,D). [resolve(5963,a,5906,a)]. 104.25/104.06 Derived: hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastLessThan(real),A),B) != hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastLessThan(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)) | ti(real,B) = ti(real,D). [resolve(5963,a,5917,a)]. 104.25/104.06 Derived: hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),A),B) != hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),C),D) | -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),A),B)) | ti(nat,B) = ti(nat,D). [resolve(5963,a,5960,a)]. 104.25/104.06 5964 -linorder(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),B),C)) | ti(A,C) = ti(A,B) # label(fact_559_linorder__antisym__conv2) # label(axiom). [clausify(4335)]. 104.25/104.06 5965 -linorder(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),B),C)) | ti(A,C) != ti(A,B) # label(fact_559_linorder__antisym__conv2) # label(axiom). [clausify(4335)]. 104.25/104.06 5966 -linorder(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),B),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),D),C))) # label(fact_4824_min__less__iff__conj) # label(axiom). [clausify(4501)]. 104.25/104.06 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(int),A),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),C),B))). [resolve(5966,a,5872,a)]. 104.25/104.06 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),C),B))). [resolve(5966,a,5879,a)]. 104.25/104.06 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),C),B))). [resolve(5966,a,5906,a)]. 104.25/104.06 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),A),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),C),B))). [resolve(5966,a,5917,a)]. 104.25/104.06 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),A),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),C),B))). [resolve(5966,a,5960,a)]. 104.25/104.06 5967 -linorder(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),B),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),D),C))) # label(fact_4824_min__less__iff__conj) # label(axiom). [clausify(4501)]. 104.25/104.12 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),A),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),C),B))). [resolve(5967,a,5872,a)]. 104.25/104.12 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),C),B))). [resolve(5967,a,5879,a)]. 104.25/104.12 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),C),B))). [resolve(5967,a,5906,a)]. 104.25/104.12 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),A),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),C),B))). [resolve(5967,a,5917,a)]. 104.25/104.12 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),C),B))). [resolve(5967,a,5960,a)]. 104.25/104.12 5968 -linorder(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),B),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),C),D))) # label(fact_4824_min__less__iff__conj) # label(axiom). [clausify(4501)]. 104.25/104.12 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),A),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),B),C))). [resolve(5968,a,5872,a)]. 104.25/104.12 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),B),C))). [resolve(5968,a,5879,a)]. 104.25/104.12 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),B),C))). [resolve(5968,a,5906,a)]. 104.25/104.12 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),A),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),B),C))). [resolve(5968,a,5917,a)]. 104.25/104.12 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),B),C))). [resolve(5968,a,5960,a)]. 104.25/104.12 5969 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),f783(A,C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),B)) # label(fact_2851_minf_I8_J) # label(axiom). [clausify(4572)]. 104.25/104.12 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),f783(int,B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),A)). [resolve(5969,a,5872,a)]. 104.25/104.12 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),f783(rat,B))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),A)). [resolve(5969,a,5879,a)]. 104.25/104.12 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),f783(code_code_numeral,B))) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),B),A)). [resolve(5969,a,5906,a)]. 104.25/104.12 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),f783(real,B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),A)). [resolve(5969,a,5917,a)]. 104.25/104.12 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),f783(nat,B))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),A)). [resolve(5969,a,5960,a)]. 104.50/104.31 5970 -linorder(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),C)),D)) | 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),D)) # label(fact_4823_min__less__iff__disj) # label(axiom). [clausify(4739)]. 104.50/104.31 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),ord_min(int),A),B)),C)) | 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),C)). [resolve(5970,a,5872,a)]. 104.50/104.31 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),ord_min(rat),A),B)),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),C)). [resolve(5970,a,5879,a)]. 104.50/104.31 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),ord_min(code_code_numeral),A),B)),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),B),C)). [resolve(5970,a,5906,a)]. 104.50/104.31 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),ord_min(real),A),B)),C)) | 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),C)). [resolve(5970,a,5917,a)]. 104.50/104.31 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),A),B)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)). [resolve(5970,a,5960,a)]. 104.50/104.31 5971 -linorder(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),C)),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),D)) # label(fact_4823_min__less__iff__disj) # label(axiom). [clausify(4739)]. 104.50/104.31 5972 -linorder(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),C)),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) # label(fact_4823_min__less__iff__disj) # label(axiom). [clausify(4739)]. 104.50/104.31 5973 -linorder(A) | hAPP(A,fun(A,bool),hAPP(A,fun(A,fun(A,bool)),ord_atLeastLessThan(A),B),C) != hAPP(A,fun(A,bool),hAPP(A,fun(A,fun(A,bool)),ord_atLeastLessThan(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)) | ti(A,B) = ti(A,D) # label(fact_4223_atLeastLessThan__inj_I1_J) # label(axiom). [clausify(4774)]. 104.50/104.31 Derived: hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),A),B) != hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(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)) | ti(int,A) = ti(int,C). [resolve(5973,a,5872,a)]. 104.50/104.31 Derived: hAPP(rat,fun(rat,bool),hAPP(rat,fun(rat,fun(rat,bool)),ord_atLeastLessThan(rat),A),B) != hAPP(rat,fun(rat,bool),hAPP(rat,fun(rat,fun(rat,bool)),ord_atLeastLessThan(rat),C),D) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),C),D)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | ti(rat,A) = ti(rat,C). [resolve(5973,a,5879,a)]. 104.50/104.31 Derived: hAPP(code_code_numeral,fun(code_code_numeral,bool),hAPP(code_code_numeral,fun(code_code_numeral,fun(code_code_numeral,bool)),ord_atLeastLessThan(code_code_numeral),A),B) != hAPP(code_code_numeral,fun(code_code_numeral,bool),hAPP(code_code_numeral,fun(code_code_numeral,fun(code_code_numeral,bool)),ord_atLeastLessThan(code_code_numeral),C),D) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),C),D)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),B)) | ti(code_code_numeral,A) = ti(code_code_numeral,C). [resolve(5973,a,5906,a)]. 104.53/104.33 Derived: hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastLessThan(real),A),B) != hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastLessThan(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)) | ti(real,A) = ti(real,C). [resolve(5973,a,5917,a)]. 104.53/104.33 Derived: hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),A),B) != hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),C),D) | -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),A),B)) | ti(nat,A) = ti(nat,C). [resolve(5973,a,5960,a)]. 104.53/104.33 5974 linorder(quickcheck_code_int) # label(arity_Quickcheck__Narrowing_Ocode__int___Orderings_Olinorder) # label(axiom). [assumption]. 104.53/104.33 Derived: hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_atMost(quickcheck_code_int),A) != hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_atMost(quickcheck_code_int),B) | ti(quickcheck_code_int,A) = ti(quickcheck_code_int,B). [resolve(5974,a,5873,a)]. 104.53/104.33 Derived: hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_atMost(quickcheck_code_int),A) = hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_atMost(quickcheck_code_int),B) | ti(quickcheck_code_int,A) != ti(quickcheck_code_int,B). [resolve(5974,a,5874,a)]. 104.53/104.33 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_max(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),B),C))),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_max(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_max(quickcheck_code_int),A),C)))). [resolve(5974,a,5875,a)]. 104.53/104.33 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),B),A)). [resolve(5974,a,5876,a)]. 104.53/104.33 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),B)) | ti(quickcheck_code_int,B) != ti(quickcheck_code_int,A) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),B)). [resolve(5974,a,5877,a)]. 104.53/104.33 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),B)) | ti(quickcheck_code_int,B) = ti(quickcheck_code_int,A) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),B)). [resolve(5974,a,5878,a)]. 104.53/104.33 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),B)) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),B). [resolve(5974,a,5881,a)]. 104.53/104.33 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_max(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_max(quickcheck_code_int),A),C)) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_max(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),B),C)). [resolve(5974,a,5882,a)]. 104.53/104.33 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),B),A)). [resolve(5974,a,5883,a)]. 104.53/104.33 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),B)) | hAPP(fun(quickcheck_code_int,bool),fun(quickcheck_code_int,bool),hAPP(fun(quickcheck_code_int,bool),fun(fun(quickcheck_code_int,bool),fun(quickcheck_code_int,bool)),minus_minus(fun(quickcheck_code_int,bool)),hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),hAPP(quickcheck_code_int,fun(quickcheck_code_int,fun(quickcheck_code_int,bool)),ord_atLeastLessThan(quickcheck_code_int),A),C)),hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),hAPP(quickcheck_code_int,fun(quickcheck_code_int,fun(quickcheck_code_int,bool)),ord_atLeastLessThan(quickcheck_code_int),A),B)) = hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),hAPP(quickcheck_code_int,fun(quickcheck_code_int,fun(quickcheck_code_int,bool)),ord_atLeastLessThan(quickcheck_code_int),B),C). [resolve(5974,a,5885,a)]. 104.53/104.33 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),B)) | hBOOL(hAPP(fun(quickcheck_code_int,bool),bool,hAPP(fun(quickcheck_code_int,bool),fun(fun(quickcheck_code_int,bool),bool),ord_less(fun(quickcheck_code_int,bool)),hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_lessThan(quickcheck_code_int),A)),hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_lessThan(quickcheck_code_int),B))). [resolve(5974,a,5888,a)]. 104.53/104.33 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),B)) | -hBOOL(hAPP(fun(quickcheck_code_int,bool),bool,hAPP(fun(quickcheck_code_int,bool),fun(fun(quickcheck_code_int,bool),bool),ord_less(fun(quickcheck_code_int,bool)),hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_lessThan(quickcheck_code_int),A)),hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_lessThan(quickcheck_code_int),B))). [resolve(5974,a,5889,a)]. 104.53/104.33 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),B)),C) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),B),C)). [resolve(5974,a,5890,a)]. 104.53/104.33 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),B)) | ti(quickcheck_code_int,B) = ti(quickcheck_code_int,A) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),B),A)). [resolve(5974,a,5891,a)]. 104.53/104.34 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_max(quickcheck_code_int),A),B)),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),C)). [resolve(5974,a,5892,a)]. 104.53/104.34 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_max(quickcheck_code_int),A),B)),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),B),C)). [resolve(5974,a,5893,a)]. 104.53/104.34 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_max(quickcheck_code_int),A),B)),C)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),C)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),B),C)). [resolve(5974,a,5894,a)]. 104.53/104.34 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),C),D)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),C)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),B),D))). [resolve(5974,a,5895,a)]. 104.53/104.34 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),B),A)). [resolve(5974,a,5896,a)]. 104.53/104.34 Derived: ti(quickcheck_code_int,A) != ti(quickcheck_code_int,B) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),B),A)). [resolve(5974,a,5897,a)]. 104.53/104.34 Derived: ti(quickcheck_code_int,A) != ti(quickcheck_code_int,B) | hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_lessThan(quickcheck_code_int),A) = hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_lessThan(quickcheck_code_int),B). [resolve(5974,a,5899,a)]. 104.53/104.34 Derived: ti(quickcheck_code_int,A) = ti(quickcheck_code_int,B) | hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_lessThan(quickcheck_code_int),A) != hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_lessThan(quickcheck_code_int),B). [resolve(5974,a,5900,a)]. 104.53/104.34 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),B),C)) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),B),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),C)). [resolve(5974,a,5901,a)]. 104.53/104.34 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),B)),B)). [resolve(5974,a,5905,a)]. 104.53/104.34 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),B),C))). [resolve(5974,a,5907,a)]. 104.53/104.34 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),B)) | hBOOL(hAPP(fun(quickcheck_code_int,bool),bool,hAPP(fun(quickcheck_code_int,bool),fun(fun(quickcheck_code_int,bool),bool),ord_less_eq(fun(quickcheck_code_int,bool)),hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_lessThan(quickcheck_code_int),A)),hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_lessThan(quickcheck_code_int),B))). [resolve(5974,a,5908,a)]. 104.53/104.34 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),B)) | -hBOOL(hAPP(fun(quickcheck_code_int,bool),bool,hAPP(fun(quickcheck_code_int,bool),fun(fun(quickcheck_code_int,bool),bool),ord_less_eq(fun(quickcheck_code_int,bool)),hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_lessThan(quickcheck_code_int),A)),hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_lessThan(quickcheck_code_int),B))). [resolve(5974,a,5909,a)]. 104.53/104.34 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),C),A)),B)). [resolve(5974,a,5910,a)]. 104.53/104.34 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),B)) | ti(quickcheck_code_int,A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),B). [resolve(5974,a,5911,a)]. 104.53/104.34 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),B)) | ti(quickcheck_code_int,A) != hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),B). [resolve(5974,a,5912,a)]. 104.53/104.34 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),B),C))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),B)). [resolve(5974,a,5913,a)]. 104.53/104.34 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),B),C))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),C)). [resolve(5974,a,5914,a)]. 104.53/104.34 Derived: -hBOOL(hAPP(fun(quickcheck_code_int,bool),bool,hAPP(fun(quickcheck_code_int,bool),fun(fun(quickcheck_code_int,bool),bool),ord_less_eq(fun(quickcheck_code_int,bool)),hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),hAPP(quickcheck_code_int,fun(quickcheck_code_int,fun(quickcheck_code_int,bool)),ord_atLeastLessThan(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),hAPP(quickcheck_code_int,fun(quickcheck_code_int,fun(quickcheck_code_int,bool)),ord_atLeastLessThan(quickcheck_code_int),C),D))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),B),A)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),B),D)). [resolve(5974,a,5915,a)]. 104.53/104.35 Derived: -hBOOL(hAPP(fun(quickcheck_code_int,bool),bool,hAPP(fun(quickcheck_code_int,bool),fun(fun(quickcheck_code_int,bool),bool),ord_less_eq(fun(quickcheck_code_int,bool)),hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),hAPP(quickcheck_code_int,fun(quickcheck_code_int,fun(quickcheck_code_int,bool)),ord_atLeastLessThan(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),hAPP(quickcheck_code_int,fun(quickcheck_code_int,fun(quickcheck_code_int,bool)),ord_atLeastLessThan(quickcheck_code_int),C),D))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),B),A)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),C),A)). [resolve(5974,a,5916,a)]. 104.53/104.35 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),B)),A)). [resolve(5974,a,5918,a)]. 104.53/104.35 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),C)),B)). [resolve(5974,a,5919,a)]. 104.53/104.35 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_max(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_max(quickcheck_code_int),C),B)) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_max(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),C)),B). [resolve(5974,a,5920,a)]. 104.53/104.35 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_max(quickcheck_code_int),B),C)) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_max(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),C)). [resolve(5974,a,5922,a)]. 104.53/104.35 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),B) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),B),A). [resolve(5974,a,5923,a)]. 104.53/104.36 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_max(quickcheck_code_int),A),B)),C) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_max(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),C)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),B),C)). [resolve(5974,a,5924,a)]. 104.53/104.36 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),B),A)). [resolve(5974,a,5926,a)]. 104.53/104.36 Derived: ti(quickcheck_code_int,A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),A). [resolve(5974,a,5927,a)]. 104.53/104.36 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),C),A)),B)). [resolve(5974,a,5928,a)]. 104.53/104.36 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),C)),B)). [resolve(5974,a,5929,a)]. 104.53/104.36 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),C),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),C),A)),B)). [resolve(5974,a,5930,a)]. 104.53/104.36 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_max(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),C))),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_max(quickcheck_code_int),B),C)))). [resolve(5974,a,5931,a)]. 104.53/104.36 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),B)) | ti(quickcheck_code_int,A) != ti(quickcheck_code_int,B). [resolve(5974,a,5936,a)]. 104.53/104.36 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),f463(quickcheck_code_int,A)),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),B),A)). [resolve(5974,a,5939,a)]. 104.53/104.37 Derived: -plus(quickcheck_code_int) | -dvd(quickcheck_code_int) | ti(fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),plus_plus(quickcheck_code_int)) = plus_plus(quickcheck_code_int). [resolve(5974,a,5940,c)]. 104.53/104.37 Derived: ti(quickcheck_code_int,A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_max(quickcheck_code_int),A),B)). [resolve(5974,a,5944,a)]. 104.53/104.37 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),f550(quickcheck_code_int,A)),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),B)). [resolve(5974,a,5945,a)]. 104.53/104.37 Derived: ti(quickcheck_code_int,A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_max(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),B)). [resolve(5974,a,5949,a)]. 104.53/104.37 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_max(quickcheck_code_int),B),C))). [resolve(5974,a,5950,a)]. 104.53/104.37 Derived: ord_min(quickcheck_code_int) = hAPP(fun(quickcheck_code_int,fun(quickcheck_code_int,bool)),fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),max(quickcheck_code_int),hAPP(fun(quickcheck_code_int,fun(quickcheck_code_int,bool)),fun(quickcheck_code_int,fun(quickcheck_code_int,bool)),combc(quickcheck_code_int,quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int))). [resolve(5974,a,5952,a)]. 104.53/104.37 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_max(quickcheck_code_int),C),B))). [resolve(5974,a,5954,a)]. 104.53/104.37 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_max(quickcheck_code_int),B),C))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),B)). [resolve(5974,a,5955,a)]. 104.53/104.37 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),f651(quickcheck_code_int,B))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),B)). [resolve(5974,a,5958,a)]. 104.53/104.37 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),B)) | ti(quickcheck_code_int,A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),B),A). [resolve(5974,a,5961,a)]. 104.53/104.37 Derived: hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),hAPP(quickcheck_code_int,fun(quickcheck_code_int,fun(quickcheck_code_int,bool)),ord_atLeastLessThan(quickcheck_code_int),A),B) != hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),hAPP(quickcheck_code_int,fun(quickcheck_code_int,fun(quickcheck_code_int,bool)),ord_atLeastLessThan(quickcheck_code_int),C),D) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),C),D)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),B)) | ti(quickcheck_code_int,B) = ti(quickcheck_code_int,D). [resolve(5974,a,5963,a)]. 104.62/104.42 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),C),B))). [resolve(5974,a,5966,a)]. 104.62/104.42 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),C),B))). [resolve(5974,a,5967,a)]. 104.62/104.42 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),B),C))). [resolve(5974,a,5968,a)]. 104.62/104.42 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),f783(quickcheck_code_int,B))) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),B),A)). [resolve(5974,a,5969,a)]. 104.62/104.42 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),ord_min(quickcheck_code_int),A),B)),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),B),C)). [resolve(5974,a,5970,a)]. 104.62/104.42 Derived: hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),hAPP(quickcheck_code_int,fun(quickcheck_code_int,fun(quickcheck_code_int,bool)),ord_atLeastLessThan(quickcheck_code_int),A),B) != hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),hAPP(quickcheck_code_int,fun(quickcheck_code_int,fun(quickcheck_code_int,bool)),ord_atLeastLessThan(quickcheck_code_int),C),D) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),C),D)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),B)) | ti(quickcheck_code_int,A) = ti(quickcheck_code_int,C). [resolve(5974,a,5973,a)]. 104.62/104.42 5975 -linorder(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),C),B)) # label(fact_570_linorder__le__less__linear) # label(axiom). [clausify(4801)]. 104.62/104.42 5976 -linorder(A) | hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),B),C) = hAPP(A,A,hAPP(A,fun(A,A),ord_min(A),C),B) # label(fact_4832_min__max_Oinf__commute) # label(axiom). [clausify(4841)]. 104.62/104.45 5977 -linorder(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)) | hAPP(A,fun(A,bool),hAPP(A,fun(A,fun(A,bool)),ord_atLeastLessThan(A),D),E) != hAPP(A,fun(A,bool),hAPP(A,fun(A,fun(A,bool)),ord_atLeastLessThan(A),B),C) | ti(A,E) = ti(A,C) # label(fact_4224_atLeastLessThan__eq__iff) # label(axiom). [clausify(4867)]. 104.62/104.45 5978 -linorder(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)) | hAPP(A,fun(A,bool),hAPP(A,fun(A,fun(A,bool)),ord_atLeastLessThan(A),D),E) != hAPP(A,fun(A,bool),hAPP(A,fun(A,fun(A,bool)),ord_atLeastLessThan(A),B),C) | ti(A,D) = ti(A,B) # label(fact_4224_atLeastLessThan__eq__iff) # label(axiom). [clausify(4867)]. 104.62/104.45 5979 -linorder(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)) | hAPP(A,fun(A,bool),hAPP(A,fun(A,fun(A,bool)),ord_atLeastLessThan(A),D),E) = hAPP(A,fun(A,bool),hAPP(A,fun(A,fun(A,bool)),ord_atLeastLessThan(A),B),C) | ti(A,E) != ti(A,C) | ti(A,D) != ti(A,B) # label(fact_4224_atLeastLessThan__eq__iff) # label(axiom). [clausify(4867)]. 104.62/104.46 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),D)) | hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),C),D) = hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),A),B) | ti(int,D) != ti(int,B) | ti(int,C) != ti(int,A). [resolve(5979,a,5872,a)]. 104.62/104.46 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),C),D)) | hAPP(rat,fun(rat,bool),hAPP(rat,fun(rat,fun(rat,bool)),ord_atLeastLessThan(rat),C),D) = hAPP(rat,fun(rat,bool),hAPP(rat,fun(rat,fun(rat,bool)),ord_atLeastLessThan(rat),A),B) | ti(rat,D) != ti(rat,B) | ti(rat,C) != ti(rat,A). [resolve(5979,a,5879,a)]. 104.62/104.46 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),C),D)) | hAPP(code_code_numeral,fun(code_code_numeral,bool),hAPP(code_code_numeral,fun(code_code_numeral,fun(code_code_numeral,bool)),ord_atLeastLessThan(code_code_numeral),C),D) = hAPP(code_code_numeral,fun(code_code_numeral,bool),hAPP(code_code_numeral,fun(code_code_numeral,fun(code_code_numeral,bool)),ord_atLeastLessThan(code_code_numeral),A),B) | ti(code_code_numeral,D) != ti(code_code_numeral,B) | ti(code_code_numeral,C) != ti(code_code_numeral,A). [resolve(5979,a,5906,a)]. 104.62/104.46 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)) | hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastLessThan(real),C),D) = hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastLessThan(real),A),B) | ti(real,D) != ti(real,B) | ti(real,C) != ti(real,A). [resolve(5979,a,5917,a)]. 104.62/104.46 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)) | hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),C),D) = hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),A),B) | ti(nat,D) != ti(nat,B) | ti(nat,C) != ti(nat,A). [resolve(5979,a,5960,a)]. 104.62/104.46 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),C),D)) | hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),hAPP(quickcheck_code_int,fun(quickcheck_code_int,fun(quickcheck_code_int,bool)),ord_atLeastLessThan(quickcheck_code_int),C),D) = hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),hAPP(quickcheck_code_int,fun(quickcheck_code_int,fun(quickcheck_code_int,bool)),ord_atLeastLessThan(quickcheck_code_int),A),B) | ti(quickcheck_code_int,D) != ti(quickcheck_code_int,B) | ti(quickcheck_code_int,C) != ti(quickcheck_code_int,A). [resolve(5979,a,5974,a)]. 107.01/106.84 5980 semiring_div(quickcheck_code_int) # label(arity_Quickcheck__Narrowing_Ocode__int___Divides_Osemiring__div) # label(axiom). [assumption]. 107.01/106.84 5981 -semiring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),C),D))),D) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),D) # label(fact_1283_mod__mult__right__eq) # label(axiom). [clausify(98)]. 107.01/106.84 5982 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),D),C))) # label(fact_1305_dvd__mod__iff) # label(axiom). [clausify(125)]. 107.01/106.84 5983 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),D),C))) # label(fact_1305_dvd__mod__iff) # label(axiom). [clausify(125)]. 107.05/106.84 5984 -semiring_div(A) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),div_div(A),zero_zero(A)),B) # label(fact_1073_div__0) # label(axiom). [clausify(179)]. 107.05/106.84 5985 -semiring_div(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),B),C)),C)) # label(fact_1410_semiring__div__class_Omod__div__equality_H) # label(axiom). [clausify(242)]. 107.05/106.84 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),B),C))),C) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B)),C). [resolve(5980,a,5981,a)]. 107.05/106.84 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),C),B))). [resolve(5980,a,5982,a)]. 107.05/106.84 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),C)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),C),B))). [resolve(5980,a,5983,a)]. 107.05/106.84 Derived: zero_zero(quickcheck_code_int) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),zero_zero(quickcheck_code_int)),A). [resolve(5980,a,5984,a)]. 107.05/106.84 Derived: ti(quickcheck_code_int,A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),A),B)),B)). [resolve(5980,a,5985,a)]. 107.15/106.97 5986 -semiring_div(A) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),one_one(A)) # label(fact_1353_mod__by__1) # label(axiom). [clausify(391)]. 107.15/106.97 Derived: zero_zero(quickcheck_code_int) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),A),one_one(quickcheck_code_int)). [resolve(5986,a,5980,a)]. 107.15/106.97 5987 -semiring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),C),D)) # label(fact_1286_mod__mult__mult1) # label(axiom). [clausify(619)]. 107.15/106.97 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),C)) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),B),C)). [resolve(5987,a,5980,a)]. 107.15/106.97 5988 -semiring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),C),D))),D) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),D) # label(fact_1295_mod__add__right__eq) # label(axiom). [clausify(627)]. 107.15/106.97 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),B),C))),C) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),B)),C). [resolve(5988,a,5980,a)]. 107.15/106.97 5989 -semiring_div(A) | ti(A,B) = zero_zero(A) | hAPP(A,A,hAPP(A,fun(A,A),div_div(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),B)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),B)) = hAPP(A,A,hAPP(A,fun(A,A),div_div(A),C),D) # label(fact_1086_div__mult__mult2) # label(axiom). [clausify(872)]. 107.15/106.97 Derived: ti(quickcheck_code_int,A) = zero_zero(quickcheck_code_int) | hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),B),A)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),C),A)) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),B),C). [resolve(5989,a,5980,a)]. 107.15/106.97 5990 -semiring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),C)),D)),C) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)),C) # label(fact_1284_mod__mult__left__eq) # label(axiom). [clausify(1114)]. 107.22/107.00 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),A),B)),C)),B) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),C)),B). [resolve(5990,a,5980,a)]. 107.22/107.00 5991 -semiring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),C) != hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),D),C) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),E),C) != hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),F),C) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),E)),C) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),F)),C) # label(fact_1289_mod__mult__cong) # label(axiom). [clausify(1236)]. 107.22/107.00 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),A),B) != hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),C),B) | hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),D),B) != hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),E),B) | hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),D)),B) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),C),E)),B). [resolve(5991,a,5980,a)]. 107.22/107.00 5992 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) | hAPP(A,A,hAPP(A,fun(A,A),div_div(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),D)),B) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),C),B)),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),D),B)) # label(fact_1096_div__add) # label(axiom). [clausify(1250)]. 107.22/107.00 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),C)) | hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),B),C)),A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),B),A)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),C),A)). [resolve(5992,a,5980,a)]. 107.22/107.06 5993 -semiring_div(A) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),C) # label(fact_1351_mod__mult__self2__is__0) # label(axiom). [clausify(1352)]. 107.22/107.06 Derived: zero_zero(quickcheck_code_int) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B)),B). [resolve(5993,a,5980,a)]. 107.22/107.06 5994 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | hAPP(A,A,hAPP(A,fun(A,A),div_div(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D)),B) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),C),B)),D) # label(fact_1094_dvd__div__mult) # label(axiom). [clausify(1365)]. 107.22/107.06 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),B)) | hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),B),C)),A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),B),A)),C). [resolve(5994,a,5980,a)]. 107.22/107.06 5995 -semiring_div(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),zero_zero(A)) # label(fact_1281_mod__by__0) # label(axiom). [clausify(1400)]. 107.22/107.06 Derived: ti(quickcheck_code_int,A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),A),zero_zero(quickcheck_code_int)). [resolve(5995,a,5980,a)]. 107.22/107.06 5996 -semiring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),C)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),D)),C) # label(fact_1287_mod__mult__mult2) # label(axiom). [clausify(1424)]. 107.22/107.06 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),C),B)) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),A),C)),B). [resolve(5996,a,5980,a)]. 107.22/107.06 5997 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),C),B) # label(fact_1356_dvd__imp__mod__0) # label(axiom). [clausify(1428)]. 107.22/107.06 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),B)) | zero_zero(quickcheck_code_int) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),B),A). [resolve(5997,a,5980,a)]. 107.22/107.06 5998 -semiring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),C)),C) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),C) # label(fact_1279_mod__mod__trivial) # label(axiom). [clausify(1477)]. 107.34/107.14 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),A),B)),B) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),A),B). [resolve(5998,a,5980,a)]. 107.34/107.14 5999 -semiring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),B),C)),C)),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),C))),D) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),D) # label(fact_1414_div__mod__equality) # label(axiom). [clausify(1641)]. 107.34/107.14 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),A),B)),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),A),B))),C) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),C). [resolve(5999,a,5980,a)]. 107.34/107.14 6000 -semiring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),C)),D)),C) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),D)),C) # label(fact_1294_mod__add__left__eq) # label(axiom). [clausify(1721)]. 107.34/107.14 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),A),B)),C)),B) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),C)),B). [resolve(6000,a,5980,a)]. 107.34/107.14 6001 -semiring_div(A) | ti(A,B) = zero_zero(A) | ti(A,C) = hAPP(A,A,hAPP(A,fun(A,A),div_div(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),B) # label(fact_1087_div__mult__self1__is__id) # label(axiom). [clausify(1780)]. 107.34/107.14 Derived: ti(quickcheck_code_int,A) = zero_zero(quickcheck_code_int) | ti(quickcheck_code_int,B) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B)),A). [resolve(6001,a,5980,a)]. 107.34/107.14 6002 -semiring_div(A) | ti(A,B) = zero_zero(A) | hAPP(A,A,hAPP(A,fun(A,A),div_div(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D))),B) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),D),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),C),B)) # label(fact_1115_div__mult__self2) # label(axiom). [clausify(1782)]. 107.34/107.14 Derived: ti(quickcheck_code_int,A) = zero_zero(quickcheck_code_int) | hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),B),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),C))),A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),C),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),B),A)). [resolve(6002,a,5980,a)]. 107.44/107.25 6003 -semiring_div(A) | ti(A,B) = zero_zero(A) | one_one(A) = hAPP(A,A,hAPP(A,fun(A,A),div_div(A),B),B) # label(fact_1090_div__self) # label(axiom). [clausify(1798)]. 107.44/107.25 Derived: ti(quickcheck_code_int,A) = zero_zero(quickcheck_code_int) | one_one(quickcheck_code_int) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),A),A). [resolve(6003,a,5980,a)]. 107.44/107.25 6004 -semiring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D))),C) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),C) # label(fact_1354_mod__mult__self2) # label(axiom). [clausify(1924)]. 107.44/107.25 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),B),C))),B) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),A),B). [resolve(6004,a,5980,a)]. 107.44/107.25 6005 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | ti(A,C) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),C),B)) # label(fact_1091_dvd__mult__div__cancel) # label(axiom). [clausify(1952)]. 107.44/107.25 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),B)) | ti(quickcheck_code_int,B) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),B),A)). [resolve(6005,a,5980,a)]. 107.44/107.25 6006 -semiring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),B) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),C),B) # label(fact_1296_mod__add__self1) # label(axiom). [clausify(1986)]. 107.44/107.25 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),B)),A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),B),A). [resolve(6006,a,5980,a)]. 107.44/107.25 6007 -semiring_div(A) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),zero_zero(A)),B) # label(fact_1282_mod__0) # label(axiom). [clausify(2107)]. 107.44/107.25 Derived: zero_zero(quickcheck_code_int) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),zero_zero(quickcheck_code_int)),A). [resolve(6007,a,5980,a)]. 107.44/107.25 6008 -semiring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),C),B))),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),C),B))),D) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),D) # label(fact_1413_div__mod__equality2) # label(axiom). [clausify(2167)]. 107.50/107.28 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),B),A))),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),B),A))),C) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),B),C). [resolve(6008,a,5980,a)]. 107.50/107.28 6009 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),C),B)),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),D),B))) # label(fact_1076_div__dvd__div) # label(axiom). [clausify(2229)]. 107.50/107.28 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),C)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),B),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),B),A)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),C),A))). [resolve(6009,a,5980,a)]. 107.50/107.28 6010 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),C),B)),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),D),B))) # label(fact_1076_div__dvd__div) # label(axiom). [clausify(2229)]. 107.50/107.28 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),B),C)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),B),A)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),C),A))). [resolve(6010,a,5980,a)]. 107.50/107.28 6011 semiring_div(code_code_numeral) # label(arity_Code__Numeral_Ocode__numeral___Divides_Osemiring__div) # label(axiom). [assumption]. 107.50/107.28 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),B),C))),C) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B)),C). [resolve(6011,a,5981,a)]. 107.50/107.29 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),C),B))). [resolve(6011,a,5982,a)]. 107.50/107.29 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),C)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),C),B))). [resolve(6011,a,5983,a)]. 107.50/107.29 Derived: zero_zero(code_code_numeral) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),zero_zero(code_code_numeral)),A). [resolve(6011,a,5984,a)]. 107.50/107.29 Derived: ti(code_code_numeral,A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),A),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),A),B)),B)). [resolve(6011,a,5985,a)]. 107.50/107.29 Derived: zero_zero(code_code_numeral) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),A),one_one(code_code_numeral)). [resolve(6011,a,5986,a)]. 107.50/107.29 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),C)) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),B),C)). [resolve(6011,a,5987,a)]. 107.50/107.29 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),B),C))),C) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),B)),C). [resolve(6011,a,5988,a)]. 107.50/107.29 Derived: ti(code_code_numeral,A) = zero_zero(code_code_numeral) | hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),B),A)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),C),A)) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),B),C). [resolve(6011,a,5989,a)]. 107.50/107.30 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),A),B)),C)),B) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),C)),B). [resolve(6011,a,5990,a)]. 107.50/107.30 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),A),B) != hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),C),B) | hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),D),B) != hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),E),B) | hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),D)),B) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),C),E)),B). [resolve(6011,a,5991,a)]. 107.50/107.30 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),C)) | hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),B),C)),A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),B),A)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),C),A)). [resolve(6011,a,5992,a)]. 107.50/107.30 Derived: zero_zero(code_code_numeral) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B)),B). [resolve(6011,a,5993,a)]. 107.50/107.30 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),B)) | hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),B),C)),A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),B),A)),C). [resolve(6011,a,5994,a)]. 107.50/107.31 Derived: ti(code_code_numeral,A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),A),zero_zero(code_code_numeral)). [resolve(6011,a,5995,a)]. 107.50/107.31 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),C),B)) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),A),C)),B). [resolve(6011,a,5996,a)]. 107.50/107.31 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),B)) | zero_zero(code_code_numeral) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),B),A). [resolve(6011,a,5997,a)]. 107.50/107.31 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),A),B)),B) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),A),B). [resolve(6011,a,5998,a)]. 107.50/107.31 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),A),B)),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),A),B))),C) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),C). [resolve(6011,a,5999,a)]. 107.50/107.31 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),A),B)),C)),B) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),C)),B). [resolve(6011,a,6000,a)]. 107.50/107.31 Derived: ti(code_code_numeral,A) = zero_zero(code_code_numeral) | ti(code_code_numeral,B) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B)),A). [resolve(6011,a,6001,a)]. 107.50/107.31 Derived: ti(code_code_numeral,A) = zero_zero(code_code_numeral) | hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),B),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),C))),A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),C),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),B),A)). [resolve(6011,a,6002,a)]. 107.50/107.32 Derived: ti(code_code_numeral,A) = zero_zero(code_code_numeral) | one_one(code_code_numeral) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),A),A). [resolve(6011,a,6003,a)]. 107.50/107.32 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),B),C))),B) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),A),B). [resolve(6011,a,6004,a)]. 107.50/107.32 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),B)) | ti(code_code_numeral,B) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),B),A)). [resolve(6011,a,6005,a)]. 107.50/107.32 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),B)),A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),B),A). [resolve(6011,a,6006,a)]. 107.50/107.32 Derived: zero_zero(code_code_numeral) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),zero_zero(code_code_numeral)),A). [resolve(6011,a,6007,a)]. 107.50/107.32 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),B),A))),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),B),A))),C) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),B),C). [resolve(6011,a,6008,a)]. 107.50/107.32 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),C)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),B),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),B),A)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),C),A))). [resolve(6011,a,6009,a)]. 107.57/107.37 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),B),C)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),B),A)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),C),A))). [resolve(6011,a,6010,a)]. 107.57/107.37 6012 -semiring_div(A) | ti(A,B) = zero_zero(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | ti(A,D) != hAPP(A,A,hAPP(A,fun(A,A),div_div(A),C),B) | ti(A,C) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),B) # label(fact_1119_dvd__div__eq__mult) # label(axiom). [clausify(2263)]. 107.57/107.37 Derived: ti(quickcheck_code_int,A) = zero_zero(quickcheck_code_int) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),B)) | ti(quickcheck_code_int,C) != hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),B),A) | ti(quickcheck_code_int,B) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),C),A). [resolve(6012,a,5980,a)]. 107.57/107.37 Derived: ti(code_code_numeral,A) = zero_zero(code_code_numeral) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),B)) | ti(code_code_numeral,C) != hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),B),A) | ti(code_code_numeral,B) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),C),A). [resolve(6012,a,6011,a)]. 107.57/107.37 6013 -semiring_div(A) | ti(A,B) = zero_zero(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | ti(A,D) = hAPP(A,A,hAPP(A,fun(A,A),div_div(A),C),B) | ti(A,C) != hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),B) # label(fact_1119_dvd__div__eq__mult) # label(axiom). [clausify(2263)]. 107.57/107.37 Derived: ti(quickcheck_code_int,A) = zero_zero(quickcheck_code_int) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),B)) | ti(quickcheck_code_int,C) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),B),A) | ti(quickcheck_code_int,B) != hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),C),A). [resolve(6013,a,5980,a)]. 107.57/107.37 Derived: ti(code_code_numeral,A) = zero_zero(code_code_numeral) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),B)) | ti(code_code_numeral,C) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),B),A) | ti(code_code_numeral,B) != hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),C),A). [resolve(6013,a,6011,a)]. 107.57/107.37 6014 semiring_div(int) # label(arity_Int_Oint___Divides_Osemiring__div) # label(axiom). [assumption]. 107.57/107.37 Derived: ti(int,A) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A),B)),B)). [resolve(6014,a,5985,a)]. 107.59/107.38 Derived: zero_zero(int) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A),one_one(int)). [resolve(6014,a,5986,a)]. 107.59/107.38 Derived: hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),C)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),C)). [resolve(6014,a,5987,a)]. 107.59/107.38 Derived: hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),C))),C) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)),C). [resolve(6014,a,5988,a)]. 107.59/107.38 Derived: ti(int,A) = zero_zero(int) | hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),A)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),A)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),C). [resolve(6014,a,5989,a)]. 107.59/107.38 Derived: hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A),B)),C)),B) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),C)),B). [resolve(6014,a,5990,a)]. 107.59/107.38 Derived: hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A),B) != hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),C),B) | hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),D),B) != hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),E),B) | hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),D)),B) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),E)),B). [resolve(6014,a,5991,a)]. 107.59/107.38 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)) | hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),C)),A) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),A)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),C),A)). [resolve(6014,a,5992,a)]. 107.59/107.38 Derived: zero_zero(int) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),B). [resolve(6014,a,5993,a)]. 107.59/107.38 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C)),A) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),A)),C). [resolve(6014,a,5994,a)]. 107.59/107.38 Derived: ti(int,A) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A),zero_zero(int)). [resolve(6014,a,5995,a)]. 107.59/107.38 Derived: hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),B)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A),C)),B). [resolve(6014,a,5996,a)]. 107.59/107.38 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | zero_zero(int) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),A). [resolve(6014,a,5997,a)]. 107.59/107.38 Derived: hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A),B)),B) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A),B). [resolve(6014,a,5998,a)]. 107.59/107.38 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A),B)),B)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A),B))),C) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),C). [resolve(6014,a,5999,a)]. 107.59/107.41 Derived: hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A),B)),C)),B) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),C)),B). [resolve(6014,a,6000,a)]. 107.59/107.41 Derived: ti(int,A) = zero_zero(int) | ti(int,B) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),A). [resolve(6014,a,6001,a)]. 107.59/107.41 Derived: ti(int,A) = zero_zero(int) | hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),C))),A) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),C),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),A)). [resolve(6014,a,6002,a)]. 107.59/107.41 Derived: hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C))),B) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A),B). [resolve(6014,a,6004,a)]. 107.59/107.41 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | ti(int,B) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),A)). [resolve(6014,a,6005,a)]. 107.59/107.41 Derived: hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)),A) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),A). [resolve(6014,a,6006,a)]. 107.59/107.41 Derived: hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),A))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),A))),C) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),C). [resolve(6014,a,6008,a)]. 107.59/107.41 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),B),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),A)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),C),A))). [resolve(6014,a,6009,a)]. 107.59/107.41 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),B),C)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),A)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),C),A))). [resolve(6014,a,6010,a)]. 107.59/107.41 Derived: ti(int,A) = zero_zero(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | ti(int,C) != hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),A) | ti(int,B) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),A). [resolve(6014,a,6012,a)]. 107.59/107.41 Derived: ti(int,A) = zero_zero(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | ti(int,C) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),A) | ti(int,B) != hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),A). [resolve(6014,a,6013,a)]. 107.59/107.41 6015 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),C),D))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) # label(fact_1302_dvd__mod__imp__dvd) # label(axiom). [clausify(2340)]. 107.59/107.41 6016 -semiring_div(A) | ti(A,B) = zero_zero(A) | hAPP(A,A,hAPP(A,fun(A,A),div_div(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),B) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),C),B)),one_one(A)) # label(fact_1118_div__add__self1) # label(axiom). [clausify(2368)]. 107.59/107.41 Derived: ti(quickcheck_code_int,A) = zero_zero(quickcheck_code_int) | hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),B)),A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),B),A)),one_one(quickcheck_code_int)). [resolve(6016,a,5980,a)]. 107.66/107.51 Derived: ti(code_code_numeral,A) = zero_zero(code_code_numeral) | hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),B)),A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),B),A)),one_one(code_code_numeral)). [resolve(6016,a,6011,a)]. 107.66/107.51 Derived: ti(int,A) = zero_zero(int) | hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)),A) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),A)),one_one(int)). [resolve(6016,a,6014,a)]. 107.66/107.51 6017 -semiring_div(A) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),B) # label(fact_1352_mod__mult__self1__is__0) # label(axiom). [clausify(2413)]. 107.66/107.51 Derived: zero_zero(quickcheck_code_int) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B)),A). [resolve(6017,a,5980,a)]. 107.66/107.51 Derived: zero_zero(code_code_numeral) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B)),A). [resolve(6017,a,6011,a)]. 107.66/107.51 Derived: zero_zero(int) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),A). [resolve(6017,a,6014,a)]. 107.66/107.51 6018 -semiring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),C),D))),D) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),D) # label(fact_1292_zmod__simps_I2_J) # label(axiom). [clausify(2541)]. 107.66/107.51 6019 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | hAPP(A,A,hAPP(A,fun(A,A),div_div(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),C)),B) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),C),B)) # label(fact_1092_div__mult__swap) # label(axiom). [clausify(2602)]. 107.66/107.51 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),B)) | hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),C),B)),A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),C),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),B),A)). [resolve(6019,a,5980,a)]. 107.66/107.51 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),B)) | hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),C),B)),A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),C),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),B),A)). [resolve(6019,a,6011,a)]. 107.74/107.55 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),B)),A) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),A)). [resolve(6019,a,6014,a)]. 107.74/107.55 6020 -semiring_div(A) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),div_div(A),B),zero_zero(A)) # label(fact_1074_div__by__0) # label(axiom). [clausify(2622)]. 107.74/107.55 Derived: zero_zero(quickcheck_code_int) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),A),zero_zero(quickcheck_code_int)). [resolve(6020,a,5980,a)]. 107.74/107.55 Derived: zero_zero(code_code_numeral) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),A),zero_zero(code_code_numeral)). [resolve(6020,a,6011,a)]. 107.74/107.55 Derived: zero_zero(int) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A),zero_zero(int)). [resolve(6020,a,6014,a)]. 107.74/107.55 6021 -semiring_div(A) | ti(A,B) = zero_zero(A) | ti(A,C) = zero_zero(A) | -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),C),E)) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),C) != hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),E) | hAPP(A,A,hAPP(A,fun(A,A),div_div(A),E),C) = hAPP(A,A,hAPP(A,fun(A,A),div_div(A),D),B) # label(fact_1120_dvd__div__div__eq__mult) # label(axiom). [clausify(2696)]. 107.74/107.55 Derived: ti(quickcheck_code_int,A) = zero_zero(quickcheck_code_int) | ti(quickcheck_code_int,B) = zero_zero(quickcheck_code_int) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),C)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),B),D)) | hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),C),B) != hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),D) | hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),D),B) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),C),A). [resolve(6021,a,5980,a)]. 107.74/107.55 Derived: ti(code_code_numeral,A) = zero_zero(code_code_numeral) | ti(code_code_numeral,B) = zero_zero(code_code_numeral) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),C)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),B),D)) | hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),C),B) != hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),D) | hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),D),B) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),C),A). [resolve(6021,a,6011,a)]. 107.89/107.68 Derived: ti(int,A) = zero_zero(int) | ti(int,B) = zero_zero(int) | -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),B),D)) | hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),B) != hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),D) | hAPP(int,int,hAPP(int,fun(int,int),div_div(int),D),B) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),C),A). [resolve(6021,a,6014,a)]. 107.89/107.68 6022 -semiring_div(A) | ti(A,B) = zero_zero(A) | ti(A,C) = zero_zero(A) | -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),C),E)) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),C) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),E) | hAPP(A,A,hAPP(A,fun(A,A),div_div(A),E),C) != hAPP(A,A,hAPP(A,fun(A,A),div_div(A),D),B) # label(fact_1120_dvd__div__div__eq__mult) # label(axiom). [clausify(2696)]. 107.89/107.68 Derived: ti(quickcheck_code_int,A) = zero_zero(quickcheck_code_int) | ti(quickcheck_code_int,B) = zero_zero(quickcheck_code_int) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),C)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),B),D)) | hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),C),B) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),D) | hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),D),B) != hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),C),A). [resolve(6022,a,5980,a)]. 107.89/107.68 Derived: ti(code_code_numeral,A) = zero_zero(code_code_numeral) | ti(code_code_numeral,B) = zero_zero(code_code_numeral) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),C)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),B),D)) | hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),C),B) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),D) | hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),D),B) != hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),C),A). [resolve(6022,a,6011,a)]. 107.89/107.68 Derived: ti(int,A) = zero_zero(int) | ti(int,B) = zero_zero(int) | -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),B),D)) | hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),B) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),D) | hAPP(int,int,hAPP(int,fun(int,int),div_div(int),D),B) != hAPP(int,int,hAPP(int,fun(int,int),div_div(int),C),A). [resolve(6022,a,6014,a)]. 107.89/107.68 6023 -semiring_div(A) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),div_div(A),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),C)),C) # label(fact_1358_mod__div__trivial) # label(axiom). [clausify(2952)]. 107.89/107.68 Derived: zero_zero(quickcheck_code_int) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),A),B)),B). [resolve(6023,a,5980,a)]. 107.89/107.68 Derived: zero_zero(code_code_numeral) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),A),B)),B). [resolve(6023,a,6011,a)]. 107.94/107.79 6024 -semiring_div(A) | ti(A,B) = zero_zero(A) | hAPP(A,A,hAPP(A,fun(A,A),div_div(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),B)),B) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),C),B)),one_one(A)) # label(fact_1117_div__add__self2) # label(axiom). [clausify(3035)]. 107.94/107.79 Derived: ti(quickcheck_code_int,A) = zero_zero(quickcheck_code_int) | hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),B),A)),A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),B),A)),one_one(quickcheck_code_int)). [resolve(6024,a,5980,a)]. 107.94/107.79 Derived: ti(code_code_numeral,A) = zero_zero(code_code_numeral) | hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),B),A)),A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),B),A)),one_one(code_code_numeral)). [resolve(6024,a,6011,a)]. 107.94/107.79 Derived: ti(int,A) = zero_zero(int) | hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),A)),A) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),A)),one_one(int)). [resolve(6024,a,6014,a)]. 107.94/107.79 6025 -semiring_div(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),B),C))),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),C)) # label(fact_1411_mod__div__equality2) # label(axiom). [clausify(3131)]. 107.94/107.79 Derived: ti(quickcheck_code_int,A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),B),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),A),B))),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),A),B)). [resolve(6025,a,5980,a)]. 107.94/107.79 Derived: ti(code_code_numeral,A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),B),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),A),B))),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),A),B)). [resolve(6025,a,6011,a)]. 107.94/107.79 Derived: ti(int,A) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A),B))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A),B)). [resolve(6025,a,6014,a)]. 107.94/107.79 6026 -semiring_div(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),div_div(A),B),one_one(A)) # label(fact_1075_div__by__1) # label(axiom). [clausify(3187)]. 108.24/108.08 Derived: ti(quickcheck_code_int,A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),A),one_one(quickcheck_code_int)). [resolve(6026,a,5980,a)]. 108.24/108.08 Derived: ti(code_code_numeral,A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),A),one_one(code_code_numeral)). [resolve(6026,a,6011,a)]. 108.24/108.08 Derived: ti(int,A) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A),one_one(int)). [resolve(6026,a,6014,a)]. 108.24/108.08 6027 -semiring_div(A) | div_div(A) = ti(fun(A,fun(A,A)),div_div(A)) # label(tsy_c_Divides_Odiv__class_Odiv_res) # label(axiom). [clausify(3383)]. 108.24/108.08 Derived: div_div(quickcheck_code_int) = ti(fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),div_div(quickcheck_code_int)). [resolve(6027,a,5980,a)]. 108.24/108.08 Derived: div_div(code_code_numeral) = ti(fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)),div_div(code_code_numeral)). [resolve(6027,a,6011,a)]. 108.24/108.08 Derived: div_div(int) = ti(fun(int,fun(int,int)),div_div(int)). [resolve(6027,a,6014,a)]. 108.24/108.08 6028 -semiring_div(A) | ti(A,B) = zero_zero(A) | hAPP(A,A,hAPP(A,fun(A,A),div_div(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)) = hAPP(A,A,hAPP(A,fun(A,A),div_div(A),C),D) # label(fact_1085_div__mult__mult1) # label(axiom). [clausify(3501)]. 108.24/108.08 Derived: ti(quickcheck_code_int,A) = zero_zero(quickcheck_code_int) | hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),C)) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),B),C). [resolve(6028,a,5980,a)]. 108.24/108.08 Derived: ti(code_code_numeral,A) = zero_zero(code_code_numeral) | hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),C)) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),B),C). [resolve(6028,a,6011,a)]. 108.24/108.08 Derived: ti(int,A) = zero_zero(int) | hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),C)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),C). [resolve(6028,a,6014,a)]. 108.24/108.08 6029 -semiring_div(A) | ti(A,B) = zero_zero(A) | hAPP(A,A,hAPP(A,fun(A,A),div_div(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),B))),B) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),D),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),C),B)) # label(fact_1116_div__mult__self1) # label(axiom). [clausify(3512)]. 108.24/108.08 Derived: ti(quickcheck_code_int,A) = zero_zero(quickcheck_code_int) | hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),B),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),C),A))),A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),C),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),B),A)). [resolve(6029,a,5980,a)]. 108.34/108.17 Derived: ti(code_code_numeral,A) = zero_zero(code_code_numeral) | hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),B),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),C),A))),A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),C),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),B),A)). [resolve(6029,a,6011,a)]. 108.34/108.17 Derived: ti(int,A) = zero_zero(int) | hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),A))),A) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),C),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),A)). [resolve(6029,a,6014,a)]. 108.34/108.17 6030 -semiring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D))),D) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),D) # label(fact_1355_mod__mult__self1) # label(axiom). [clausify(3576)]. 108.34/108.17 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),B),C))),C) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),A),C). [resolve(6030,a,5980,a)]. 108.34/108.17 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),B),C))),C) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),A),C). [resolve(6030,a,6011,a)]. 108.34/108.17 Derived: hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C))),C) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A),C). [resolve(6030,a,6014,a)]. 108.34/108.17 6031 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),C),B)),D) = hAPP(A,A,hAPP(A,fun(A,A),div_div(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),D)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),D)) # label(fact_1097_div__power) # label(axiom). [clausify(3667)]. 108.34/108.17 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),B)) | hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),B),A)),C) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),B),C)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),C)). [resolve(6031,a,5980,a)]. 108.41/108.27 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),B)) | hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),B),A)),C) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),B),C)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),C)). [resolve(6031,a,6011,a)]. 108.41/108.27 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),A)),C) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),C)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),C)). [resolve(6031,a,6014,a)]. 108.41/108.27 6032 -semiring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),C)),D)),C) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)),C) # label(fact_1288_zmod__simps_I4_J) # label(axiom). [clausify(3670)]. 108.41/108.27 6033 -semiring_div(A) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),B) # label(fact_1280_mod__self) # label(axiom). [clausify(3743)]. 108.41/108.27 Derived: zero_zero(quickcheck_code_int) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),A),A). [resolve(6033,a,5980,a)]. 108.41/108.27 Derived: zero_zero(code_code_numeral) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),A),A). [resolve(6033,a,6011,a)]. 108.41/108.27 6034 -semiring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),C) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),C) # label(fact_1297_mod__add__self2) # label(axiom). [clausify(3755)]. 108.41/108.27 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),B)),B) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),A),B). [resolve(6034,a,5980,a)]. 108.41/108.27 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),B)),B) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),A),B). [resolve(6034,a,6011,a)]. 108.41/108.27 Derived: hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)),B) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A),B). [resolve(6034,a,6014,a)]. 108.41/108.27 6035 -semiring_div(A) | ti(A,B) = zero_zero(A) | ti(A,C) = hAPP(A,A,hAPP(A,fun(A,A),div_div(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),B)),B) # label(fact_1088_div__mult__self2__is__id) # label(axiom). [clausify(3815)]. 108.41/108.27 Derived: ti(quickcheck_code_int,A) = zero_zero(quickcheck_code_int) | ti(quickcheck_code_int,B) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),B),A)),A). [resolve(6035,a,5980,a)]. 108.82/108.60 Derived: ti(code_code_numeral,A) = zero_zero(code_code_numeral) | ti(code_code_numeral,B) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),B),A)),A). [resolve(6035,a,6011,a)]. 108.82/108.60 Derived: ti(int,A) = zero_zero(int) | ti(int,B) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),A)),A). [resolve(6035,a,6014,a)]. 108.82/108.60 6036 -number(A) | -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(int,A,number_number_of(A),B)),hAPP(int,A,number_number_of(A),C))) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(int,A,number_number_of(A),C)),hAPP(int,A,number_number_of(A),B)) # label(fact_1409_dvd__eq__mod__eq__0__number__of) # label(axiom). [clausify(4015)]. 108.82/108.60 6037 -number(A) | -semiring_div(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(int,A,number_number_of(A),B)),hAPP(int,A,number_number_of(A),C))) | zero_zero(A) != hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(int,A,number_number_of(A),C)),hAPP(int,A,number_number_of(A),B)) # label(fact_1409_dvd__eq__mod__eq__0__number__of) # label(axiom). [clausify(4015)]. 108.82/108.60 Derived: -number(quickcheck_code_int) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),A)),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),B))) | zero_zero(quickcheck_code_int) != hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),B)),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),A)). [resolve(6037,b,5980,a)]. 108.82/108.60 Derived: -number(code_code_numeral) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),A)),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),B))) | zero_zero(code_code_numeral) != hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),B)),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),A)). [resolve(6037,b,6011,a)]. 108.82/108.60 6038 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),C),D))) # label(fact_1303_dvd__mod) # label(axiom). [clausify(4175)]. 108.82/108.60 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),C))). [resolve(6038,a,6014,a)]. 108.82/108.60 6039 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),D),C)),B) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),D),B) # label(fact_1304_mod__mod__cancel) # label(axiom). [clausify(4295)]. 108.82/108.60 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),B)) | hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),C),B)),A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),C),A). [resolve(6039,a,5980,a)]. 108.94/108.77 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),B)) | hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),C),B)),A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),C),A). [resolve(6039,a,6011,a)]. 108.94/108.77 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),C),B)),A) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),C),A). [resolve(6039,a,6014,a)]. 108.94/108.77 6040 -semiring_div(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),B),C)),C)),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),C)) # label(fact_1412_mod__div__equality) # label(axiom). [clausify(4413)]. 108.94/108.77 Derived: ti(quickcheck_code_int,A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),A),B)),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),A),B)). [resolve(6040,a,5980,a)]. 108.94/108.77 Derived: ti(code_code_numeral,A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),A),B)),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),A),B)). [resolve(6040,a,6011,a)]. 108.94/108.77 Derived: ti(int,A) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A),B)),B)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A),B)). [resolve(6040,a,6014,a)]. 108.94/108.77 6041 -semiring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),C) != hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),D),C) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),E),C) != hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),F),C) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),E)),C) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),D),F)),C) # label(fact_1290_mod__add__cong) # label(axiom). [clausify(4537)]. 108.94/108.77 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),A),B) != hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),C),B) | hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),D),B) != hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),E),B) | hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),D)),B) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),C),E)),B). [resolve(6041,a,5980,a)]. 109.03/108.87 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),A),B) != hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),C),B) | hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),D),B) != hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),E),B) | hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),D)),B) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),C),E)),B). [resolve(6041,a,6011,a)]. 109.03/108.87 Derived: hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A),B) != hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),C),B) | hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),D),B) != hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),E),B) | hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),D)),B) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),C),E)),B). [resolve(6041,a,6014,a)]. 109.03/108.87 6042 -semiring_div(A) | div_mod(A) = ti(fun(A,fun(A,A)),div_mod(A)) # label(tsy_c_Divides_Odiv__class_Omod_res) # label(axiom). [clausify(4584)]. 109.03/108.87 Derived: div_mod(quickcheck_code_int) = ti(fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),div_mod(quickcheck_code_int)). [resolve(6042,a,5980,a)]. 109.03/108.87 Derived: div_mod(code_code_numeral) = ti(fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)),div_mod(code_code_numeral)). [resolve(6042,a,6011,a)]. 109.03/108.87 Derived: div_mod(int) = ti(fun(int,fun(int,int)),div_mod(int)). [resolve(6042,a,6014,a)]. 109.03/108.87 6043 -semiring_div(A) | ti(A,B) = zero_zero(A) | hAPP(A,A,hAPP(A,fun(A,A),div_div(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)) = hAPP(A,A,hAPP(A,fun(A,A),div_div(A),C),D) # label(fact_1089_div__mult__mult1__if) # label(axiom). [clausify(4684)]. 109.03/108.87 6044 -semiring_div(A) | ti(A,B) != zero_zero(A) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),div_div(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)) # label(fact_1089_div__mult__mult1__if) # label(axiom). [clausify(4684)]. 109.03/108.87 Derived: ti(quickcheck_code_int,A) != zero_zero(quickcheck_code_int) | zero_zero(quickcheck_code_int) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),C)). [resolve(6044,a,5980,a)]. 109.03/108.87 Derived: ti(code_code_numeral,A) != zero_zero(code_code_numeral) | zero_zero(code_code_numeral) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),C)). [resolve(6044,a,6011,a)]. 109.11/108.96 Derived: ti(int,A) != zero_zero(int) | zero_zero(int) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),C)). [resolve(6044,a,6014,a)]. 109.11/108.96 6045 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),C),B) # label(fact_1357_dvd__eq__mod__eq__0) # label(axiom). [clausify(4771)]. 109.11/108.96 6046 -semiring_div(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | zero_zero(A) != hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),C),B) # label(fact_1357_dvd__eq__mod__eq__0) # label(axiom). [clausify(4771)]. 109.11/108.96 Derived: hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),B)) | zero_zero(quickcheck_code_int) != hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),B),A). [resolve(6046,a,5980,a)]. 109.11/108.96 Derived: hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),B)) | zero_zero(code_code_numeral) != hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),B),A). [resolve(6046,a,6011,a)]. 109.11/108.96 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | zero_zero(int) != hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),A). [resolve(6046,a,6014,a)]. 109.11/108.96 6047 semiring_div(nat) # label(arity_Nat_Onat___Divides_Osemiring__div) # label(axiom). [assumption]. 109.11/108.96 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),B),C))),C) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),C). [resolve(6047,a,5981,a)]. 109.11/108.96 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),C),B))). [resolve(6047,a,5982,a)]. 109.11/108.96 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),C),B))). [resolve(6047,a,5983,a)]. 109.11/108.96 Derived: zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),zero_zero(nat)),A). [resolve(6047,a,5984,a)]. 109.11/108.96 Derived: ti(nat,A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A),B)),B)). [resolve(6047,a,5985,a)]. 109.11/108.96 Derived: zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A),one_one(nat)). [resolve(6047,a,5986,a)]. 109.11/108.96 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),C)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),B),C)). [resolve(6047,a,5987,a)]. 109.11/108.96 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),B),C))),C) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),B)),C). [resolve(6047,a,5988,a)]. 109.11/108.96 Derived: ti(nat,A) = zero_zero(nat) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),A)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),A)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),C). [resolve(6047,a,5989,a)]. 109.11/108.97 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A),B)),C)),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),C)),B). [resolve(6047,a,5990,a)]. 109.11/108.97 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A),B) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),C),B) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),D),B) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),E),B) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),D)),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),E)),B). [resolve(6047,a,5991,a)]. 109.11/108.97 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),C)) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),C)),A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),A)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),C),A)). [resolve(6047,a,5992,a)]. 109.11/108.97 Derived: zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),B). [resolve(6047,a,5993,a)]. 109.11/108.97 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C)),A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),A)),C). [resolve(6047,a,5994,a)]. 109.11/108.97 Derived: ti(nat,A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A),zero_zero(nat)). [resolve(6047,a,5995,a)]. 109.11/108.97 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),B)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A),C)),B). [resolve(6047,a,5996,a)]. 109.11/108.97 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),B),A). [resolve(6047,a,5997,a)]. 109.11/108.97 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A),B)),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A),B). [resolve(6047,a,5998,a)]. 109.11/108.97 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A),B)),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A),B))),C) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),C). [resolve(6047,a,5999,a)]. 109.11/108.97 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A),B)),C)),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),C)),B). [resolve(6047,a,6000,a)]. 109.11/108.97 Derived: ti(nat,A) = zero_zero(nat) | ti(nat,B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),A). [resolve(6047,a,6001,a)]. 109.11/108.97 Derived: ti(nat,A) = zero_zero(nat) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),C))),A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),A)). [resolve(6047,a,6002,a)]. 109.11/108.97 Derived: ti(nat,A) = zero_zero(nat) | one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A),A). [resolve(6047,a,6003,a)]. 109.11/108.97 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C))),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A),B). [resolve(6047,a,6004,a)]. 109.20/108.98 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | ti(nat,B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),A)). [resolve(6047,a,6005,a)]. 109.20/108.98 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),B)),A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),B),A). [resolve(6047,a,6006,a)]. 109.20/108.98 Derived: zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),zero_zero(nat)),A). [resolve(6047,a,6007,a)]. 109.20/108.98 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),A))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),B),A))),C) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),C). [resolve(6047,a,6008,a)]. 109.20/108.98 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),A)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),C),A))). [resolve(6047,a,6009,a)]. 109.20/108.98 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),A)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),C),A))). [resolve(6047,a,6010,a)]. 109.20/108.98 Derived: ti(nat,A) = zero_zero(nat) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | ti(nat,C) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),A) | ti(nat,B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),A). [resolve(6047,a,6012,a)]. 109.20/108.98 Derived: ti(nat,A) = zero_zero(nat) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | ti(nat,C) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),A) | ti(nat,B) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),A). [resolve(6047,a,6013,a)]. 109.20/108.98 Derived: ti(nat,A) = zero_zero(nat) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),B)),A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),A)),one_one(nat)). [resolve(6047,a,6016,a)]. 109.20/108.98 Derived: zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),A). [resolve(6047,a,6017,a)]. 109.20/108.98 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),B)),A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),A)). [resolve(6047,a,6019,a)]. 109.20/108.98 Derived: zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A),zero_zero(nat)). [resolve(6047,a,6020,a)]. 109.20/108.98 Derived: ti(nat,A) = zero_zero(nat) | ti(nat,B) = zero_zero(nat) | -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),B),D)) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),B) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),D) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),D),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),C),A). [resolve(6047,a,6021,a)]. 109.20/108.98 Derived: ti(nat,A) = zero_zero(nat) | ti(nat,B) = zero_zero(nat) | -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),B),D)) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),D) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),D),B) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),C),A). [resolve(6047,a,6022,a)]. 109.21/108.99 Derived: zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A),B)),B). [resolve(6047,a,6023,a)]. 109.21/108.99 Derived: ti(nat,A) = zero_zero(nat) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),A)),A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),A)),one_one(nat)). [resolve(6047,a,6024,a)]. 109.21/108.99 Derived: ti(nat,A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A),B))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A),B)). [resolve(6047,a,6025,a)]. 109.21/108.99 Derived: ti(nat,A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A),one_one(nat)). [resolve(6047,a,6026,a)]. 109.21/108.99 Derived: div_div(nat) = ti(fun(nat,fun(nat,nat)),div_div(nat)). [resolve(6047,a,6027,a)]. 109.21/108.99 Derived: ti(nat,A) = zero_zero(nat) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),C)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),C). [resolve(6047,a,6028,a)]. 109.21/108.99 Derived: ti(nat,A) = zero_zero(nat) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),A))),A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),A)). [resolve(6047,a,6029,a)]. 109.21/108.99 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C))),C) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A),C). [resolve(6047,a,6030,a)]. 109.21/108.99 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),A)),C) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),C)). [resolve(6047,a,6031,a)]. 109.21/108.99 Derived: zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A),A). [resolve(6047,a,6033,a)]. 109.21/108.99 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),B)),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A),B). [resolve(6047,a,6034,a)]. 109.21/108.99 Derived: ti(nat,A) = zero_zero(nat) | ti(nat,B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),A)),A). [resolve(6047,a,6035,a)]. 109.21/108.99 Derived: -number(nat) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(int,nat,number_number_of(nat),A)),hAPP(int,nat,number_number_of(nat),B))) | zero_zero(nat) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(int,nat,number_number_of(nat),B)),hAPP(int,nat,number_number_of(nat),A)). [resolve(6047,a,6037,b)]. 109.21/108.99 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),C),B)),A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),C),A). [resolve(6047,a,6039,a)]. 109.21/108.99 Derived: ti(nat,A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A),B)),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A),B)). [resolve(6047,a,6040,a)]. 109.21/108.99 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A),B) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),C),B) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),D),B) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),E),B) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),D)),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),C),E)),B). [resolve(6047,a,6041,a)]. 109.25/109.03 Derived: div_mod(nat) = ti(fun(nat,fun(nat,nat)),div_mod(nat)). [resolve(6047,a,6042,a)]. 109.25/109.03 Derived: ti(nat,A) != zero_zero(nat) | zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),C)). [resolve(6047,a,6044,a)]. 109.25/109.03 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | zero_zero(nat) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),B),A). [resolve(6047,a,6046,a)]. 109.25/109.03 6048 -semiring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),C)),D)),C) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),D)),C) # label(fact_1291_zmod__simps_I1_J) # label(axiom). [clausify(4782)]. 109.25/109.03 6049 -semiring_div(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),B),C)),C)),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),C)) # label(fact_1520_mod__div__decomp) # label(axiom). [clausify(4797)]. 109.25/109.03 6050 -semiring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),D),C))),C) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),D)),C) # label(fact_1293_mod__add__eq) # label(axiom). [clausify(4816)]. 109.25/109.03 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),C),B))),B) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),C)),B). [resolve(6050,a,5980,a)]. 109.25/109.03 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),A),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),C),B))),B) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),C)),B). [resolve(6050,a,6011,a)]. 109.25/109.03 Derived: hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),C),B))),B) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),C)),B). [resolve(6050,a,6014,a)]. 109.25/109.03 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),C),B))),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),C)),B). [resolve(6050,a,6047,a)]. 109.25/109.03 6051 -semiring_div(A) | hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),D),C))),C) = hAPP(A,A,hAPP(A,fun(A,A),div_mod(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)),C) # label(fact_1285_mod__mult__eq) # label(axiom). [clausify(4825)]. 109.25/109.04 Derived: hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),C),B))),B) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),C)),B). [resolve(6051,a,5980,a)]. 109.25/109.04 Derived: hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),A),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),C),B))),B) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),C)),B). [resolve(6051,a,6011,a)]. 109.25/109.04 Derived: hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),C),B))),B) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),C)),B). [resolve(6051,a,6014,a)]. 109.25/109.04 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),C),B))),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),C)),B). [resolve(6051,a,6047,a)]. 109.25/109.04 6052 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | ti(A,C) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),C),B)),B) # label(fact_1093_dvd__div__mult__self) # label(axiom). [clausify(4829)]. 109.25/109.04 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),B)) | ti(quickcheck_code_int,B) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),B),A)),A). [resolve(6052,a,5980,a)]. 109.25/109.04 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),B)) | ti(code_code_numeral,B) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),B),A)),A). [resolve(6052,a,6011,a)]. 109.25/109.04 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | ti(int,B) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),A)),A). [resolve(6052,a,6014,a)]. 109.25/109.07 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | ti(nat,B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),A)),A). [resolve(6052,a,6047,a)]. 109.25/109.07 6053 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),D),E)) | hAPP(A,A,hAPP(A,fun(A,A),div_div(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),E)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),C),B)),hAPP(A,A,hAPP(A,fun(A,A),div_div(A),E),D)) # label(fact_1095_div__mult__div__if__dvd) # label(axiom). [clausify(4868)]. 109.25/109.07 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),C),D)) | hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),B),D)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),C)) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),B),A)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),D),C)). [resolve(6053,a,5980,a)]. 109.25/109.07 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),C),D)) | hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),B),D)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),C)) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),B),A)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),D),C)). [resolve(6053,a,6011,a)]. 109.25/109.07 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),C),D)) | hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),D)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),C)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),A)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),D),C)). [resolve(6053,a,6014,a)]. 109.25/109.07 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)) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),D)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),C)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),A)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),D),C)). [resolve(6053,a,6047,a)]. 111.83/111.68 6054 monoid_mult(nat) # label(arity_Nat_Onat___Groups_Omonoid__mult) # label(axiom). [assumption]. 111.83/111.68 6055 -monoid_mult(A) | -number(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(int,A,number_number_of(A),B)),hAPP(int,A,number_number_of(A),B)) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(int,A,number_number_of(A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) # label(fact_551_power2__eq__square__number__of) # label(axiom). [clausify(108)]. 111.83/111.68 6056 -monoid_mult(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),C))) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) # label(fact_1505_power__odd__eq) # label(axiom). [clausify(486)]. 111.83/111.68 6057 -monoid_mult(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,C))) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(int,nat,number_number_of(nat),C))),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(int,nat,number_number_of(nat),C))) # label(fact_3881_power__number__of__even) # label(axiom). [clausify(560)]. 111.83/111.68 6058 -monoid_mult(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(nat,nat,suc,C)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C)),B) # label(fact_1370_power__Suc2) # label(axiom). [clausify(592)]. 111.83/111.68 Derived: -number(nat) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),A)),hAPP(int,nat,number_number_of(nat),A)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,number_number_of(nat),A)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))). [resolve(6054,a,6055,a)]. 111.83/111.68 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))). [resolve(6054,a,6056,a)]. 111.83/111.68 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,B))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),hAPP(int,nat,number_number_of(nat),B))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),hAPP(int,nat,number_number_of(nat),B))). [resolve(6054,a,6057,a)]. 111.83/111.68 6059 monoid_mult(quickcheck_code_int) # label(arity_Quickcheck__Narrowing_Ocode__int___Groups_Omonoid__mult) # label(axiom). [assumption]. 111.83/111.68 Derived: -number(quickcheck_code_int) | hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),A)),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),A)) = hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),A)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))). [resolve(6059,a,6055,a)]. 111.83/111.68 Derived: hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))). [resolve(6059,a,6056,a)]. 112.13/111.93 Derived: hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,B))) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),hAPP(int,nat,number_number_of(nat),B))),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),hAPP(int,nat,number_number_of(nat),B))). [resolve(6059,a,6057,a)]. 112.13/111.93 6060 -monoid_mult(A) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),B) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),B),one_one(nat)))),C) # label(fact_883_realpow__minus__mult) # label(axiom). [clausify(1025)]. 112.13/111.93 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),A) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A),one_one(nat)))),B). [resolve(6060,a,6054,a)]. 112.13/111.93 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),B),A) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A),one_one(nat)))),B). [resolve(6060,a,6059,a)]. 112.13/111.93 6061 -monoid_mult(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),B) # label(fact_547_power2__eq__square) # label(axiom). [clausify(1438)]. 112.13/111.93 6062 -monoid_mult(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),one_one(A)) # label(fact_390_mult__1__right) # label(axiom). [clausify(1868)]. 112.13/111.93 6063 -comm_ring(A) | -monoid_mult(A) | hAPP(fun(nat,bool),A,hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),hAPP(A,fun(nat,A),power_power(A),B))),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),minus_minus(A)),hAPP(fun(nat,nat),fun(nat,A),hAPP(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),combb(nat,A,nat),hAPP(A,fun(nat,A),power_power(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C))),hAPP(nat,fun(nat,nat),minus_minus(nat),D)))),hAPP(fun(nat,nat),fun(nat,A),hAPP(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),combb(nat,A,nat),hAPP(A,fun(nat,A),power_power(A),B)),hAPP(nat,fun(nat,nat),minus_minus(nat),D))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),D)) = hAPP(fun(nat,bool),A,hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),hAPP(A,fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(A,fun(nat,A)),combc(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),minus_minus(A)),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),hAPP(fun(nat,nat),fun(nat,A),hAPP(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),combb(nat,A,nat),hAPP(A,fun(nat,A),power_power(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C))),hAPP(nat,fun(nat,nat),minus_minus(nat),D)))),hAPP(A,fun(nat,A),power_power(A),B)))),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),D))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),D)) # label(fact_4253_lemma__termdiff1) # label(axiom). [clausify(2052)]. 112.13/111.94 Derived: -comm_ring(nat) | hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,nat)),combs(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),times_times(nat)),hAPP(nat,fun(nat,nat),power_power(nat),A))),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,nat)),combs(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),minus_minus(nat)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),hAPP(nat,fun(nat,nat),power_power(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),B))),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),hAPP(nat,fun(nat,nat),power_power(nat),A)),hAPP(nat,fun(nat,nat),minus_minus(nat),C))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),C)) = hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),minus_minus(nat)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,nat)),combs(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),times_times(nat)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),hAPP(nat,fun(nat,nat),power_power(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),B))),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(nat,fun(nat,nat),power_power(nat),A)))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),C))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),C)). [resolve(6063,b,6054,a)]. 112.13/111.94 Derived: -comm_ring(quickcheck_code_int) | hAPP(fun(nat,bool),quickcheck_code_int,hAPP(fun(nat,quickcheck_code_int),fun(fun(nat,bool),quickcheck_code_int),big_co1399186613setsum(nat,quickcheck_code_int),hAPP(fun(nat,quickcheck_code_int),fun(nat,quickcheck_code_int),hAPP(fun(nat,fun(quickcheck_code_int,quickcheck_code_int)),fun(fun(nat,quickcheck_code_int),fun(nat,quickcheck_code_int)),combs(nat,quickcheck_code_int,quickcheck_code_int),hAPP(fun(nat,quickcheck_code_int),fun(nat,fun(quickcheck_code_int,quickcheck_code_int)),hAPP(fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),fun(fun(nat,quickcheck_code_int),fun(nat,fun(quickcheck_code_int,quickcheck_code_int))),combb(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),nat),times_times(quickcheck_code_int)),hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A))),hAPP(fun(nat,quickcheck_code_int),fun(nat,quickcheck_code_int),hAPP(fun(nat,fun(quickcheck_code_int,quickcheck_code_int)),fun(fun(nat,quickcheck_code_int),fun(nat,quickcheck_code_int)),combs(nat,quickcheck_code_int,quickcheck_code_int),hAPP(fun(nat,quickcheck_code_int),fun(nat,fun(quickcheck_code_int,quickcheck_code_int)),hAPP(fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),fun(fun(nat,quickcheck_code_int),fun(nat,fun(quickcheck_code_int,quickcheck_code_int))),combb(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),nat),minus_minus(quickcheck_code_int)),hAPP(fun(nat,nat),fun(nat,quickcheck_code_int),hAPP(fun(nat,quickcheck_code_int),fun(fun(nat,nat),fun(nat,quickcheck_code_int)),combb(nat,quickcheck_code_int,nat),hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),B))),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(fun(nat,nat),fun(nat,quickcheck_code_int),hAPP(fun(nat,quickcheck_code_int),fun(fun(nat,nat),fun(nat,quickcheck_code_int)),combb(nat,quickcheck_code_int,nat),hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A)),hAPP(nat,fun(nat,nat),minus_minus(nat),C))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),C)) = hAPP(fun(nat,bool),quickcheck_code_int,hAPP(fun(nat,quickcheck_code_int),fun(fun(nat,bool),quickcheck_code_int),big_co1399186613setsum(nat,quickcheck_code_int),hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),hAPP(fun(nat,fun(quickcheck_code_int,quickcheck_code_int)),fun(quickcheck_code_int,fun(nat,quickcheck_code_int)),combc(nat,quickcheck_code_int,quickcheck_code_int),hAPP(fun(nat,quickcheck_code_int),fun(nat,fun(quickcheck_code_int,quickcheck_code_int)),hAPP(fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),fun(fun(nat,quickcheck_code_int),fun(nat,fun(quickcheck_code_int,quickcheck_code_int))),combb(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),nat),minus_minus(quickcheck_code_int)),hAPP(fun(nat,quickcheck_code_int),fun(nat,quickcheck_code_int),hAPP(fun(nat,fun(quickcheck_code_int,quickcheck_code_int)),fun(fun(nat,quickcheck_code_int),fun(nat,quickcheck_code_int)),combs(nat,quickcheck_code_int,quickcheck_code_int),hAPP(fun(nat,quickcheck_code_int),fun(nat,fun(quickcheck_code_int,quickcheck_code_int)),hAPP(fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),fun(fun(nat,quickcheck_code_int),fun(nat,fun(quickcheck_code_int,quickcheck_code_int))),combb(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),nat),times_times(quickcheck_code_int)),hAPP(fun(nat,nat),fun(nat,quickcheck_code_int),hAPP(fun(nat,quickcheck_code_int),fun(fun(nat,nat),fun(nat,quickcheck_code_int)),combb(nat,quickcheck_code_int,nat),hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),B))),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A)))),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),C))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),C)). [resolve(6063,b,6059,a)]. 112.13/111.95 6064 -monoid_mult(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),B))) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,B))) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),hAPP(int,nat,number_number_of(nat),B)))),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),hAPP(int,nat,number_number_of(nat),B))) # label(fact_3927_power__number__of__odd) # label(axiom). [clausify(2099)]. 112.13/111.95 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,A))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),hAPP(int,nat,number_number_of(nat),A)))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),hAPP(int,nat,number_number_of(nat),A))). [resolve(6064,a,6054,a)]. 112.13/111.97 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,A))) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),B),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),B),hAPP(int,nat,number_number_of(nat),A)))),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),B),hAPP(int,nat,number_number_of(nat),A))). [resolve(6064,a,6059,a)]. 112.13/111.97 6065 -monoid_mult(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),B))) | one_one(A) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,B))) # label(fact_3927_power__number__of__odd) # label(axiom). [clausify(2099)]. 112.13/111.97 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,A))). [resolve(6065,a,6054,a)]. 112.13/111.97 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | one_one(quickcheck_code_int) = hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,A))). [resolve(6065,a,6059,a)]. 112.13/111.97 6066 monoid_mult(real) # label(arity_RealDef_Oreal___Groups_Omonoid__mult) # label(axiom). [assumption]. 112.13/111.97 Derived: -number(real) | hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),A)),hAPP(int,real,number_number_of(real),A)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),A)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))). [resolve(6066,a,6055,a)]. 112.13/111.97 Derived: hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))). [resolve(6066,a,6056,a)]. 112.13/111.97 Derived: hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,B))) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(int,nat,number_number_of(nat),B))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(int,nat,number_number_of(nat),B))). [resolve(6066,a,6057,a)]. 112.13/111.97 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),A) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A),one_one(nat)))),B). [resolve(6066,a,6060,a)]. 112.13/111.99 Derived: -comm_ring(real) | hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),A))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),minus_minus(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B))),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),A)),hAPP(nat,fun(nat,nat),minus_minus(nat),C))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),C)) = hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),minus_minus(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B))),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(real,fun(nat,real),power_power(real),A)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),C))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),C)). [resolve(6066,a,6063,b)]. 112.13/111.99 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,A))) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),hAPP(int,nat,number_number_of(nat),A)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),hAPP(int,nat,number_number_of(nat),A))). [resolve(6066,a,6064,a)]. 112.13/111.99 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | one_one(real) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,A))). [resolve(6066,a,6065,a)]. 112.13/111.99 6067 monoid_mult(complex) # label(arity_Complex_Ocomplex___Groups_Omonoid__mult) # label(axiom). [assumption]. 112.13/111.99 Derived: -number(complex) | hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(int,complex,number_number_of(complex),A)),hAPP(int,complex,number_number_of(complex),A)) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(int,complex,number_number_of(complex),A)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))). [resolve(6067,a,6055,a)]. 112.13/111.99 Derived: hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))). [resolve(6067,a,6056,a)]. 112.13/111.99 Derived: hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,B))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),hAPP(int,nat,number_number_of(nat),B))),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),hAPP(int,nat,number_number_of(nat),B))). [resolve(6067,a,6057,a)]. 112.13/111.99 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),B),A) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A),one_one(nat)))),B). [resolve(6067,a,6060,a)]. 112.13/111.99 Derived: -comm_ring(complex) | hAPP(fun(nat,bool),complex,hAPP(fun(nat,complex),fun(fun(nat,bool),complex),big_co1399186613setsum(nat,complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),hAPP(complex,fun(nat,complex),power_power(complex),A))),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),minus_minus(complex)),hAPP(fun(nat,nat),fun(nat,complex),hAPP(fun(nat,complex),fun(fun(nat,nat),fun(nat,complex)),combb(nat,complex,nat),hAPP(complex,fun(nat,complex),power_power(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),B))),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(fun(nat,nat),fun(nat,complex),hAPP(fun(nat,complex),fun(fun(nat,nat),fun(nat,complex)),combb(nat,complex,nat),hAPP(complex,fun(nat,complex),power_power(complex),A)),hAPP(nat,fun(nat,nat),minus_minus(nat),C))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),C)) = hAPP(fun(nat,bool),complex,hAPP(fun(nat,complex),fun(fun(nat,bool),complex),big_co1399186613setsum(nat,complex),hAPP(complex,fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(complex,fun(nat,complex)),combc(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),minus_minus(complex)),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),hAPP(fun(nat,nat),fun(nat,complex),hAPP(fun(nat,complex),fun(fun(nat,nat),fun(nat,complex)),combb(nat,complex,nat),hAPP(complex,fun(nat,complex),power_power(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A),B))),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(complex,fun(nat,complex),power_power(complex),A)))),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),C))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),C)). [resolve(6067,a,6063,b)]. 112.23/112.08 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,A))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),B),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),B),hAPP(int,nat,number_number_of(nat),A)))),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),B),hAPP(int,nat,number_number_of(nat),A))). [resolve(6067,a,6064,a)]. 112.23/112.08 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | one_one(complex) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,A))). [resolve(6067,a,6065,a)]. 112.23/112.08 6068 monoid_mult(int) # label(arity_Int_Oint___Groups_Omonoid__mult) # label(axiom). [assumption]. 112.23/112.08 Derived: -number(int) | hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),A)),hAPP(int,int,number_number_of(int),A)) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),A)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))). [resolve(6068,a,6055,a)]. 112.23/112.08 Derived: hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))). [resolve(6068,a,6056,a)]. 112.23/112.08 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),A) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A),one_one(nat)))),B). [resolve(6068,a,6060,a)]. 112.23/112.08 Derived: -comm_ring(int) | hAPP(fun(nat,bool),int,hAPP(fun(nat,int),fun(fun(nat,bool),int),big_co1399186613setsum(nat,int),hAPP(fun(nat,int),fun(nat,int),hAPP(fun(nat,fun(int,int)),fun(fun(nat,int),fun(nat,int)),combs(nat,int,int),hAPP(fun(nat,int),fun(nat,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(nat,int),fun(nat,fun(int,int))),combb(int,fun(int,int),nat),times_times(int)),hAPP(int,fun(nat,int),power_power(int),A))),hAPP(fun(nat,int),fun(nat,int),hAPP(fun(nat,fun(int,int)),fun(fun(nat,int),fun(nat,int)),combs(nat,int,int),hAPP(fun(nat,int),fun(nat,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(nat,int),fun(nat,fun(int,int))),combb(int,fun(int,int),nat),minus_minus(int)),hAPP(fun(nat,nat),fun(nat,int),hAPP(fun(nat,int),fun(fun(nat,nat),fun(nat,int)),combb(nat,int,nat),hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B))),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(fun(nat,nat),fun(nat,int),hAPP(fun(nat,int),fun(fun(nat,nat),fun(nat,int)),combb(nat,int,nat),hAPP(int,fun(nat,int),power_power(int),A)),hAPP(nat,fun(nat,nat),minus_minus(nat),C))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),C)) = hAPP(fun(nat,bool),int,hAPP(fun(nat,int),fun(fun(nat,bool),int),big_co1399186613setsum(nat,int),hAPP(int,fun(nat,int),hAPP(fun(nat,fun(int,int)),fun(int,fun(nat,int)),combc(nat,int,int),hAPP(fun(nat,int),fun(nat,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(nat,int),fun(nat,fun(int,int))),combb(int,fun(int,int),nat),minus_minus(int)),hAPP(fun(nat,int),fun(nat,int),hAPP(fun(nat,fun(int,int)),fun(fun(nat,int),fun(nat,int)),combs(nat,int,int),hAPP(fun(nat,int),fun(nat,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(nat,int),fun(nat,fun(int,int))),combb(int,fun(int,int),nat),times_times(int)),hAPP(fun(nat,nat),fun(nat,int),hAPP(fun(nat,int),fun(fun(nat,nat),fun(nat,int)),combb(nat,int,nat),hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B))),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(int,fun(nat,int),power_power(int),A)))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),C))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),C)). [resolve(6068,a,6063,b)]. 112.61/112.39 6069 -monoid_mult(A) | -linordered_ring(A) | -hBOOL(hAPP(nat,bool,even_odd_even(nat),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),B))) # label(fact_2489_zero__le__even__power) # label(axiom). [clausify(2468)]. 112.61/112.39 Derived: -linordered_ring(quickcheck_code_int) | -hBOOL(hAPP(nat,bool,even_odd_even(nat),A)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),B),A))). [resolve(6069,a,6059,a)]. 112.61/112.39 Derived: -linordered_ring(real) | -hBOOL(hAPP(nat,bool,even_odd_even(nat),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),A))). [resolve(6069,a,6066,a)]. 112.61/112.39 Derived: -linordered_ring(complex) | -hBOOL(hAPP(nat,bool,even_odd_even(nat),A)) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),ord_less_eq(complex),zero_zero(complex)),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),B),A))). [resolve(6069,a,6067,a)]. 112.61/112.39 Derived: -linordered_ring(int) | -hBOOL(hAPP(nat,bool,even_odd_even(nat),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),A))). [resolve(6069,a,6068,a)]. 112.61/112.39 6070 -monoid_mult(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),D)) # label(fact_465_power__add) # label(axiom). [clausify(2561)]. 112.61/112.39 6071 -monoid_mult(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C)),D) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),D)) # label(fact_608_power__mult) # label(axiom). [clausify(2719)]. 112.61/112.39 6072 -monoid_mult(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C)),B) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C)) # label(fact_393_power__commutes) # label(axiom). [clausify(2740)]. 112.61/112.39 6073 -monoid_mult(A) | ti(A,B) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),one_one(A)),B) # label(fact_387_mult__1__left) # label(axiom). [clausify(2920)]. 112.61/112.39 6074 monoid_mult(rat) # label(arity_Rat_Orat___Groups_Omonoid__mult) # label(axiom). [assumption]. 112.61/112.39 Derived: -number(rat) | hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(int,rat,number_number_of(rat),A)),hAPP(int,rat,number_number_of(rat),A)) = hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),hAPP(int,rat,number_number_of(rat),A)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))). [resolve(6074,a,6055,a)]. 112.61/112.39 Derived: hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))). [resolve(6074,a,6056,a)]. 112.61/112.39 Derived: hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,B))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(int,nat,number_number_of(nat),B))),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(int,nat,number_number_of(nat),B))). [resolve(6074,a,6057,a)]. 112.65/112.44 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),B),A) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A),one_one(nat)))),B). [resolve(6074,a,6060,a)]. 112.65/112.44 Derived: -comm_ring(rat) | hAPP(fun(nat,bool),rat,hAPP(fun(nat,rat),fun(fun(nat,bool),rat),big_co1399186613setsum(nat,rat),hAPP(fun(nat,rat),fun(nat,rat),hAPP(fun(nat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,rat)),combs(nat,rat,rat),hAPP(fun(nat,rat),fun(nat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,fun(rat,rat))),combb(rat,fun(rat,rat),nat),times_times(rat)),hAPP(rat,fun(nat,rat),power_power(rat),A))),hAPP(fun(nat,rat),fun(nat,rat),hAPP(fun(nat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,rat)),combs(nat,rat,rat),hAPP(fun(nat,rat),fun(nat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,fun(rat,rat))),combb(rat,fun(rat,rat),nat),minus_minus(rat)),hAPP(fun(nat,nat),fun(nat,rat),hAPP(fun(nat,rat),fun(fun(nat,nat),fun(nat,rat)),combb(nat,rat,nat),hAPP(rat,fun(nat,rat),power_power(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B))),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(fun(nat,nat),fun(nat,rat),hAPP(fun(nat,rat),fun(fun(nat,nat),fun(nat,rat)),combb(nat,rat,nat),hAPP(rat,fun(nat,rat),power_power(rat),A)),hAPP(nat,fun(nat,nat),minus_minus(nat),C))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),C)) = hAPP(fun(nat,bool),rat,hAPP(fun(nat,rat),fun(fun(nat,bool),rat),big_co1399186613setsum(nat,rat),hAPP(rat,fun(nat,rat),hAPP(fun(nat,fun(rat,rat)),fun(rat,fun(nat,rat)),combc(nat,rat,rat),hAPP(fun(nat,rat),fun(nat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,fun(rat,rat))),combb(rat,fun(rat,rat),nat),minus_minus(rat)),hAPP(fun(nat,rat),fun(nat,rat),hAPP(fun(nat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,rat)),combs(nat,rat,rat),hAPP(fun(nat,rat),fun(nat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,fun(rat,rat))),combb(rat,fun(rat,rat),nat),times_times(rat)),hAPP(fun(nat,nat),fun(nat,rat),hAPP(fun(nat,rat),fun(fun(nat,nat),fun(nat,rat)),combb(nat,rat,nat),hAPP(rat,fun(nat,rat),power_power(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B))),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(rat,fun(nat,rat),power_power(rat),A)))),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),C))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),C)). [resolve(6074,a,6063,b)]. 112.65/112.44 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,A))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),B),hAPP(int,nat,number_number_of(nat),A)))),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),B),hAPP(int,nat,number_number_of(nat),A))). [resolve(6074,a,6064,a)]. 112.65/112.44 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | one_one(rat) = hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,A))). [resolve(6074,a,6065,a)]. 112.65/112.44 Derived: -linordered_ring(rat) | -hBOOL(hAPP(nat,bool,even_odd_even(nat),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),B),A))). [resolve(6074,a,6069,a)]. 112.65/112.44 6075 -comm_ring(A) | -monoid_mult(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C)),hAPP(fun(nat,bool),A,hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),hAPP(A,fun(nat,A),power_power(A),B))),hAPP(fun(nat,nat),fun(nat,A),hAPP(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),combb(nat,A,nat),hAPP(A,fun(nat,A),power_power(A),C)),hAPP(nat,fun(nat,nat),minus_minus(nat),D)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,D)))) = hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(nat,nat,suc,D))),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),C),hAPP(nat,nat,suc,D))) # label(fact_4254_lemma__realpow__diff__sumr2) # label(axiom). [clausify(3119)]. 112.65/112.45 Derived: -comm_ring(nat) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A),B)),hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,nat)),combs(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),times_times(nat)),hAPP(nat,fun(nat,nat),power_power(nat),A))),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),hAPP(nat,fun(nat,nat),power_power(nat),B)),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,C)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),hAPP(nat,nat,suc,C))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),hAPP(nat,nat,suc,C))). [resolve(6075,b,6054,a)]. 112.65/112.45 Derived: -comm_ring(quickcheck_code_int) | hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),minus_minus(quickcheck_code_int),A),B)),hAPP(fun(nat,bool),quickcheck_code_int,hAPP(fun(nat,quickcheck_code_int),fun(fun(nat,bool),quickcheck_code_int),big_co1399186613setsum(nat,quickcheck_code_int),hAPP(fun(nat,quickcheck_code_int),fun(nat,quickcheck_code_int),hAPP(fun(nat,fun(quickcheck_code_int,quickcheck_code_int)),fun(fun(nat,quickcheck_code_int),fun(nat,quickcheck_code_int)),combs(nat,quickcheck_code_int,quickcheck_code_int),hAPP(fun(nat,quickcheck_code_int),fun(nat,fun(quickcheck_code_int,quickcheck_code_int)),hAPP(fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),fun(fun(nat,quickcheck_code_int),fun(nat,fun(quickcheck_code_int,quickcheck_code_int))),combb(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),nat),times_times(quickcheck_code_int)),hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A))),hAPP(fun(nat,nat),fun(nat,quickcheck_code_int),hAPP(fun(nat,quickcheck_code_int),fun(fun(nat,nat),fun(nat,quickcheck_code_int)),combb(nat,quickcheck_code_int,nat),hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),B)),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,C)))) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),minus_minus(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),hAPP(nat,nat,suc,C))),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),B),hAPP(nat,nat,suc,C))). [resolve(6075,b,6059,a)]. 112.65/112.45 Derived: -comm_ring(real) | hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B)),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),A))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),B)),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,C)))) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(nat,nat,suc,C))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),hAPP(nat,nat,suc,C))). [resolve(6075,b,6066,a)]. 112.65/112.45 Derived: -comm_ring(complex) | hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),A),B)),hAPP(fun(nat,bool),complex,hAPP(fun(nat,complex),fun(fun(nat,bool),complex),big_co1399186613setsum(nat,complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),hAPP(complex,fun(nat,complex),power_power(complex),A))),hAPP(fun(nat,nat),fun(nat,complex),hAPP(fun(nat,complex),fun(fun(nat,nat),fun(nat,complex)),combb(nat,complex,nat),hAPP(complex,fun(nat,complex),power_power(complex),B)),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,C)))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),hAPP(nat,nat,suc,C))),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),B),hAPP(nat,nat,suc,C))). [resolve(6075,b,6067,a)]. 112.65/112.45 Derived: -comm_ring(int) | hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B)),hAPP(fun(nat,bool),int,hAPP(fun(nat,int),fun(fun(nat,bool),int),big_co1399186613setsum(nat,int),hAPP(fun(nat,int),fun(nat,int),hAPP(fun(nat,fun(int,int)),fun(fun(nat,int),fun(nat,int)),combs(nat,int,int),hAPP(fun(nat,int),fun(nat,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(nat,int),fun(nat,fun(int,int))),combb(int,fun(int,int),nat),times_times(int)),hAPP(int,fun(nat,int),power_power(int),A))),hAPP(fun(nat,nat),fun(nat,int),hAPP(fun(nat,int),fun(fun(nat,nat),fun(nat,int)),combb(nat,int,nat),hAPP(int,fun(nat,int),power_power(int),B)),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,C)))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),hAPP(nat,nat,suc,C))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(nat,nat,suc,C))). [resolve(6075,b,6068,a)]. 112.65/112.45 Derived: -comm_ring(rat) | hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B)),hAPP(fun(nat,bool),rat,hAPP(fun(nat,rat),fun(fun(nat,bool),rat),big_co1399186613setsum(nat,rat),hAPP(fun(nat,rat),fun(nat,rat),hAPP(fun(nat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,rat)),combs(nat,rat,rat),hAPP(fun(nat,rat),fun(nat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,fun(rat,rat))),combb(rat,fun(rat,rat),nat),times_times(rat)),hAPP(rat,fun(nat,rat),power_power(rat),A))),hAPP(fun(nat,nat),fun(nat,rat),hAPP(fun(nat,rat),fun(fun(nat,nat),fun(nat,rat)),combb(nat,rat,nat),hAPP(rat,fun(nat,rat),power_power(rat),B)),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,C)))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(nat,nat,suc,C))),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),B),hAPP(nat,nat,suc,C))). [resolve(6075,b,6074,a)]. 112.75/112.59 6076 -monoid_mult(A) | -comm_semiring_0(A) | hAPP(fun(nat,bool),A,hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),hAPP(A,fun(nat,A),power_power(A),B))),hAPP(fun(nat,nat),fun(nat,A),hAPP(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),combb(nat,A,nat),hAPP(A,fun(nat,A),power_power(A),C)),hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,D))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,D))) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),hAPP(fun(nat,bool),A,hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),hAPP(fun(nat,A),fun(nat,A),hAPP(fun(nat,fun(A,A)),fun(fun(nat,A),fun(nat,A)),combs(nat,A,A),hAPP(fun(nat,A),fun(nat,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,fun(A,A))),combb(A,fun(A,A),nat),times_times(A)),hAPP(A,fun(nat,A),power_power(A),B))),hAPP(fun(nat,nat),fun(nat,A),hAPP(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),combb(nat,A,nat),hAPP(A,fun(nat,A),power_power(A),C)),hAPP(nat,fun(nat,nat),minus_minus(nat),D)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,D)))) # label(fact_4202_lemma__realpow__diff__sumr) # label(axiom). [clausify(3379)]. 112.75/112.59 Derived: -comm_semiring_0(nat) | hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,nat)),combs(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),times_times(nat)),hAPP(nat,fun(nat,nat),power_power(nat),A))),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),hAPP(nat,fun(nat,nat),power_power(nat),B)),hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,C))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,C))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,nat)),combs(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),times_times(nat)),hAPP(nat,fun(nat,nat),power_power(nat),A))),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),hAPP(nat,fun(nat,nat),power_power(nat),B)),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,C)))). [resolve(6076,a,6054,a)]. 112.75/112.59 Derived: -comm_semiring_0(quickcheck_code_int) | hAPP(fun(nat,bool),quickcheck_code_int,hAPP(fun(nat,quickcheck_code_int),fun(fun(nat,bool),quickcheck_code_int),big_co1399186613setsum(nat,quickcheck_code_int),hAPP(fun(nat,quickcheck_code_int),fun(nat,quickcheck_code_int),hAPP(fun(nat,fun(quickcheck_code_int,quickcheck_code_int)),fun(fun(nat,quickcheck_code_int),fun(nat,quickcheck_code_int)),combs(nat,quickcheck_code_int,quickcheck_code_int),hAPP(fun(nat,quickcheck_code_int),fun(nat,fun(quickcheck_code_int,quickcheck_code_int)),hAPP(fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),fun(fun(nat,quickcheck_code_int),fun(nat,fun(quickcheck_code_int,quickcheck_code_int))),combb(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),nat),times_times(quickcheck_code_int)),hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A))),hAPP(fun(nat,nat),fun(nat,quickcheck_code_int),hAPP(fun(nat,quickcheck_code_int),fun(fun(nat,nat),fun(nat,quickcheck_code_int)),combb(nat,quickcheck_code_int,nat),hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),B)),hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,C))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,C))) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),B),hAPP(fun(nat,bool),quickcheck_code_int,hAPP(fun(nat,quickcheck_code_int),fun(fun(nat,bool),quickcheck_code_int),big_co1399186613setsum(nat,quickcheck_code_int),hAPP(fun(nat,quickcheck_code_int),fun(nat,quickcheck_code_int),hAPP(fun(nat,fun(quickcheck_code_int,quickcheck_code_int)),fun(fun(nat,quickcheck_code_int),fun(nat,quickcheck_code_int)),combs(nat,quickcheck_code_int,quickcheck_code_int),hAPP(fun(nat,quickcheck_code_int),fun(nat,fun(quickcheck_code_int,quickcheck_code_int)),hAPP(fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),fun(fun(nat,quickcheck_code_int),fun(nat,fun(quickcheck_code_int,quickcheck_code_int))),combb(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),nat),times_times(quickcheck_code_int)),hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A))),hAPP(fun(nat,nat),fun(nat,quickcheck_code_int),hAPP(fun(nat,quickcheck_code_int),fun(fun(nat,nat),fun(nat,quickcheck_code_int)),combb(nat,quickcheck_code_int,nat),hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),B)),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,C)))). [resolve(6076,a,6059,a)]. 112.75/112.59 Derived: -comm_semiring_0(real) | hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),A))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),B)),hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,C))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,C))) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),A))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),B)),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,C)))). [resolve(6076,a,6066,a)]. 112.75/112.59 Derived: -comm_semiring_0(complex) | hAPP(fun(nat,bool),complex,hAPP(fun(nat,complex),fun(fun(nat,bool),complex),big_co1399186613setsum(nat,complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),hAPP(complex,fun(nat,complex),power_power(complex),A))),hAPP(fun(nat,nat),fun(nat,complex),hAPP(fun(nat,complex),fun(fun(nat,nat),fun(nat,complex)),combb(nat,complex,nat),hAPP(complex,fun(nat,complex),power_power(complex),B)),hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,C))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,C))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),B),hAPP(fun(nat,bool),complex,hAPP(fun(nat,complex),fun(fun(nat,bool),complex),big_co1399186613setsum(nat,complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(nat,fun(complex,complex)),fun(fun(nat,complex),fun(nat,complex)),combs(nat,complex,complex),hAPP(fun(nat,complex),fun(nat,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(nat,complex),fun(nat,fun(complex,complex))),combb(complex,fun(complex,complex),nat),times_times(complex)),hAPP(complex,fun(nat,complex),power_power(complex),A))),hAPP(fun(nat,nat),fun(nat,complex),hAPP(fun(nat,complex),fun(fun(nat,nat),fun(nat,complex)),combb(nat,complex,nat),hAPP(complex,fun(nat,complex),power_power(complex),B)),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,C)))). [resolve(6076,a,6067,a)]. 112.75/112.59 Derived: -comm_semiring_0(int) | hAPP(fun(nat,bool),int,hAPP(fun(nat,int),fun(fun(nat,bool),int),big_co1399186613setsum(nat,int),hAPP(fun(nat,int),fun(nat,int),hAPP(fun(nat,fun(int,int)),fun(fun(nat,int),fun(nat,int)),combs(nat,int,int),hAPP(fun(nat,int),fun(nat,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(nat,int),fun(nat,fun(int,int))),combb(int,fun(int,int),nat),times_times(int)),hAPP(int,fun(nat,int),power_power(int),A))),hAPP(fun(nat,nat),fun(nat,int),hAPP(fun(nat,int),fun(fun(nat,nat),fun(nat,int)),combb(nat,int,nat),hAPP(int,fun(nat,int),power_power(int),B)),hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,C))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,C))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),hAPP(fun(nat,bool),int,hAPP(fun(nat,int),fun(fun(nat,bool),int),big_co1399186613setsum(nat,int),hAPP(fun(nat,int),fun(nat,int),hAPP(fun(nat,fun(int,int)),fun(fun(nat,int),fun(nat,int)),combs(nat,int,int),hAPP(fun(nat,int),fun(nat,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(nat,int),fun(nat,fun(int,int))),combb(int,fun(int,int),nat),times_times(int)),hAPP(int,fun(nat,int),power_power(int),A))),hAPP(fun(nat,nat),fun(nat,int),hAPP(fun(nat,int),fun(fun(nat,nat),fun(nat,int)),combb(nat,int,nat),hAPP(int,fun(nat,int),power_power(int),B)),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,C)))). [resolve(6076,a,6068,a)]. 112.75/112.59 Derived: -comm_semiring_0(rat) | hAPP(fun(nat,bool),rat,hAPP(fun(nat,rat),fun(fun(nat,bool),rat),big_co1399186613setsum(nat,rat),hAPP(fun(nat,rat),fun(nat,rat),hAPP(fun(nat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,rat)),combs(nat,rat,rat),hAPP(fun(nat,rat),fun(nat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,fun(rat,rat))),combb(rat,fun(rat,rat),nat),times_times(rat)),hAPP(rat,fun(nat,rat),power_power(rat),A))),hAPP(fun(nat,nat),fun(nat,rat),hAPP(fun(nat,rat),fun(fun(nat,nat),fun(nat,rat)),combb(nat,rat,nat),hAPP(rat,fun(nat,rat),power_power(rat),B)),hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,C))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,C))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),hAPP(fun(nat,bool),rat,hAPP(fun(nat,rat),fun(fun(nat,bool),rat),big_co1399186613setsum(nat,rat),hAPP(fun(nat,rat),fun(nat,rat),hAPP(fun(nat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,rat)),combs(nat,rat,rat),hAPP(fun(nat,rat),fun(nat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,fun(rat,rat))),combb(rat,fun(rat,rat),nat),times_times(rat)),hAPP(rat,fun(nat,rat),power_power(rat),A))),hAPP(fun(nat,nat),fun(nat,rat),hAPP(fun(nat,rat),fun(fun(nat,nat),fun(nat,rat)),combb(nat,rat,nat),hAPP(rat,fun(nat,rat),power_power(rat),B)),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,C)))). [resolve(6076,a,6074,a)]. 113.28/113.11 6077 monoid_mult(code_code_numeral) # label(arity_Code__Numeral_Ocode__numeral___Groups_Omonoid__mult) # label(axiom). [assumption]. 113.28/113.11 Derived: -number(code_code_numeral) | hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),A)),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),A)) = hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),A)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))). [resolve(6077,a,6055,a)]. 113.28/113.11 Derived: hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B))) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))). [resolve(6077,a,6056,a)]. 113.28/113.11 Derived: hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,B))) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),hAPP(int,nat,number_number_of(nat),B))),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),hAPP(int,nat,number_number_of(nat),B))). [resolve(6077,a,6057,a)]. 113.28/113.11 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),B),A) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A),one_one(nat)))),B). [resolve(6077,a,6060,a)]. 113.28/113.11 Derived: -comm_ring(code_code_numeral) | hAPP(fun(nat,bool),code_code_numeral,hAPP(fun(nat,code_code_numeral),fun(fun(nat,bool),code_code_numeral),big_co1399186613setsum(nat,code_code_numeral),hAPP(fun(nat,code_code_numeral),fun(nat,code_code_numeral),hAPP(fun(nat,fun(code_code_numeral,code_code_numeral)),fun(fun(nat,code_code_numeral),fun(nat,code_code_numeral)),combs(nat,code_code_numeral,code_code_numeral),hAPP(fun(nat,code_code_numeral),fun(nat,fun(code_code_numeral,code_code_numeral)),hAPP(fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)),fun(fun(nat,code_code_numeral),fun(nat,fun(code_code_numeral,code_code_numeral))),combb(code_code_numeral,fun(code_code_numeral,code_code_numeral),nat),times_times(code_code_numeral)),hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A))),hAPP(fun(nat,code_code_numeral),fun(nat,code_code_numeral),hAPP(fun(nat,fun(code_code_numeral,code_code_numeral)),fun(fun(nat,code_code_numeral),fun(nat,code_code_numeral)),combs(nat,code_code_numeral,code_code_numeral),hAPP(fun(nat,code_code_numeral),fun(nat,fun(code_code_numeral,code_code_numeral)),hAPP(fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)),fun(fun(nat,code_code_numeral),fun(nat,fun(code_code_numeral,code_code_numeral))),combb(code_code_numeral,fun(code_code_numeral,code_code_numeral),nat),minus_minus(code_code_numeral)),hAPP(fun(nat,nat),fun(nat,code_code_numeral),hAPP(fun(nat,code_code_numeral),fun(fun(nat,nat),fun(nat,code_code_numeral)),combb(nat,code_code_numeral,nat),hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),B))),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(fun(nat,nat),fun(nat,code_code_numeral),hAPP(fun(nat,code_code_numeral),fun(fun(nat,nat),fun(nat,code_code_numeral)),combb(nat,code_code_numeral,nat),hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A)),hAPP(nat,fun(nat,nat),minus_minus(nat),C))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),C)) = hAPP(fun(nat,bool),code_code_numeral,hAPP(fun(nat,code_code_numeral),fun(fun(nat,bool),code_code_numeral),big_co1399186613setsum(nat,code_code_numeral),hAPP(code_code_numeral,fun(nat,code_code_numeral),hAPP(fun(nat,fun(code_code_numeral,code_code_numeral)),fun(code_code_numeral,fun(nat,code_code_numeral)),combc(nat,code_code_numeral,code_code_numeral),hAPP(fun(nat,code_code_numeral),fun(nat,fun(code_code_numeral,code_code_numeral)),hAPP(fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)),fun(fun(nat,code_code_numeral),fun(nat,fun(code_code_numeral,code_code_numeral))),combb(code_code_numeral,fun(code_code_numeral,code_code_numeral),nat),minus_minus(code_code_numeral)),hAPP(fun(nat,code_code_numeral),fun(nat,code_code_numeral),hAPP(fun(nat,fun(code_code_numeral,code_code_numeral)),fun(fun(nat,code_code_numeral),fun(nat,code_code_numeral)),combs(nat,code_code_numeral,code_code_numeral),hAPP(fun(nat,code_code_numeral),fun(nat,fun(code_code_numeral,code_code_numeral)),hAPP(fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)),fun(fun(nat,code_code_numeral),fun(nat,fun(code_code_numeral,code_code_numeral))),combb(code_code_numeral,fun(code_code_numeral,code_code_numeral),nat),times_times(code_code_numeral)),hAPP(fun(nat,nat),fun(nat,code_code_numeral),hAPP(fun(nat,code_code_numeral),fun(fun(nat,nat),fun(nat,code_code_numeral)),combb(nat,code_code_numeral,nat),hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),B))),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A)))),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),C))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),C)). [resolve(6077,a,6063,b)]. 113.28/113.11 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,A))) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),B),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),B),hAPP(int,nat,number_number_of(nat),A)))),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),B),hAPP(int,nat,number_number_of(nat),A))). [resolve(6077,a,6064,a)]. 113.36/113.12 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | one_one(code_code_numeral) = hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,A))). [resolve(6077,a,6065,a)]. 113.36/113.12 Derived: -linordered_ring(code_code_numeral) | -hBOOL(hAPP(nat,bool,even_odd_even(nat),A)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),B),A))). [resolve(6077,a,6069,a)]. 113.36/113.12 Derived: -comm_ring(code_code_numeral) | hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),minus_minus(code_code_numeral),A),B)),hAPP(fun(nat,bool),code_code_numeral,hAPP(fun(nat,code_code_numeral),fun(fun(nat,bool),code_code_numeral),big_co1399186613setsum(nat,code_code_numeral),hAPP(fun(nat,code_code_numeral),fun(nat,code_code_numeral),hAPP(fun(nat,fun(code_code_numeral,code_code_numeral)),fun(fun(nat,code_code_numeral),fun(nat,code_code_numeral)),combs(nat,code_code_numeral,code_code_numeral),hAPP(fun(nat,code_code_numeral),fun(nat,fun(code_code_numeral,code_code_numeral)),hAPP(fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)),fun(fun(nat,code_code_numeral),fun(nat,fun(code_code_numeral,code_code_numeral))),combb(code_code_numeral,fun(code_code_numeral,code_code_numeral),nat),times_times(code_code_numeral)),hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A))),hAPP(fun(nat,nat),fun(nat,code_code_numeral),hAPP(fun(nat,code_code_numeral),fun(fun(nat,nat),fun(nat,code_code_numeral)),combb(nat,code_code_numeral,nat),hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),B)),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,C)))) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),minus_minus(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),hAPP(nat,nat,suc,C))),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),B),hAPP(nat,nat,suc,C))). [resolve(6077,a,6075,b)]. 113.36/113.12 Derived: -comm_semiring_0(code_code_numeral) | hAPP(fun(nat,bool),code_code_numeral,hAPP(fun(nat,code_code_numeral),fun(fun(nat,bool),code_code_numeral),big_co1399186613setsum(nat,code_code_numeral),hAPP(fun(nat,code_code_numeral),fun(nat,code_code_numeral),hAPP(fun(nat,fun(code_code_numeral,code_code_numeral)),fun(fun(nat,code_code_numeral),fun(nat,code_code_numeral)),combs(nat,code_code_numeral,code_code_numeral),hAPP(fun(nat,code_code_numeral),fun(nat,fun(code_code_numeral,code_code_numeral)),hAPP(fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)),fun(fun(nat,code_code_numeral),fun(nat,fun(code_code_numeral,code_code_numeral))),combb(code_code_numeral,fun(code_code_numeral,code_code_numeral),nat),times_times(code_code_numeral)),hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A))),hAPP(fun(nat,nat),fun(nat,code_code_numeral),hAPP(fun(nat,code_code_numeral),fun(fun(nat,nat),fun(nat,code_code_numeral)),combb(nat,code_code_numeral,nat),hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),B)),hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,C))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,C))) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),B),hAPP(fun(nat,bool),code_code_numeral,hAPP(fun(nat,code_code_numeral),fun(fun(nat,bool),code_code_numeral),big_co1399186613setsum(nat,code_code_numeral),hAPP(fun(nat,code_code_numeral),fun(nat,code_code_numeral),hAPP(fun(nat,fun(code_code_numeral,code_code_numeral)),fun(fun(nat,code_code_numeral),fun(nat,code_code_numeral)),combs(nat,code_code_numeral,code_code_numeral),hAPP(fun(nat,code_code_numeral),fun(nat,fun(code_code_numeral,code_code_numeral)),hAPP(fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)),fun(fun(nat,code_code_numeral),fun(nat,fun(code_code_numeral,code_code_numeral))),combb(code_code_numeral,fun(code_code_numeral,code_code_numeral),nat),times_times(code_code_numeral)),hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A))),hAPP(fun(nat,nat),fun(nat,code_code_numeral),hAPP(fun(nat,code_code_numeral),fun(fun(nat,nat),fun(nat,code_code_numeral)),combb(nat,code_code_numeral,nat),hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),B)),hAPP(nat,fun(nat,nat),minus_minus(nat),C)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,C)))). [resolve(6077,a,6076,a)]. 113.42/113.22 6078 -monoid_mult(A) | one_one(A) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),one_one(A)),B) # label(fact_169_power__one) # label(axiom). [clausify(4034)]. 113.42/113.22 Derived: one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),one_one(nat)),A). [resolve(6078,a,6054,a)]. 113.42/113.22 Derived: one_one(quickcheck_code_int) = hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),one_one(quickcheck_code_int)),A). [resolve(6078,a,6059,a)]. 113.42/113.22 Derived: one_one(real) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),one_one(real)),A). [resolve(6078,a,6066,a)]. 113.42/113.22 Derived: one_one(complex) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),one_one(complex)),A). [resolve(6078,a,6067,a)]. 113.42/113.22 Derived: one_one(int) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),one_one(int)),A). [resolve(6078,a,6068,a)]. 113.42/113.22 Derived: one_one(rat) = hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),one_one(rat)),A). [resolve(6078,a,6074,a)]. 113.42/113.22 Derived: one_one(code_code_numeral) = hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),one_one(code_code_numeral)),A). [resolve(6078,a,6077,a)]. 113.42/113.22 6079 -number(A) | -monoid_mult(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),B))) | one_one(A) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(int,A,number_number_of(A),C)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,B))) # label(fact_3934_power__number__of__odd__number__of) # label(axiom). [clausify(4094)]. 113.42/113.22 Derived: -number(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | one_one(int) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,A))). [resolve(6079,b,6068,a)]. 113.42/113.22 6080 -number(A) | -monoid_mult(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),B))) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(int,A,number_number_of(A),C)),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(int,A,number_number_of(A),C)),hAPP(int,nat,number_number_of(nat),B)))),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(int,A,number_number_of(A),C)),hAPP(int,nat,number_number_of(nat),B))) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(int,A,number_number_of(A),C)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,B))) # label(fact_3934_power__number__of__odd__number__of) # label(axiom). [clausify(4094)]. 113.42/113.22 Derived: -number(nat) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,number_number_of(nat),B)),hAPP(int,nat,number_number_of(nat),A)))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,number_number_of(nat),B)),hAPP(int,nat,number_number_of(nat),A))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,number_number_of(nat),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,A))). [resolve(6080,b,6054,a)]. 113.42/113.22 Derived: -number(quickcheck_code_int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),B)),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),B)),hAPP(int,nat,number_number_of(nat),A)))),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),B)),hAPP(int,nat,number_number_of(nat),A))) = hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,A))). [resolve(6080,b,6059,a)]. 113.42/113.22 Derived: -number(real) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),B)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),B)),hAPP(int,nat,number_number_of(nat),A)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),B)),hAPP(int,nat,number_number_of(nat),A))) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,A))). [resolve(6080,b,6066,a)]. 113.42/113.22 Derived: -number(complex) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(int,complex,number_number_of(complex),B)),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(int,complex,number_number_of(complex),B)),hAPP(int,nat,number_number_of(nat),A)))),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(int,complex,number_number_of(complex),B)),hAPP(int,nat,number_number_of(nat),A))) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(int,complex,number_number_of(complex),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,A))). [resolve(6080,b,6067,a)]. 113.42/113.22 Derived: -number(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),B)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),B)),hAPP(int,nat,number_number_of(nat),A)))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),B)),hAPP(int,nat,number_number_of(nat),A))) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,A))). [resolve(6080,b,6068,a)]. 113.53/113.32 Derived: -number(rat) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(int,rat,number_number_of(rat),B)),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),hAPP(int,rat,number_number_of(rat),B)),hAPP(int,nat,number_number_of(nat),A)))),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),hAPP(int,rat,number_number_of(rat),B)),hAPP(int,nat,number_number_of(nat),A))) = hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),hAPP(int,rat,number_number_of(rat),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,A))). [resolve(6080,b,6074,a)]. 113.53/113.32 Derived: -number(code_code_numeral) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),A))) | hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),B)),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),B)),hAPP(int,nat,number_number_of(nat),A)))),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),B)),hAPP(int,nat,number_number_of(nat),A))) = hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,A))). [resolve(6080,b,6077,a)]. 113.53/113.32 6081 -monoid_mult(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),B)),B) # label(fact_540_power3__eq__cube) # label(axiom). [clausify(4201)]. 113.53/113.32 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),A)),A). [resolve(6081,a,6054,a)]. 113.53/113.32 Derived: hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),A)),A). [resolve(6081,a,6059,a)]. 113.53/113.32 Derived: hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),A)),A). [resolve(6081,a,6066,a)]. 113.53/113.32 Derived: hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),A)),A). [resolve(6081,a,6067,a)]. 113.53/113.32 Derived: hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),A)),A). [resolve(6081,a,6068,a)]. 113.70/113.50 Derived: hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),A)),A). [resolve(6081,a,6074,a)]. 113.70/113.50 Derived: hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),A)),A). [resolve(6081,a,6077,a)]. 113.70/113.50 6082 -monoid_mult(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),C)) # label(fact_645_power__even__eq) # label(axiom). [clausify(4445)]. 113.70/113.50 Derived: hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B)). [resolve(6082,a,6054,a)]. 113.70/113.50 Derived: hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B)). [resolve(6082,a,6059,a)]. 113.70/113.50 Derived: hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B)). [resolve(6082,a,6066,a)]. 113.70/113.50 Derived: hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B)). [resolve(6082,a,6067,a)]. 113.70/113.50 Derived: hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B)). [resolve(6082,a,6068,a)]. 113.70/113.50 Derived: hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B)). [resolve(6082,a,6074,a)]. 113.70/113.55 Derived: hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),B)). [resolve(6082,a,6077,a)]. 113.70/113.55 6083 -number(A) | -monoid_mult(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(int,A,number_number_of(A),B)),hAPP(int,nat,number_number_of(nat),C))),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(int,A,number_number_of(A),B)),hAPP(int,nat,number_number_of(nat),C))) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(int,A,number_number_of(A),B)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,C))) # label(fact_3882_power__number__of__even__number__of) # label(axiom). [clausify(4506)]. 113.70/113.55 Derived: -number(nat) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,number_number_of(nat),A)),hAPP(int,nat,number_number_of(nat),B))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,number_number_of(nat),A)),hAPP(int,nat,number_number_of(nat),B))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,number_number_of(nat),A)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,B))). [resolve(6083,b,6054,a)]. 113.70/113.55 Derived: -number(quickcheck_code_int) | hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),A)),hAPP(int,nat,number_number_of(nat),B))),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),A)),hAPP(int,nat,number_number_of(nat),B))) = hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),A)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,B))). [resolve(6083,b,6059,a)]. 113.70/113.55 Derived: -number(real) | hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),A)),hAPP(int,nat,number_number_of(nat),B))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),A)),hAPP(int,nat,number_number_of(nat),B))) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),A)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,B))). [resolve(6083,b,6066,a)]. 113.70/113.55 Derived: -number(complex) | hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(int,complex,number_number_of(complex),A)),hAPP(int,nat,number_number_of(nat),B))),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(int,complex,number_number_of(complex),A)),hAPP(int,nat,number_number_of(nat),B))) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(int,complex,number_number_of(complex),A)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,B))). [resolve(6083,b,6067,a)]. 113.70/113.55 Derived: -number(int) | hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),A)),hAPP(int,nat,number_number_of(nat),B))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),A)),hAPP(int,nat,number_number_of(nat),B))) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),A)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,B))). [resolve(6083,b,6068,a)]. 113.94/113.72 Derived: -number(rat) | hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),hAPP(int,rat,number_number_of(rat),A)),hAPP(int,nat,number_number_of(nat),B))),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),hAPP(int,rat,number_number_of(rat),A)),hAPP(int,nat,number_number_of(nat),B))) = hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),hAPP(int,rat,number_number_of(rat),A)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,B))). [resolve(6083,b,6074,a)]. 113.94/113.72 Derived: -number(code_code_numeral) | hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),A)),hAPP(int,nat,number_number_of(nat),B))),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),A)),hAPP(int,nat,number_number_of(nat),B))) = hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),A)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,B))). [resolve(6083,b,6077,a)]. 113.94/113.72 6084 -monoid_mult(A) | ti(A,B) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),one_one(nat)) # label(fact_160_power__one__right) # label(axiom). [clausify(4614)]. 113.94/113.72 6085 -monoid_mult(A) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),D),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,C)),B)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),D),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),C),B))),D) # label(fact_1463_lemma__realpow__diff) # label(axiom). [clausify(4724)]. 113.94/113.72 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,B)),A)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),B),A))),C). [resolve(6085,a,6054,a)]. 113.94/113.72 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,B)),A)) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),B),A))),C). [resolve(6085,a,6059,a)]. 113.94/113.72 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,B)),A)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),B),A))),C). [resolve(6085,a,6066,a)]. 113.94/113.72 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,B)),A)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),B),A))),C). [resolve(6085,a,6067,a)]. 117.24/117.00 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,B)),A)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),B),A))),C). [resolve(6085,a,6068,a)]. 117.24/117.00 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,B)),A)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),B),A))),C). [resolve(6085,a,6074,a)]. 117.24/117.00 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,B)),A)) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),B),A))),C). [resolve(6085,a,6077,a)]. 117.24/117.00 6086 ordere216010020id_add(nat) # label(arity_Nat_Onat___Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 117.24/117.00 6087 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C))) # label(fact_199_add__pos__pos) # label(axiom). [clausify(132)]. 117.24/117.00 6088 -ordere216010020id_add(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | -hBOOL(hAPP(fun(B,bool),bool,hAPP(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),D),C)) | hBOOL(hAPP(fun(B,bool),bool,hAPP(B,fun(fun(B,bool),bool),member(B),f85(B,A,E,D,C)),hAPP(fun(B,bool),fun(B,bool),hAPP(fun(B,bool),fun(fun(B,bool),fun(B,bool)),minus_minus(fun(B,bool)),C),D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),E),D)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),E),C))) # label(fact_4320_setsum__mono2) # label(axiom). [clausify(380)]. 117.24/117.00 6089 -ordere216010020id_add(A) | -hBOOL(hAPP(fun(B,bool),bool,finite_finite_1(B),C)) | -hBOOL(hAPP(fun(B,bool),bool,hAPP(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),D),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(B,A,E,f85(B,A,E,D,C)))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),E),D)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),E),C))) # label(fact_4320_setsum__mono2) # label(axiom). [clausify(380)]. 117.24/117.00 6090 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),D))) # label(fact_228_add__increasing) # label(axiom). [clausify(542)]. 117.24/117.00 6091 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),D),B))) # label(fact_227_add__increasing2) # label(axiom). [clausify(874)]. 117.24/117.00 6092 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),zero_zero(A))) # label(fact_242_add__nonpos__neg) # label(axiom). [clausify(925)]. 117.24/117.01 6093 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),zero_zero(A))) # label(fact_243_add__neg__nonpos) # label(axiom). [clausify(1248)]. 117.24/117.01 6094 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),zero_zero(A))) # label(fact_226_add__nonpos__nonpos) # label(axiom). [clausify(1371)]. 117.24/117.01 6095 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),D))) # label(fact_245_add__strict__increasing) # label(axiom). [clausify(1556)]. 117.24/117.01 6096 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C))) # label(fact_230_add__nonneg__nonneg) # label(axiom). [clausify(2022)]. 117.24/117.01 6097 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),D))) # label(fact_244_add__strict__increasing2) # label(axiom). [clausify(2255)]. 117.24/117.01 6098 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),zero_zero(A))) # label(fact_200_add__neg__neg) # label(axiom). [clausify(2258)]. 117.24/117.01 6099 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C))) # label(fact_246_add__nonneg__pos) # label(axiom). [clausify(2265)]. 117.24/117.01 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f85(A,nat,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co1399186613setsum(A,nat),D),C)),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co1399186613setsum(A,nat),D),B))). [resolve(6086,a,6088,a)]. 117.24/117.01 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),hAPP(A,nat,D,f85(A,nat,D,C,B)))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co1399186613setsum(A,nat),D),C)),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co1399186613setsum(A,nat),D),B))). [resolve(6086,a,6089,a)]. 117.24/117.01 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),zero_zero(nat))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),zero_zero(nat))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),B)),zero_zero(nat))). [resolve(6086,a,6094,a)]. 117.31/117.16 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),C))). [resolve(6086,a,6095,a)]. 117.31/117.16 6100 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C))) # label(fact_247_add__pos__nonneg) # label(axiom). [clausify(2882)]. 117.31/117.16 6101 ordere216010020id_add(rat) # label(arity_Rat_Orat___Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 117.31/117.16 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f85(A,rat,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),D),C)),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),D),B))). [resolve(6101,a,6088,a)]. 117.31/117.16 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(A,rat,D,f85(A,rat,D,C,B)))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),D),C)),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),D),B))). [resolve(6101,a,6089,a)]. 117.31/117.16 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),C))). [resolve(6101,a,6090,a)]. 117.31/117.16 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),C),A))). [resolve(6101,a,6091,a)]. 117.31/117.16 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),zero_zero(rat))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B)),zero_zero(rat))). [resolve(6101,a,6092,a)]. 117.31/117.16 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),zero_zero(rat))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B)),zero_zero(rat))). [resolve(6101,a,6093,a)]. 117.31/117.16 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),zero_zero(rat))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B)),zero_zero(rat))). [resolve(6101,a,6094,a)]. 117.31/117.16 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),B),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),C))). [resolve(6101,a,6095,a)]. 117.31/117.16 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),C)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),C))). [resolve(6101,a,6097,a)]. 117.76/117.54 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),zero_zero(rat))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),zero_zero(rat))) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B)),zero_zero(rat))). [resolve(6101,a,6098,a)]. 117.76/117.54 6102 ordere216010020id_add(code_code_numeral) # label(arity_Code__Numeral_Ocode__numeral___Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 117.76/117.54 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f85(A,code_code_numeral,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),D),C)),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),D),B))). [resolve(6102,a,6088,a)]. 117.76/117.54 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),hAPP(A,code_code_numeral,D,f85(A,code_code_numeral,D,C,B)))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),D),C)),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),D),B))). [resolve(6102,a,6089,a)]. 117.76/117.54 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),A)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),B),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),B),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),C))). [resolve(6102,a,6090,a)]. 117.76/117.54 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),A)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),B),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),B),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),C),A))). [resolve(6102,a,6091,a)]. 117.76/117.54 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),zero_zero(code_code_numeral))) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),B),zero_zero(code_code_numeral))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),B)),zero_zero(code_code_numeral))). [resolve(6102,a,6092,a)]. 118.01/117.78 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),zero_zero(code_code_numeral))) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),B),zero_zero(code_code_numeral))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),B)),zero_zero(code_code_numeral))). [resolve(6102,a,6093,a)]. 118.01/117.78 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),A),zero_zero(code_code_numeral))) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),B),zero_zero(code_code_numeral))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),B)),zero_zero(code_code_numeral))). [resolve(6102,a,6094,a)]. 118.01/117.78 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),zero_zero(code_code_numeral)),A)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),B),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),B),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),C))). [resolve(6102,a,6095,a)]. 118.01/117.78 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),A)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),B),C)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),B),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),C))). [resolve(6102,a,6097,a)]. 118.01/117.78 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),zero_zero(code_code_numeral))) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),B),zero_zero(code_code_numeral))) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),B)),zero_zero(code_code_numeral))). [resolve(6102,a,6098,a)]. 118.01/117.78 6103 ordere216010020id_add(int) # label(arity_Int_Oint___Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 118.01/117.78 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f85(A,int,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),D),C)),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),D),B))). [resolve(6103,a,6088,a)]. 118.01/117.78 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(A,int,D,f85(A,int,D,C,B)))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),D),C)),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),D),B))). [resolve(6103,a,6089,a)]. 118.24/118.06 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),B),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),C))). [resolve(6103,a,6090,a)]. 118.24/118.06 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),B),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),C),A))). [resolve(6103,a,6091,a)]. 118.24/118.06 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)),zero_zero(int))). [resolve(6103,a,6092,a)]. 118.24/118.06 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)),zero_zero(int))). [resolve(6103,a,6093,a)]. 118.24/118.06 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)),zero_zero(int))). [resolve(6103,a,6094,a)]. 118.24/118.06 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_eq(int),B),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),C))). [resolve(6103,a,6095,a)]. 118.24/118.06 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(int),B),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),C))). [resolve(6103,a,6097,a)]. 118.24/118.06 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)),zero_zero(int))). [resolve(6103,a,6098,a)]. 118.24/118.06 6104 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | zero_zero(A) != hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C) | ti(A,B) = zero_zero(A) # label(fact_229_add__nonneg__eq__0__iff) # label(axiom). [clausify(4360)]. 118.24/118.06 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | zero_zero(nat) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),B) | ti(nat,A) = zero_zero(nat). [resolve(6104,a,6086,a)]. 118.24/118.06 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),B)) | zero_zero(rat) != hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B) | ti(rat,A) = zero_zero(rat). [resolve(6104,a,6101,a)]. 118.24/118.06 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),A)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),B)) | zero_zero(code_code_numeral) != hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),B) | ti(code_code_numeral,A) = zero_zero(code_code_numeral). [resolve(6104,a,6102,a)]. 118.24/118.07 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | zero_zero(int) != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B) | ti(int,A) = zero_zero(int). [resolve(6104,a,6103,a)]. 118.24/118.07 6105 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | zero_zero(A) != hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C) | ti(A,C) = zero_zero(A) # label(fact_229_add__nonneg__eq__0__iff) # label(axiom). [clausify(4360)]. 118.24/118.07 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | zero_zero(nat) != hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),B) | ti(nat,B) = zero_zero(nat). [resolve(6105,a,6086,a)]. 118.24/118.07 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),B)) | zero_zero(rat) != hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B) | ti(rat,B) = zero_zero(rat). [resolve(6105,a,6101,a)]. 118.24/118.07 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),A)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),B)) | zero_zero(code_code_numeral) != hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),B) | ti(code_code_numeral,B) = zero_zero(code_code_numeral). [resolve(6105,a,6102,a)]. 118.24/118.07 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | zero_zero(int) != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B) | ti(int,B) = zero_zero(int). [resolve(6105,a,6103,a)]. 118.24/118.07 6106 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C) | ti(A,B) != zero_zero(A) | ti(A,C) != zero_zero(A) # label(fact_229_add__nonneg__eq__0__iff) # label(axiom). [clausify(4360)]. 118.24/118.07 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A),B) | ti(nat,A) != zero_zero(nat) | ti(nat,B) != zero_zero(nat). [resolve(6106,a,6086,a)]. 118.24/118.07 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),B)) | zero_zero(rat) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B) | ti(rat,A) != zero_zero(rat) | ti(rat,B) != zero_zero(rat). [resolve(6106,a,6101,a)]. 118.24/118.07 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),A)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),B)) | zero_zero(code_code_numeral) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),A),B) | ti(code_code_numeral,A) != zero_zero(code_code_numeral) | ti(code_code_numeral,B) != zero_zero(code_code_numeral). [resolve(6106,a,6102,a)]. 118.24/118.07 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | zero_zero(int) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B) | ti(int,A) != zero_zero(int) | ti(int,B) != zero_zero(int). [resolve(6106,a,6103,a)]. 118.44/118.24 6107 ordere216010020id_add(quickcheck_code_int) # label(arity_Quickcheck__Narrowing_Ocode__int___Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 118.44/118.24 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f85(A,quickcheck_code_int,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),D),C)),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),D),B))). [resolve(6107,a,6088,a)]. 118.44/118.24 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(A,quickcheck_code_int,D,f85(A,quickcheck_code_int,D,C,B)))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),D),C)),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),D),B))). [resolve(6107,a,6089,a)]. 118.44/118.24 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),A)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),B),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),B),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),C))). [resolve(6107,a,6090,a)]. 118.44/118.24 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),A)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),B),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),B),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),C),A))). [resolve(6107,a,6091,a)]. 118.44/118.24 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),zero_zero(quickcheck_code_int))) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),B),zero_zero(quickcheck_code_int))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),B)),zero_zero(quickcheck_code_int))). [resolve(6107,a,6092,a)]. 118.44/118.24 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),zero_zero(quickcheck_code_int))) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),B),zero_zero(quickcheck_code_int))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),B)),zero_zero(quickcheck_code_int))). [resolve(6107,a,6093,a)]. 118.44/118.25 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),A),zero_zero(quickcheck_code_int))) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),B),zero_zero(quickcheck_code_int))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),B)),zero_zero(quickcheck_code_int))). [resolve(6107,a,6094,a)]. 118.44/118.25 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),zero_zero(quickcheck_code_int)),A)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),B),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),B),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),C))). [resolve(6107,a,6095,a)]. 118.44/118.25 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),A)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),B),C)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),B),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),C))). [resolve(6107,a,6097,a)]. 118.44/118.25 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),zero_zero(quickcheck_code_int))) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),B),zero_zero(quickcheck_code_int))) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),B)),zero_zero(quickcheck_code_int))). [resolve(6107,a,6098,a)]. 118.44/118.25 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),A)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),B)) | zero_zero(quickcheck_code_int) != hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),B) | ti(quickcheck_code_int,A) = zero_zero(quickcheck_code_int). [resolve(6107,a,6104,a)]. 118.44/118.25 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),A)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),B)) | zero_zero(quickcheck_code_int) != hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),B) | ti(quickcheck_code_int,B) = zero_zero(quickcheck_code_int). [resolve(6107,a,6105,a)]. 118.44/118.29 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),A)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),B)) | zero_zero(quickcheck_code_int) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),A),B) | ti(quickcheck_code_int,A) != zero_zero(quickcheck_code_int) | ti(quickcheck_code_int,B) != zero_zero(quickcheck_code_int). [resolve(6107,a,6106,a)]. 118.44/118.29 6108 ordere216010020id_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 118.44/118.29 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | hBOOL(hAPP(fun(A,bool),bool,hAPP(A,fun(fun(A,bool),bool),member(A),f85(A,real,D,C,B)),hAPP(fun(A,bool),fun(A,bool),hAPP(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),B),C))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),D),C)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),D),B))). [resolve(6108,a,6088,a)]. 118.44/118.29 Derived: -hBOOL(hAPP(fun(A,bool),bool,finite_finite_1(A),B)) | -hBOOL(hAPP(fun(A,bool),bool,hAPP(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),C),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(A,real,D,f85(A,real,D,C,B)))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),D),C)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),D),B))). [resolve(6108,a,6089,a)]. 118.44/118.29 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),C))). [resolve(6108,a,6090,a)]. 118.44/118.29 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),C),A))). [resolve(6108,a,6091,a)]. 118.44/118.29 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B)),zero_zero(real))). [resolve(6108,a,6092,a)]. 118.44/118.29 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B)),zero_zero(real))). [resolve(6108,a,6093,a)]. 118.44/118.29 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B)),zero_zero(real))). [resolve(6108,a,6094,a)]. 118.44/118.29 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),C))). [resolve(6108,a,6095,a)]. 122.25/122.06 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),C))). [resolve(6108,a,6097,a)]. 122.25/122.06 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B)),zero_zero(real))). [resolve(6108,a,6098,a)]. 122.25/122.06 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | zero_zero(real) != hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B) | ti(real,A) = zero_zero(real). [resolve(6108,a,6104,a)]. 122.25/122.06 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | zero_zero(real) != hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B) | ti(real,B) = zero_zero(real). [resolve(6108,a,6105,a)]. 122.25/122.06 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | zero_zero(real) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B) | ti(real,A) != zero_zero(real) | ti(real,B) != zero_zero(real). [resolve(6108,a,6106,a)]. 122.25/122.06 6109 -sgn_if(A) | zero_zero(A) = hAPP(A,A,sgn_sgn(A),zero_zero(A)) # label(fact_2260_sgn0) # label(axiom). [clausify(669)]. 122.25/122.06 6110 sgn_if(rat) # label(arity_Rat_Orat___Groups_Osgn__if) # label(axiom). [assumption]. 122.25/122.06 Derived: zero_zero(rat) = hAPP(rat,rat,sgn_sgn(rat),zero_zero(rat)). [resolve(6109,a,6110,a)]. 122.25/122.06 6111 sgn_if(int) # label(arity_Int_Oint___Groups_Osgn__if) # label(axiom). [assumption]. 122.25/122.06 Derived: zero_zero(int) = hAPP(int,int,sgn_sgn(int),zero_zero(int)). [resolve(6111,a,6109,a)]. 122.25/122.06 6112 sgn_if(real) # label(arity_RealDef_Oreal___Groups_Osgn__if) # label(axiom). [assumption]. 122.25/122.06 Derived: zero_zero(real) = hAPP(real,real,sgn_sgn(real),zero_zero(real)). [resolve(6112,a,6109,a)]. 122.25/122.06 6113 -sgn_if(A) | ti(A,B) = zero_zero(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | hAPP(A,A,sgn_sgn(A),B) = hAPP(A,A,uminus_uminus(A),one_one(A)) # label(fact_2315_sgn__if) # label(axiom). [clausify(3500)]. 122.25/122.06 Derived: ti(rat,A) = zero_zero(rat) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | hAPP(rat,rat,sgn_sgn(rat),A) = hAPP(rat,rat,uminus_uminus(rat),one_one(rat)). [resolve(6113,a,6110,a)]. 122.25/122.06 Derived: ti(int,A) = zero_zero(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | hAPP(int,int,sgn_sgn(int),A) = hAPP(int,int,uminus_uminus(int),one_one(int)). [resolve(6113,a,6111,a)]. 122.25/122.06 Derived: ti(real,A) = zero_zero(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | hAPP(real,real,sgn_sgn(real),A) = hAPP(real,real,uminus_uminus(real),one_one(real)). [resolve(6113,a,6112,a)]. 122.25/122.06 6114 -sgn_if(A) | ti(A,B) = zero_zero(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | one_one(A) = hAPP(A,A,sgn_sgn(A),B) # label(fact_2315_sgn__if) # label(axiom). [clausify(3500)]. 122.25/122.06 Derived: ti(rat,A) = zero_zero(rat) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | one_one(rat) = hAPP(rat,rat,sgn_sgn(rat),A). [resolve(6114,a,6110,a)]. 122.25/122.06 Derived: ti(int,A) = zero_zero(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | one_one(int) = hAPP(int,int,sgn_sgn(int),A). [resolve(6114,a,6111,a)]. 130.45/130.23 Derived: ti(real,A) = zero_zero(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | one_one(real) = hAPP(real,real,sgn_sgn(real),A). [resolve(6114,a,6112,a)]. 130.45/130.23 6115 -sgn_if(A) | ti(A,B) != zero_zero(A) | zero_zero(A) = hAPP(A,A,sgn_sgn(A),B) # label(fact_2315_sgn__if) # label(axiom). [clausify(3500)]. 130.45/130.23 Derived: ti(rat,A) != zero_zero(rat) | zero_zero(rat) = hAPP(rat,rat,sgn_sgn(rat),A). [resolve(6115,a,6110,a)]. 130.45/130.23 Derived: ti(int,A) != zero_zero(int) | zero_zero(int) = hAPP(int,int,sgn_sgn(int),A). [resolve(6115,a,6111,a)]. 130.45/130.23 Derived: ti(real,A) != zero_zero(real) | zero_zero(real) = hAPP(real,real,sgn_sgn(real),A). [resolve(6115,a,6112,a)]. 130.45/130.23 6116 -sgn_if(A) | sgn_sgn(A) = ti(fun(A,A),sgn_sgn(A)) # label(tsy_c_Groups_Osgn__class_Osgn_1_res) # label(axiom). [clausify(3844)]. 130.45/130.23 Derived: sgn_sgn(rat) = ti(fun(rat,rat),sgn_sgn(rat)). [resolve(6116,a,6110,a)]. 130.45/130.23 Derived: sgn_sgn(int) = ti(fun(int,int),sgn_sgn(int)). [resolve(6116,a,6111,a)]. 130.45/130.23 Derived: sgn_sgn(real) = ti(fun(real,real),sgn_sgn(real)). [resolve(6116,a,6112,a)]. 130.45/130.23 6117 -one(A) | ti(A,one_one(A)) = one_one(A) # label(tsy_c_Groups_Oone__class_Oone_res) # label(axiom). [clausify(1373)]. 130.45/130.23 6118 one(real) # label(arity_RealDef_Oreal___Groups_Oone) # label(axiom). [assumption]. 130.45/130.23 6119 one(rat) # label(arity_Rat_Orat___Groups_Oone) # label(axiom). [assumption]. 130.45/130.23 6120 one(quickcheck_code_int) # label(arity_Quickcheck__Narrowing_Ocode__int___Groups_Oone) # label(axiom). [assumption]. 130.45/130.23 Derived: ti(real,one_one(real)) = one_one(real). [resolve(6117,a,6118,a)]. 130.45/130.23 Derived: ti(rat,one_one(rat)) = one_one(rat). [resolve(6117,a,6119,a)]. 130.45/130.23 Derived: ti(quickcheck_code_int,one_one(quickcheck_code_int)) = one_one(quickcheck_code_int). [resolve(6117,a,6120,a)]. 130.45/130.23 6121 one(complex) # label(arity_Complex_Ocomplex___Groups_Oone) # label(axiom). [assumption]. 130.45/130.23 Derived: ti(complex,one_one(complex)) = one_one(complex). [resolve(6121,a,6117,a)]. 130.45/130.23 6122 one(code_code_numeral) # label(arity_Code__Numeral_Ocode__numeral___Groups_Oone) # label(axiom). [assumption]. 130.45/130.23 Derived: ti(code_code_numeral,one_one(code_code_numeral)) = one_one(code_code_numeral). [resolve(6122,a,6117,a)]. 130.45/130.23 6123 one(nat) # label(arity_Nat_Onat___Groups_Oone) # label(axiom). [assumption]. 130.45/130.23 Derived: ti(nat,one_one(nat)) = one_one(nat). [resolve(6123,a,6117,a)]. 130.45/130.23 6124 one(int) # label(arity_Int_Oint___Groups_Oone) # label(axiom). [assumption]. 130.45/130.23 Derived: ti(int,one_one(int)) = one_one(int). [resolve(6124,a,6117,a)]. 130.45/130.23 6125 -linordered_semiring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) # label(fact_477_mult__left__less__imp__less) # label(axiom). [clausify(1022)]. 130.45/130.23 6126 linordered_semiring(quickcheck_code_int) # label(arity_Quickcheck__Narrowing_Ocode__int___Rings_Olinordered__semiring) # label(axiom). [assumption]. 130.45/130.23 6127 linordered_semiring(code_code_numeral) # label(arity_Code__Numeral_Ocode__numeral___Rings_Olinordered__semiring) # label(axiom). [assumption]. 130.45/130.23 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),C))) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),A)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),B),C)). [resolve(6125,a,6126,a)]. 130.45/130.23 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),C))) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),A)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),B),C)). [resolve(6125,a,6127,a)]. 130.83/130.59 6128 linordered_semiring(rat) # label(arity_Rat_Orat___Rings_Olinordered__semiring) # label(axiom). [assumption]. 130.83/130.59 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),C))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),B),C)). [resolve(6128,a,6125,a)]. 130.83/130.59 6129 linordered_semiring(int) # label(arity_Int_Oint___Rings_Olinordered__semiring) # label(axiom). [assumption]. 130.83/130.59 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),C))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)). [resolve(6129,a,6125,a)]. 130.83/130.59 6130 linordered_semiring(nat) # label(arity_Nat_Onat___Rings_Olinordered__semiring) # label(axiom). [assumption]. 130.83/130.59 6131 -linordered_semiring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),D)) # label(fact_479_mult__right__less__imp__less) # label(axiom). [clausify(2391)]. 130.83/130.59 Derived: -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),B)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),C),B))) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),zero_zero(quickcheck_code_int)),B)) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),A),C)). [resolve(6131,a,6126,a)]. 130.83/130.59 Derived: -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),B)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),C),B))) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less_eq(code_code_numeral),zero_zero(code_code_numeral)),B)) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),ord_less(code_code_numeral),A),C)). [resolve(6131,a,6127,a)]. 130.83/130.59 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),B))) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),A),C)). [resolve(6131,a,6128,a)]. 130.83/130.59 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),C)). [resolve(6131,a,6129,a)]. 139.52/139.37 6132 linordered_semiring(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring) # label(axiom). [assumption]. 139.52/139.37 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),C))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),C)). [resolve(6132,a,6125,a)]. 139.52/139.37 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),C)). [resolve(6132,a,6131,a)]. 139.52/139.37 6133 real_algebra(real) # label(arity_RealDef_Oreal___RealVector_Oreal__algebra) # label(axiom). [assumption]. 139.52/139.37 6134 -real_algebra(A) | hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),D)) # label(fact_4719_mult__scaleR__right) # label(axiom). [clausify(161)]. 139.52/139.37 6135 -real_algebra(A) | hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D)) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),C)),D) # label(fact_4721_mult__scaleR__left) # label(axiom). [clausify(2667)]. 139.52/139.37 6136 real_algebra(complex) # label(arity_Complex_Ocomplex___RealVector_Oreal__algebra) # label(axiom). [assumption]. 139.52/139.37 6137 semiring_0(complex) # label(arity_Complex_Ocomplex___Rings_Osemiring__0) # label(axiom). [assumption]. 139.52/139.37 6138 -semiring_0(A) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),hAPP(fun(C,bool),fun(B,A),hAPP(fun(B,fun(fun(C,bool),A)),fun(fun(C,bool),fun(B,A)),combc(B,fun(C,bool),A),hAPP(fun(B,fun(C,A)),fun(B,fun(fun(C,bool),A)),hAPP(fun(fun(C,A),fun(fun(C,bool),A)),fun(fun(B,fun(C,A)),fun(B,fun(fun(C,bool),A))),combb(fun(C,A),fun(fun(C,bool),A),B),big_co1399186613setsum(C,A)),hAPP(fun(C,A),fun(B,fun(C,A)),hAPP(fun(B,fun(fun(C,A),fun(C,A))),fun(fun(C,A),fun(B,fun(C,A))),combc(B,fun(C,A),fun(C,A)),hAPP(fun(B,fun(A,A)),fun(B,fun(fun(C,A),fun(C,A))),hAPP(fun(fun(A,A),fun(fun(C,A),fun(C,A))),fun(fun(B,fun(A,A)),fun(B,fun(fun(C,A),fun(C,A)))),combb(fun(A,A),fun(fun(C,A),fun(C,A)),B),combb(A,A,C)),hAPP(fun(B,A),fun(B,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),combb(A,fun(A,A),B),times_times(A)),D))),E))),F)),V6) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),D),V6)),hAPP(fun(C,bool),A,hAPP(fun(C,A),fun(fun(C,bool),A),big_co1399186613setsum(C,A),E),F)) # label(fact_4240_setsum__product) # label(axiom). [clausify(165)]. 139.52/139.37 6139 -semiring_0(A) | -dvd(A) | -hBOOL(hAPP(A,bool,B,C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),D),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),zero_zero(A)))) | hBOOL(hAPP(A,bool,B,hAPP(A,A,hAPP(A,fun(A,A),times_times(A),D),f57(A,B,D)))) # label(fact_771_unity__coeff__ex) # label(axiom). [clausify(266)]. 139.52/139.37 6140 -semiring_0(A) | -dvd(A) | hBOOL(hAPP(A,bool,B,f58(A,B,C))) | -hBOOL(hAPP(A,bool,B,hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D))) # label(fact_771_unity__coeff__ex) # label(axiom). [clausify(266)]. 139.52/139.37 6141 -semiring_0(A) | -dvd(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),f58(A,C,B)),zero_zero(A)))) | -hBOOL(hAPP(A,bool,C,hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D))) # label(fact_771_unity__coeff__ex) # label(axiom). [clausify(266)]. 139.52/139.37 6142 -semiring_0(A) | -power(A) | zero_zero(A) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),zero_zero(A)),hAPP(nat,nat,suc,B)) # label(fact_1364_power__0__Suc) # label(axiom). [clausify(522)]. 139.66/139.45 Derived: hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co1399186613setsum(A,complex),hAPP(fun(B,bool),fun(A,complex),hAPP(fun(A,fun(fun(B,bool),complex)),fun(fun(B,bool),fun(A,complex)),combc(A,fun(B,bool),complex),hAPP(fun(A,fun(B,complex)),fun(A,fun(fun(B,bool),complex)),hAPP(fun(fun(B,complex),fun(fun(B,bool),complex)),fun(fun(A,fun(B,complex)),fun(A,fun(fun(B,bool),complex))),combb(fun(B,complex),fun(fun(B,bool),complex),A),big_co1399186613setsum(B,complex)),hAPP(fun(B,complex),fun(A,fun(B,complex)),hAPP(fun(A,fun(fun(B,complex),fun(B,complex))),fun(fun(B,complex),fun(A,fun(B,complex))),combc(A,fun(B,complex),fun(B,complex)),hAPP(fun(A,fun(complex,complex)),fun(A,fun(fun(B,complex),fun(B,complex))),hAPP(fun(fun(complex,complex),fun(fun(B,complex),fun(B,complex))),fun(fun(A,fun(complex,complex)),fun(A,fun(fun(B,complex),fun(B,complex)))),combb(fun(complex,complex),fun(fun(B,complex),fun(B,complex)),A),combb(complex,complex,B)),hAPP(fun(A,complex),fun(A,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(A,complex),fun(A,fun(complex,complex))),combb(complex,fun(complex,complex),A),times_times(complex)),C))),D))),E)),F) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(fun(A,bool),complex,hAPP(fun(A,complex),fun(fun(A,bool),complex),big_co1399186613setsum(A,complex),C),F)),hAPP(fun(B,bool),complex,hAPP(fun(B,complex),fun(fun(B,bool),complex),big_co1399186613setsum(B,complex),D),E)). [resolve(6137,a,6138,a)]. 139.66/139.45 Derived: -dvd(complex) | -hBOOL(hAPP(complex,bool,A,B)) | -hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),C),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),B),zero_zero(complex)))) | hBOOL(hAPP(complex,bool,A,hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),C),f57(complex,A,C)))). [resolve(6137,a,6139,a)]. 139.66/139.45 Derived: -dvd(complex) | hBOOL(hAPP(complex,bool,A,f58(complex,A,B))) | -hBOOL(hAPP(complex,bool,A,hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),B),C))). [resolve(6137,a,6140,a)]. 139.66/139.45 Derived: -dvd(complex) | hBOOL(hAPP(complex,bool,hAPP(complex,fun(complex,bool),dvd_dvd(complex),A),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),f58(complex,B,A)),zero_zero(complex)))) | -hBOOL(hAPP(complex,bool,B,hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),C))). [resolve(6137,a,6141,a)]. 139.66/139.45 Derived: -power(complex) | zero_zero(complex) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),zero_zero(complex)),hAPP(nat,nat,suc,A)). [resolve(6137,a,6142,a)]. 139.66/139.45 6143 -semiring_0(A) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),hAPP(fun(B,A),fun(B,A),hAPP(fun(A,A),fun(fun(B,A),fun(B,A)),combb(A,A,B),hAPP(A,fun(A,A),times_times(A),C)),D)),E) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),D),E)) # label(fact_4242_setsum__right__distrib) # label(axiom). [clausify(1042)]. 139.66/139.45 6144 semiring_0(real) # label(arity_RealDef_Oreal___Rings_Osemiring__0) # label(axiom). [assumption]. 139.66/139.45 Derived: hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),hAPP(fun(B,bool),fun(A,real),hAPP(fun(A,fun(fun(B,bool),real)),fun(fun(B,bool),fun(A,real)),combc(A,fun(B,bool),real),hAPP(fun(A,fun(B,real)),fun(A,fun(fun(B,bool),real)),hAPP(fun(fun(B,real),fun(fun(B,bool),real)),fun(fun(A,fun(B,real)),fun(A,fun(fun(B,bool),real))),combb(fun(B,real),fun(fun(B,bool),real),A),big_co1399186613setsum(B,real)),hAPP(fun(B,real),fun(A,fun(B,real)),hAPP(fun(A,fun(fun(B,real),fun(B,real))),fun(fun(B,real),fun(A,fun(B,real))),combc(A,fun(B,real),fun(B,real)),hAPP(fun(A,fun(real,real)),fun(A,fun(fun(B,real),fun(B,real))),hAPP(fun(fun(real,real),fun(fun(B,real),fun(B,real))),fun(fun(A,fun(real,real)),fun(A,fun(fun(B,real),fun(B,real)))),combb(fun(real,real),fun(fun(B,real),fun(B,real)),A),combb(real,real,B)),hAPP(fun(A,real),fun(A,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(A,real),fun(A,fun(real,real))),combb(real,fun(real,real),A),times_times(real)),C))),D))),E)),F) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),C),F)),hAPP(fun(B,bool),real,hAPP(fun(B,real),fun(fun(B,bool),real),big_co1399186613setsum(B,real),D),E)). [resolve(6144,a,6138,a)]. 140.23/140.00 Derived: -dvd(real) | -hBOOL(hAPP(real,bool,A,B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),C),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),zero_zero(real)))) | hBOOL(hAPP(real,bool,A,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),f57(real,A,C)))). [resolve(6144,a,6139,a)]. 140.23/140.00 Derived: -dvd(real) | hBOOL(hAPP(real,bool,A,f58(real,A,B))) | -hBOOL(hAPP(real,bool,A,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),C))). [resolve(6144,a,6140,a)]. 140.23/140.00 Derived: -dvd(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),f58(real,B,A)),zero_zero(real)))) | -hBOOL(hAPP(real,bool,B,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),C))). [resolve(6144,a,6141,a)]. 140.23/140.00 Derived: -power(real) | zero_zero(real) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),zero_zero(real)),hAPP(nat,nat,suc,A)). [resolve(6144,a,6142,a)]. 140.23/140.00 6145 semiring_0(nat) # label(arity_Nat_Onat___Rings_Osemiring__0) # label(axiom). [assumption]. 140.23/140.00 Derived: hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co1399186613setsum(A,nat),hAPP(fun(B,bool),fun(A,nat),hAPP(fun(A,fun(fun(B,bool),nat)),fun(fun(B,bool),fun(A,nat)),combc(A,fun(B,bool),nat),hAPP(fun(A,fun(B,nat)),fun(A,fun(fun(B,bool),nat)),hAPP(fun(fun(B,nat),fun(fun(B,bool),nat)),fun(fun(A,fun(B,nat)),fun(A,fun(fun(B,bool),nat))),combb(fun(B,nat),fun(fun(B,bool),nat),A),big_co1399186613setsum(B,nat)),hAPP(fun(B,nat),fun(A,fun(B,nat)),hAPP(fun(A,fun(fun(B,nat),fun(B,nat))),fun(fun(B,nat),fun(A,fun(B,nat))),combc(A,fun(B,nat),fun(B,nat)),hAPP(fun(A,fun(nat,nat)),fun(A,fun(fun(B,nat),fun(B,nat))),hAPP(fun(fun(nat,nat),fun(fun(B,nat),fun(B,nat))),fun(fun(A,fun(nat,nat)),fun(A,fun(fun(B,nat),fun(B,nat)))),combb(fun(nat,nat),fun(fun(B,nat),fun(B,nat)),A),combb(nat,nat,B)),hAPP(fun(A,nat),fun(A,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(A,nat),fun(A,fun(nat,nat))),combb(nat,fun(nat,nat),A),times_times(nat)),C))),D))),E)),F) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co1399186613setsum(A,nat),C),F)),hAPP(fun(B,bool),nat,hAPP(fun(B,nat),fun(fun(B,bool),nat),big_co1399186613setsum(B,nat),D),E)). [resolve(6145,a,6138,a)]. 140.23/140.00 Derived: -dvd(nat) | -hBOOL(hAPP(nat,bool,A,B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),C),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B),zero_zero(nat)))) | hBOOL(hAPP(nat,bool,A,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),f57(nat,A,C)))). [resolve(6145,a,6139,a)]. 140.23/140.00 Derived: -dvd(nat) | hBOOL(hAPP(nat,bool,A,f58(nat,A,B))) | -hBOOL(hAPP(nat,bool,A,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C))). [resolve(6145,a,6140,a)]. 140.23/140.00 Derived: -dvd(nat) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),f58(nat,B,A)),zero_zero(nat)))) | -hBOOL(hAPP(nat,bool,B,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A),C))). [resolve(6145,a,6141,a)]. 140.23/140.00 Derived: -power(nat) | zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),zero_zero(nat)),hAPP(nat,nat,suc,A)). [resolve(6145,a,6142,a)]. 140.23/140.00 Derived: hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co1399186613setsum(A,nat),hAPP(fun(A,nat),fun(A,nat),hAPP(fun(nat,nat),fun(fun(A,nat),fun(A,nat)),combb(nat,nat,A),hAPP(nat,fun(nat,nat),times_times(nat),B)),C)),D) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co1399186613setsum(A,nat),C),D)). [resolve(6145,a,6143,a)]. 140.23/140.00 6146 semiring_0(code_code_numeral) # label(arity_Code__Numeral_Ocode__numeral___Rings_Osemiring__0) # label(axiom). [assumption]. 140.23/140.00 Derived: hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),hAPP(fun(B,bool),fun(A,code_code_numeral),hAPP(fun(A,fun(fun(B,bool),code_code_numeral)),fun(fun(B,bool),fun(A,code_code_numeral)),combc(A,fun(B,bool),code_code_numeral),hAPP(fun(A,fun(B,code_code_numeral)),fun(A,fun(fun(B,bool),code_code_numeral)),hAPP(fun(fun(B,code_code_numeral),fun(fun(B,bool),code_code_numeral)),fun(fun(A,fun(B,code_code_numeral)),fun(A,fun(fun(B,bool),code_code_numeral))),combb(fun(B,code_code_numeral),fun(fun(B,bool),code_code_numeral),A),big_co1399186613setsum(B,code_code_numeral)),hAPP(fun(B,code_code_numeral),fun(A,fun(B,code_code_numeral)),hAPP(fun(A,fun(fun(B,code_code_numeral),fun(B,code_code_numeral))),fun(fun(B,code_code_numeral),fun(A,fun(B,code_code_numeral))),combc(A,fun(B,code_code_numeral),fun(B,code_code_numeral)),hAPP(fun(A,fun(code_code_numeral,code_code_numeral)),fun(A,fun(fun(B,code_code_numeral),fun(B,code_code_numeral))),hAPP(fun(fun(code_code_numeral,code_code_numeral),fun(fun(B,code_code_numeral),fun(B,code_code_numeral))),fun(fun(A,fun(code_code_numeral,code_code_numeral)),fun(A,fun(fun(B,code_code_numeral),fun(B,code_code_numeral)))),combb(fun(code_code_numeral,code_code_numeral),fun(fun(B,code_code_numeral),fun(B,code_code_numeral)),A),combb(code_code_numeral,code_code_numeral,B)),hAPP(fun(A,code_code_numeral),fun(A,fun(code_code_numeral,code_code_numeral)),hAPP(fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)),fun(fun(A,code_code_numeral),fun(A,fun(code_code_numeral,code_code_numeral))),combb(code_code_numeral,fun(code_code_numeral,code_code_numeral),A),times_times(code_code_numeral)),C))),D))),E)),F) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),C),F)),hAPP(fun(B,bool),code_code_numeral,hAPP(fun(B,code_code_numeral),fun(fun(B,bool),code_code_numeral),big_co1399186613setsum(B,code_code_numeral),D),E)). [resolve(6146,a,6138,a)]. 140.23/140.00 Derived: -dvd(code_code_numeral) | -hBOOL(hAPP(code_code_numeral,bool,A,B)) | -hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),C),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),B),zero_zero(code_code_numeral)))) | hBOOL(hAPP(code_code_numeral,bool,A,hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),C),f57(code_code_numeral,A,C)))). [resolve(6146,a,6139,a)]. 140.23/140.00 Derived: -dvd(code_code_numeral) | hBOOL(hAPP(code_code_numeral,bool,A,f58(code_code_numeral,A,B))) | -hBOOL(hAPP(code_code_numeral,bool,A,hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),B),C))). [resolve(6146,a,6140,a)]. 140.23/140.00 Derived: -dvd(code_code_numeral) | hBOOL(hAPP(code_code_numeral,bool,hAPP(code_code_numeral,fun(code_code_numeral,bool),dvd_dvd(code_code_numeral),A),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),plus_plus(code_code_numeral),f58(code_code_numeral,B,A)),zero_zero(code_code_numeral)))) | -hBOOL(hAPP(code_code_numeral,bool,B,hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),A),C))). [resolve(6146,a,6141,a)]. 140.23/140.00 Derived: -power(code_code_numeral) | zero_zero(code_code_numeral) = hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),zero_zero(code_code_numeral)),hAPP(nat,nat,suc,A)). [resolve(6146,a,6142,a)]. 140.23/140.00 Derived: hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),hAPP(fun(A,code_code_numeral),fun(A,code_code_numeral),hAPP(fun(code_code_numeral,code_code_numeral),fun(fun(A,code_code_numeral),fun(A,code_code_numeral)),combb(code_code_numeral,code_code_numeral,A),hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),B)),C)),D) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),B),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),C),D)). [resolve(6146,a,6143,a)]. 140.34/140.19 6147 semiring_0(rat) # label(arity_Rat_Orat___Rings_Osemiring__0) # label(axiom). [assumption]. 140.34/140.19 Derived: hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),hAPP(fun(B,bool),fun(A,rat),hAPP(fun(A,fun(fun(B,bool),rat)),fun(fun(B,bool),fun(A,rat)),combc(A,fun(B,bool),rat),hAPP(fun(A,fun(B,rat)),fun(A,fun(fun(B,bool),rat)),hAPP(fun(fun(B,rat),fun(fun(B,bool),rat)),fun(fun(A,fun(B,rat)),fun(A,fun(fun(B,bool),rat))),combb(fun(B,rat),fun(fun(B,bool),rat),A),big_co1399186613setsum(B,rat)),hAPP(fun(B,rat),fun(A,fun(B,rat)),hAPP(fun(A,fun(fun(B,rat),fun(B,rat))),fun(fun(B,rat),fun(A,fun(B,rat))),combc(A,fun(B,rat),fun(B,rat)),hAPP(fun(A,fun(rat,rat)),fun(A,fun(fun(B,rat),fun(B,rat))),hAPP(fun(fun(rat,rat),fun(fun(B,rat),fun(B,rat))),fun(fun(A,fun(rat,rat)),fun(A,fun(fun(B,rat),fun(B,rat)))),combb(fun(rat,rat),fun(fun(B,rat),fun(B,rat)),A),combb(rat,rat,B)),hAPP(fun(A,rat),fun(A,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(A,rat),fun(A,fun(rat,rat))),combb(rat,fun(rat,rat),A),times_times(rat)),C))),D))),E)),F) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),C),F)),hAPP(fun(B,bool),rat,hAPP(fun(B,rat),fun(fun(B,bool),rat),big_co1399186613setsum(B,rat),D),E)). [resolve(6147,a,6138,a)]. 140.34/140.19 Derived: -dvd(rat) | -hBOOL(hAPP(rat,bool,A,B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),C),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),B),zero_zero(rat)))) | hBOOL(hAPP(rat,bool,A,hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),C),f57(rat,A,C)))). [resolve(6147,a,6139,a)]. 140.34/140.19 Derived: -dvd(rat) | hBOOL(hAPP(rat,bool,A,f58(rat,A,B))) | -hBOOL(hAPP(rat,bool,A,hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),C))). [resolve(6147,a,6140,a)]. 140.34/140.19 Derived: -dvd(rat) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),dvd_dvd(rat),A),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),f58(rat,B,A)),zero_zero(rat)))) | -hBOOL(hAPP(rat,bool,B,hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),C))). [resolve(6147,a,6141,a)]. 140.34/140.19 Derived: -power(rat) | zero_zero(rat) = hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),zero_zero(rat)),hAPP(nat,nat,suc,A)). [resolve(6147,a,6142,a)]. 140.34/140.19 Derived: hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),hAPP(fun(A,rat),fun(A,rat),hAPP(fun(rat,rat),fun(fun(A,rat),fun(A,rat)),combb(rat,rat,A),hAPP(rat,fun(rat,rat),times_times(rat),B)),C)),D) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),B),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),C),D)). [resolve(6147,a,6143,a)]. 140.34/140.19 6148 -semiring_0(A) | hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),hAPP(A,fun(B,A),hAPP(fun(B,fun(A,A)),fun(A,fun(B,A)),combc(B,A,A),hAPP(fun(B,A),fun(B,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),combb(A,fun(A,A),B),times_times(A)),C)),D)),E) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),C),E)),D) # label(fact_4241_setsum__left__distrib) # label(axiom). [clausify(3330)]. 140.34/140.19 Derived: hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co1399186613setsum(A,nat),hAPP(nat,fun(A,nat),hAPP(fun(A,fun(nat,nat)),fun(nat,fun(A,nat)),combc(A,nat,nat),hAPP(fun(A,nat),fun(A,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(A,nat),fun(A,fun(nat,nat))),combb(nat,fun(nat,nat),A),times_times(nat)),B)),C)),D) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(fun(A,bool),nat,hAPP(fun(A,nat),fun(fun(A,bool),nat),big_co1399186613setsum(A,nat),B),D)),C). [resolve(6148,a,6145,a)]. 140.64/140.46 Derived: hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),hAPP(code_code_numeral,fun(A,code_code_numeral),hAPP(fun(A,fun(code_code_numeral,code_code_numeral)),fun(code_code_numeral,fun(A,code_code_numeral)),combc(A,code_code_numeral,code_code_numeral),hAPP(fun(A,code_code_numeral),fun(A,fun(code_code_numeral,code_code_numeral)),hAPP(fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)),fun(fun(A,code_code_numeral),fun(A,fun(code_code_numeral,code_code_numeral))),combb(code_code_numeral,fun(code_code_numeral,code_code_numeral),A),times_times(code_code_numeral)),B)),C)),D) = hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),times_times(code_code_numeral),hAPP(fun(A,bool),code_code_numeral,hAPP(fun(A,code_code_numeral),fun(fun(A,bool),code_code_numeral),big_co1399186613setsum(A,code_code_numeral),B),D)),C). [resolve(6148,a,6146,a)]. 140.64/140.46 Derived: hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),hAPP(rat,fun(A,rat),hAPP(fun(A,fun(rat,rat)),fun(rat,fun(A,rat)),combc(A,rat,rat),hAPP(fun(A,rat),fun(A,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(A,rat),fun(A,fun(rat,rat))),combb(rat,fun(rat,rat),A),times_times(rat)),B)),C)),D) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),B),D)),C). [resolve(6148,a,6147,a)]. 140.64/140.46 6149 -semiring_0(A) | -power(A) | zero_zero(nat) = hAPP(int,nat,number_number_of(nat),B) | zero_zero(A) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),zero_zero(A)),hAPP(int,nat,number_number_of(nat),B)) # label(fact_11_power__0__left__number__of) # label(axiom). [clausify(3615)]. 140.64/140.46 Derived: -power(complex) | zero_zero(nat) = hAPP(int,nat,number_number_of(nat),A) | zero_zero(complex) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),zero_zero(complex)),hAPP(int,nat,number_number_of(nat),A)). [resolve(6149,a,6137,a)]. 140.64/140.46 Derived: -power(real) | zero_zero(nat) = hAPP(int,nat,number_number_of(nat),A) | zero_zero(real) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),zero_zero(real)),hAPP(int,nat,number_number_of(nat),A)). [resolve(6149,a,6144,a)]. 140.64/140.46 Derived: -power(nat) | zero_zero(nat) = hAPP(int,nat,number_number_of(nat),A) | zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),zero_zero(nat)),hAPP(int,nat,number_number_of(nat),A)). [resolve(6149,a,6145,a)]. 140.64/140.46 Derived: -power(code_code_numeral) | zero_zero(nat) = hAPP(int,nat,number_number_of(nat),A) | zero_zero(code_code_numeral) = hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),zero_zero(code_code_numeral)),hAPP(int,nat,number_number_of(nat),A)). [resolve(6149,a,6146,a)]. 140.64/140.46 Derived: -power(rat) | zero_zero(nat) = hAPP(int,nat,number_number_of(nat),A) | zero_zero(rat) = hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),zero_zero(rat)),hAPP(int,nat,number_number_of(nat),A)). [resolve(6149,a,6147,a)]. 140.64/140.46 6150 -semiring_0(A) | -power(A) | zero_zero(nat) != hAPP(int,nat,number_number_of(nat),B) | one_one(A) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),zero_zero(A)),hAPP(int,nat,number_number_of(nat),B)) # label(fact_11_power__0__left__number__of) # label(axiom). [clausify(3615)]. 140.64/140.46 Derived: -power(complex) | zero_zero(nat) != hAPP(int,nat,number_number_of(nat),A) | one_one(complex) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),zero_zero(complex)),hAPP(int,nat,number_number_of(nat),A)). [resolve(6150,a,6137,a)]. 140.64/140.46 Derived: -power(real) | zero_zero(nat) != hAPP(int,nat,number_number_of(nat),A) | one_one(real) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),zero_zero(real)),hAPP(int,nat,number_number_of(nat),A)). [resolve(6150,a,6144,a)]. 140.73/140.51 Derived: -power(code_code_numeral) | zero_zero(nat) != hAPP(int,nat,number_number_of(nat),A) | one_one(code_code_numeral) = hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),zero_zero(code_code_numeral)),hAPP(int,nat,number_number_of(nat),A)). [resolve(6150,a,6146,a)]. 140.73/140.51 Derived: -power(rat) | zero_zero(nat) != hAPP(int,nat,number_number_of(nat),A) | one_one(rat) = hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),zero_zero(rat)),hAPP(int,nat,number_number_of(nat),A)). [resolve(6150,a,6147,a)]. 140.73/140.51 6151 semiring_0(int) # label(arity_Int_Oint___Rings_Osemiring__0) # label(axiom). [assumption]. 140.73/140.51 Derived: hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),hAPP(fun(B,bool),fun(A,int),hAPP(fun(A,fun(fun(B,bool),int)),fun(fun(B,bool),fun(A,int)),combc(A,fun(B,bool),int),hAPP(fun(A,fun(B,int)),fun(A,fun(fun(B,bool),int)),hAPP(fun(fun(B,int),fun(fun(B,bool),int)),fun(fun(A,fun(B,int)),fun(A,fun(fun(B,bool),int))),combb(fun(B,int),fun(fun(B,bool),int),A),big_co1399186613setsum(B,int)),hAPP(fun(B,int),fun(A,fun(B,int)),hAPP(fun(A,fun(fun(B,int),fun(B,int))),fun(fun(B,int),fun(A,fun(B,int))),combc(A,fun(B,int),fun(B,int)),hAPP(fun(A,fun(int,int)),fun(A,fun(fun(B,int),fun(B,int))),hAPP(fun(fun(int,int),fun(fun(B,int),fun(B,int))),fun(fun(A,fun(int,int)),fun(A,fun(fun(B,int),fun(B,int)))),combb(fun(int,int),fun(fun(B,int),fun(B,int)),A),combb(int,int,B)),hAPP(fun(A,int),fun(A,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(A,int),fun(A,fun(int,int))),combb(int,fun(int,int),A),times_times(int)),C))),D))),E)),F) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),C),F)),hAPP(fun(B,bool),int,hAPP(fun(B,int),fun(fun(B,bool),int),big_co1399186613setsum(B,int),D),E)). [resolve(6151,a,6138,a)]. 140.73/140.51 Derived: -dvd(int) | -hBOOL(hAPP(int,bool,A,B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),C),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),zero_zero(int)))) | hBOOL(hAPP(int,bool,A,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),f57(int,A,C)))). [resolve(6151,a,6139,a)]. 140.73/140.51 Derived: -dvd(int) | hBOOL(hAPP(int,bool,A,f58(int,A,B))) | -hBOOL(hAPP(int,bool,A,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C))). [resolve(6151,a,6140,a)]. 140.73/140.51 Derived: -dvd(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),f58(int,B,A)),zero_zero(int)))) | -hBOOL(hAPP(int,bool,B,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),C))). [resolve(6151,a,6141,a)]. 140.73/140.51 Derived: -power(int) | zero_zero(int) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),zero_zero(int)),hAPP(nat,nat,suc,A)). [resolve(6151,a,6142,a)]. 140.73/140.51 Derived: hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),hAPP(fun(A,int),fun(A,int),hAPP(fun(int,int),fun(fun(A,int),fun(A,int)),combb(int,int,A),hAPP(int,fun(int,int),times_times(int),B)),C)),D) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),C),D)). [resolve(6151,a,6143,a)]. 140.73/140.51 Derived: hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),hAPP(int,fun(A,int),hAPP(fun(A,fun(int,int)),fun(int,fun(A,int)),combc(A,int,int),hAPP(fun(A,int),fun(A,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(A,int),fun(A,fun(int,int))),combb(int,fun(int,int),A),times_times(int)),B)),C)),D) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),B),D)),C). [resolve(6151,a,6148,a)]. 140.73/140.51 Derived: -power(int) | zero_zero(nat) = hAPP(int,nat,number_number_of(nat),A) | zero_zero(int) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),zero_zero(int)),hAPP(int,nat,number_number_of(nat),A)). [resolve(6151,a,6149,a)]. 140.73/140.51 Derived: -power(int) | zero_zero(nat) != hAPP(int,nat,number_number_of(nat),A) | one_one(int) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),zero_zero(int)),hAPP(int,nat,number_number_of(nat),A)). [resolve(6151,a,6150,a)]. 141.45/141.24 6152 -semiring_0(A) | -power(A) | zero_zero(nat) = B | zero_zero(A) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),zero_zero(A)),B) # label(fact_112_power__0__left) # label(axiom). [clausify(4367)]. 141.45/141.24 Derived: -power(complex) | zero_zero(nat) = A | zero_zero(complex) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),zero_zero(complex)),A). [resolve(6152,a,6137,a)]. 141.45/141.24 Derived: -power(real) | zero_zero(nat) = A | zero_zero(real) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),zero_zero(real)),A). [resolve(6152,a,6144,a)]. 141.45/141.24 Derived: -power(nat) | zero_zero(nat) = A | zero_zero(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),zero_zero(nat)),A). [resolve(6152,a,6145,a)]. 141.45/141.24 Derived: -power(code_code_numeral) | zero_zero(nat) = A | zero_zero(code_code_numeral) = hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),zero_zero(code_code_numeral)),A). [resolve(6152,a,6146,a)]. 141.45/141.25 Derived: -power(rat) | zero_zero(nat) = A | zero_zero(rat) = hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),zero_zero(rat)),A). [resolve(6152,a,6147,a)]. 141.45/141.25 Derived: -power(int) | zero_zero(nat) = A | zero_zero(int) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),zero_zero(int)),A). [resolve(6152,a,6151,a)]. 141.45/141.25 6153 -semiring_0(A) | -power(A) | zero_zero(nat) != B | one_one(A) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),zero_zero(A)),B) # label(fact_112_power__0__left) # label(axiom). [clausify(4367)]. 141.45/141.25 Derived: -power(complex) | zero_zero(nat) != A | one_one(complex) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),zero_zero(complex)),A). [resolve(6153,a,6137,a)]. 141.45/141.25 Derived: -power(real) | zero_zero(nat) != A | one_one(real) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),zero_zero(real)),A). [resolve(6153,a,6144,a)]. 141.45/141.25 Derived: -power(nat) | zero_zero(nat) != A | one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),zero_zero(nat)),A). [resolve(6153,a,6145,a)]. 141.45/141.25 Derived: -power(code_code_numeral) | zero_zero(nat) != A | one_one(code_code_numeral) = hAPP(nat,code_code_numeral,hAPP(code_code_numeral,fun(nat,code_code_numeral),power_power(code_code_numeral),zero_zero(code_code_numeral)),A). [resolve(6153,a,6146,a)]. 141.45/141.25 Derived: -power(rat) | zero_zero(nat) != A | one_one(rat) = hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),zero_zero(rat)),A). [resolve(6153,a,6147,a)]. 141.45/141.25 Derived: -power(int) | zero_zero(nat) != A | one_one(int) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),zero_zero(int)),A). [resolve(6153,a,6151,a)]. 141.45/141.25 6154 semiring_0(quickcheck_code_int) # label(arity_Quickcheck__Narrowing_Ocode__int___Rings_Osemiring__0) # label(axiom). [assumption]. 141.45/141.25 Derived: hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),hAPP(fun(B,bool),fun(A,quickcheck_code_int),hAPP(fun(A,fun(fun(B,bool),quickcheck_code_int)),fun(fun(B,bool),fun(A,quickcheck_code_int)),combc(A,fun(B,bool),quickcheck_code_int),hAPP(fun(A,fun(B,quickcheck_code_int)),fun(A,fun(fun(B,bool),quickcheck_code_int)),hAPP(fun(fun(B,quickcheck_code_int),fun(fun(B,bool),quickcheck_code_int)),fun(fun(A,fun(B,quickcheck_code_int)),fun(A,fun(fun(B,bool),quickcheck_code_int))),combb(fun(B,quickcheck_code_int),fun(fun(B,bool),quickcheck_code_int),A),big_co1399186613setsum(B,quickcheck_code_int)),hAPP(fun(B,quickcheck_code_int),fun(A,fun(B,quickcheck_code_int)),hAPP(fun(A,fun(fun(B,quickcheck_code_int),fun(B,quickcheck_code_int))),fun(fun(B,quickcheck_code_int),fun(A,fun(B,quickcheck_code_int))),combc(A,fun(B,quickcheck_code_int),fun(B,quickcheck_code_int)),hAPP(fun(A,fun(quickcheck_code_int,quickcheck_code_int)),fun(A,fun(fun(B,quickcheck_code_int),fun(B,quickcheck_code_int))),hAPP(fun(fun(quickcheck_code_int,quickcheck_code_int),fun(fun(B,quickcheck_code_int),fun(B,quickcheck_code_int))),fun(fun(A,fun(quickcheck_code_int,quickcheck_code_int)),fun(A,fun(fun(B,quickcheck_code_int),fun(B,quickcheck_code_int)))),combb(fun(quickcheck_code_int,quickcheck_code_int),fun(fun(B,quickcheck_code_int),fun(B,quickcheck_code_int)),A),combb(quickcheck_code_int,quickcheck_code_int,B)),hAPP(fun(A,quickcheck_code_int),fun(A,fun(quickcheck_code_int,quickcheck_code_int)),hAPP(fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),fun(fun(A,quickcheck_code_int),fun(A,fun(quickcheck_code_int,quickcheck_code_int))),combb(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),A),times_times(quickcheck_code_int)),C))),D))),E)),F) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),C),F)),hAPP(fun(B,bool),quickcheck_code_int,hAPP(fun(B,quickcheck_code_int),fun(fun(B,bool),quickcheck_code_int),big_co1399186613setsum(B,quickcheck_code_int),D),E)). [resolve(6154,a,6138,a)]. 141.45/141.25 Derived: -dvd(quickcheck_code_int) | -hBOOL(hAPP(quickcheck_code_int,bool,A,B)) | -hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),C),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),B),zero_zero(quickcheck_code_int)))) | hBOOL(hAPP(quickcheck_code_int,bool,A,hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),C),f57(quickcheck_code_int,A,C)))). [resolve(6154,a,6139,a)]. 141.45/141.25 Derived: -dvd(quickcheck_code_int) | hBOOL(hAPP(quickcheck_code_int,bool,A,f58(quickcheck_code_int,A,B))) | -hBOOL(hAPP(quickcheck_code_int,bool,A,hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),B),C))). [resolve(6154,a,6140,a)]. 141.45/141.25 Derived: -dvd(quickcheck_code_int) | hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),dvd_dvd(quickcheck_code_int),A),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),f58(quickcheck_code_int,B,A)),zero_zero(quickcheck_code_int)))) | -hBOOL(hAPP(quickcheck_code_int,bool,B,hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),A),C))). [resolve(6154,a,6141,a)]. 141.45/141.25 Derived: -power(quickcheck_code_int) | zero_zero(quickcheck_code_int) = hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(nat,nat,suc,A)). [resolve(6154,a,6142,a)]. 141.45/141.25 Derived: hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),hAPP(fun(A,quickcheck_code_int),fun(A,quickcheck_code_int),hAPP(fun(quickcheck_code_int,quickcheck_code_int),fun(fun(A,quickcheck_code_int),fun(A,quickcheck_code_int)),combb(quickcheck_code_int,quickcheck_code_int,A),hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),B)),C)),D) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),B),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),C),D)). [resolve(6154,a,6143,a)]. 141.45/141.25 Derived: hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),hAPP(quickcheck_code_int,fun(A,quickcheck_code_int),hAPP(fun(A,fun(quickcheck_code_int,quickcheck_code_int)),fun(quickcheck_code_int,fun(A,quickcheck_code_int)),combc(A,quickcheck_code_int,quickcheck_code_int),hAPP(fun(A,quickcheck_code_int),fun(A,fun(quickcheck_code_int,quickcheck_code_int)),hAPP(fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),fun(fun(A,quickcheck_code_int),fun(A,fun(quickcheck_code_int,quickcheck_code_int))),combb(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),A),times_times(quickcheck_code_int)),B)),C)),D) = hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),hAPP(fun(A,bool),quickcheck_code_int,hAPP(fun(A,quickcheck_code_int),fun(fun(A,bool),quickcheck_code_int),big_co1399186613setsum(A,quickcheck_code_int),B),D)),C). [resolve(6154,a,6148,a)]. 144.75/144.56 Derived: -power(quickcheck_code_int) | zero_zero(nat) = hAPP(int,nat,number_number_of(nat),A) | zero_zero(quickcheck_code_int) = hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(int,nat,number_number_of(nat),A)). [resolve(6154,a,6149,a)]. 144.75/144.56 Derived: -power(quickcheck_code_int) | zero_zero(nat) != hAPP(int,nat,number_number_of(nat),A) | one_one(quickcheck_code_int) = hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),zero_zero(quickcheck_code_int)),hAPP(int,nat,number_number_of(nat),A)). [resolve(6154,a,6150,a)]. 144.75/144.56 Derived: -power(quickcheck_code_int) | zero_zero(nat) = A | zero_zero(quickcheck_code_int) = hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),zero_zero(quickcheck_code_int)),A). [resolve(6154,a,6152,a)]. 144.75/144.56 Derived: -power(quickcheck_code_int) | zero_zero(nat) != A | one_one(quickcheck_code_int) = hAPP(nat,quickcheck_code_int,hAPP(quickcheck_code_int,fun(nat,quickcheck_code_int),power_power(quickcheck_code_int),zero_zero(quickcheck_code_int)),A). [resolve(6154,a,6153,a)]. 144.75/144.56 6155 -inverse(A) | -real_normed_vector(A) | -hBOOL(hAPP(filter(A),bool,hAPP(A,fun(filter(A),bool),hAPP(fun(A,A),fun(A,fun(filter(A),bool)),tendsto(A,A),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,A)),combs(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),inverse_divide(A)),hAPP(A,fun(A,A),hAPP(fun(A,fun(A,A)),fun(A,fun(A,A)),combc(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),minus_minus(A)),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,A),fun(fun(A,A),fun(A,A)),combb(A,A,A),B),hAPP(A,fun(A,A),plus_plus(A),C)))),hAPP(A,A,B,C)))),combi(A))),D),hAPP(A,filter(A),at(A),zero_zero(A)))) | hBOOL(hAPP(filter(A),bool,hAPP(A,fun(filter(A),bool),hAPP(fun(A,A),fun(A,fun(filter(A),bool)),tendsto(A,A),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,A)),combs(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),inverse_divide(A)),hAPP(A,fun(A,A),hAPP(fun(A,fun(A,A)),fun(A,fun(A,A)),combc(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),minus_minus(A)),B)),hAPP(A,A,B,C)))),hAPP(A,fun(A,A),hAPP(fun(A,fun(A,A)),fun(A,fun(A,A)),combc(A,A,A),minus_minus(A)),C))),D),hAPP(A,filter(A),at(A),C))) # label(fact_4632_DERIV__LIM__iff) # label(axiom). [clausify(2645)]. 144.75/144.56 6156 inverse(complex) # label(arity_Complex_Ocomplex___Fields_Oinverse) # label(axiom). [assumption]. 144.75/144.56 6157 inverse(rat) # label(arity_Rat_Orat___Fields_Oinverse) # label(axiom). [assumption]. 144.75/144.56 Derived: -real_normed_vector(complex) | -hBOOL(hAPP(filter(complex),bool,hAPP(complex,fun(filter(complex),bool),hAPP(fun(complex,complex),fun(complex,fun(filter(complex),bool)),tendsto(complex,complex),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,complex)),combs(complex,complex,complex),hAPP(fun(complex,complex),fun(complex,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,fun(complex,complex))),combb(complex,fun(complex,complex),complex),inverse_divide(complex)),hAPP(complex,fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(complex,fun(complex,complex)),combc(complex,complex,complex),hAPP(fun(complex,complex),fun(complex,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,fun(complex,complex))),combb(complex,fun(complex,complex),complex),minus_minus(complex)),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,complex),fun(fun(complex,complex),fun(complex,complex)),combb(complex,complex,complex),A),hAPP(complex,fun(complex,complex),plus_plus(complex),B)))),hAPP(complex,complex,A,B)))),combi(complex))),C),hAPP(complex,filter(complex),at(complex),zero_zero(complex)))) | hBOOL(hAPP(filter(complex),bool,hAPP(complex,fun(filter(complex),bool),hAPP(fun(complex,complex),fun(complex,fun(filter(complex),bool)),tendsto(complex,complex),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,complex)),combs(complex,complex,complex),hAPP(fun(complex,complex),fun(complex,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,fun(complex,complex))),combb(complex,fun(complex,complex),complex),inverse_divide(complex)),hAPP(complex,fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(complex,fun(complex,complex)),combc(complex,complex,complex),hAPP(fun(complex,complex),fun(complex,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,fun(complex,complex))),combb(complex,fun(complex,complex),complex),minus_minus(complex)),A)),hAPP(complex,complex,A,B)))),hAPP(complex,fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(complex,fun(complex,complex)),combc(complex,complex,complex),minus_minus(complex)),B))),C),hAPP(complex,filter(complex),at(complex),B))). [resolve(6155,a,6156,a)]. 144.75/144.56 Derived: -real_normed_vector(rat) | -hBOOL(hAPP(filter(rat),bool,hAPP(rat,fun(filter(rat),bool),hAPP(fun(rat,rat),fun(rat,fun(filter(rat),bool)),tendsto(rat,rat),hAPP(fun(rat,rat),fun(rat,rat),hAPP(fun(rat,fun(rat,rat)),fun(fun(rat,rat),fun(rat,rat)),combs(rat,rat,rat),hAPP(fun(rat,rat),fun(rat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(rat,rat),fun(rat,fun(rat,rat))),combb(rat,fun(rat,rat),rat),inverse_divide(rat)),hAPP(rat,fun(rat,rat),hAPP(fun(rat,fun(rat,rat)),fun(rat,fun(rat,rat)),combc(rat,rat,rat),hAPP(fun(rat,rat),fun(rat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(rat,rat),fun(rat,fun(rat,rat))),combb(rat,fun(rat,rat),rat),minus_minus(rat)),hAPP(fun(rat,rat),fun(rat,rat),hAPP(fun(rat,rat),fun(fun(rat,rat),fun(rat,rat)),combb(rat,rat,rat),A),hAPP(rat,fun(rat,rat),plus_plus(rat),B)))),hAPP(rat,rat,A,B)))),combi(rat))),C),hAPP(rat,filter(rat),at(rat),zero_zero(rat)))) | hBOOL(hAPP(filter(rat),bool,hAPP(rat,fun(filter(rat),bool),hAPP(fun(rat,rat),fun(rat,fun(filter(rat),bool)),tendsto(rat,rat),hAPP(fun(rat,rat),fun(rat,rat),hAPP(fun(rat,fun(rat,rat)),fun(fun(rat,rat),fun(rat,rat)),combs(rat,rat,rat),hAPP(fun(rat,rat),fun(rat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(rat,rat),fun(rat,fun(rat,rat))),combb(rat,fun(rat,rat),rat),inverse_divide(rat)),hAPP(rat,fun(rat,rat),hAPP(fun(rat,fun(rat,rat)),fun(rat,fun(rat,rat)),combc(rat,rat,rat),hAPP(fun(rat,rat),fun(rat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(rat,rat),fun(rat,fun(rat,rat))),combb(rat,fun(rat,rat),rat),minus_minus(rat)),A)),hAPP(rat,rat,A,B)))),hAPP(rat,fun(rat,rat),hAPP(fun(rat,fun(rat,rat)),fun(rat,fun(rat,rat)),combc(rat,rat,rat),minus_minus(rat)),B))),C),hAPP(rat,filter(rat),at(rat),B))). [resolve(6155,a,6157,a)]. 144.75/144.56 6158 -inverse(A) | -real_normed_vector(A) | hBOOL(hAPP(filter(A),bool,hAPP(A,fun(filter(A),bool),hAPP(fun(A,A),fun(A,fun(filter(A),bool)),tendsto(A,A),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,A)),combs(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),inverse_divide(A)),hAPP(A,fun(A,A),hAPP(fun(A,fun(A,A)),fun(A,fun(A,A)),combc(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),minus_minus(A)),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,A),fun(fun(A,A),fun(A,A)),combb(A,A,A),B),hAPP(A,fun(A,A),plus_plus(A),C)))),hAPP(A,A,B,C)))),combi(A))),D),hAPP(A,filter(A),at(A),zero_zero(A)))) | -hBOOL(hAPP(filter(A),bool,hAPP(A,fun(filter(A),bool),hAPP(fun(A,A),fun(A,fun(filter(A),bool)),tendsto(A,A),hAPP(fun(A,A),fun(A,A),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,A)),combs(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),inverse_divide(A)),hAPP(A,fun(A,A),hAPP(fun(A,fun(A,A)),fun(A,fun(A,A)),combc(A,A,A),hAPP(fun(A,A),fun(A,fun(A,A)),hAPP(fun(A,fun(A,A)),fun(fun(A,A),fun(A,fun(A,A))),combb(A,fun(A,A),A),minus_minus(A)),B)),hAPP(A,A,B,C)))),hAPP(A,fun(A,A),hAPP(fun(A,fun(A,A)),fun(A,fun(A,A)),combc(A,A,A),minus_minus(A)),C))),D),hAPP(A,filter(A),at(A),C))) # label(fact_4632_DERIV__LIM__iff) # label(axiom). [clausify(2645)]. 144.75/144.56 Derived: -real_normed_vector(complex) | hBOOL(hAPP(filter(complex),bool,hAPP(complex,fun(filter(complex),bool),hAPP(fun(complex,complex),fun(complex,fun(filter(complex),bool)),tendsto(complex,complex),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,complex)),combs(complex,complex,complex),hAPP(fun(complex,complex),fun(complex,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,fun(complex,complex))),combb(complex,fun(complex,complex),complex),inverse_divide(complex)),hAPP(complex,fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(complex,fun(complex,complex)),combc(complex,complex,complex),hAPP(fun(complex,complex),fun(complex,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,fun(complex,complex))),combb(complex,fun(complex,complex),complex),minus_minus(complex)),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,complex),fun(fun(complex,complex),fun(complex,complex)),combb(complex,complex,complex),A),hAPP(complex,fun(complex,complex),plus_plus(complex),B)))),hAPP(complex,complex,A,B)))),combi(complex))),C),hAPP(complex,filter(complex),at(complex),zero_zero(complex)))) | -hBOOL(hAPP(filter(complex),bool,hAPP(complex,fun(filter(complex),bool),hAPP(fun(complex,complex),fun(complex,fun(filter(complex),bool)),tendsto(complex,complex),hAPP(fun(complex,complex),fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,complex)),combs(complex,complex,complex),hAPP(fun(complex,complex),fun(complex,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,fun(complex,complex))),combb(complex,fun(complex,complex),complex),inverse_divide(complex)),hAPP(complex,fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(complex,fun(complex,complex)),combc(complex,complex,complex),hAPP(fun(complex,complex),fun(complex,fun(complex,complex)),hAPP(fun(complex,fun(complex,complex)),fun(fun(complex,complex),fun(complex,fun(complex,complex))),combb(complex,fun(complex,complex),complex),minus_minus(complex)),A)),hAPP(complex,complex,A,B)))),hAPP(complex,fun(complex,complex),hAPP(fun(complex,fun(complex,complex)),fun(complex,fun(complex,complex)),combc(complex,complex,complex),minus_minus(complex)),B))),C),hAPP(complex,filter(complex),at(complex),B))). [resolve(6158,a,6156,a)]. 144.75/144.56 Derived: -real_normed_vector(rat) | hBOOL(hAPP(filter(rat),bool,hAPP(rat,fun(filter(rat),bool),hAPP(fun(rat,rat),fun(rat,fun(filter(rat),bool)),tendsto(rat,rat),hAPP(fun(rat,rat),fun(rat,rat),hAPP(fun(rat,fun(rat,rat)),fun(fun(rat,rat),fun(rat,rat)),combs(rat,rat,rat),hAPP(fun(rat,rat),fun(rat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(rat,rat),fun(rat,fun(rat,rat))),combb(rat,fun(rat,rat),rat),inverse_divide(rat)),hAPP(rat,fun(rat,rat),hAPP(fun(rat,fun(rat,rat)),fun(rat,fun(rat,rat)),combc(rat,rat,rat),hAPP(fun(rat,rat),fun(rat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(rat,rat),fun(rat,fun(rat,rat))),combb(rat,fun(rat,rat),rat),minus_minus(rat)),hAPP(fun(rat,rat),fun(rat,rat),hAPP(fun(rat,rat),fun(fun(rat,rat),fun(rat,rat)),combb(rat,rat,rat),A),hAPP(rat,fun(rat,rat),plus_plus(rat),B)))),hAPP(rat,rat,A,B)))),combi(rat))),C),hAPP(rat,filter(rat),at(rat),zero_zero(rat)))) | -hBOOL(hAPP(filter(rat),bool,hAPP(rat,fun(filter(rat),bool),hAPP(fun(rat,rat),fun(rat,fun(filter(rat),bool)),tendsto(rat,rat),hAPP(fun(rat,rat),fun(rat,rat),hAPP(fun(rat,fun(rat,rat)),fun(fun(rat,rat),fun(rat,rat)),combs(rat,rat,rat),hAPP(fun(rat,rat),fun(rat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(rat,rat),fun(rat,fun(rat,rat))),combb(rat,fun(rat,rat),rat),inverse_divide(rat)),hAPP(rat,fun(rat,rat),hAPP(fun(rat,fun(rat,rat)),fun(rat,fun(rat,rat)),combc(rat,rat,rat),hAPP(fun(rat,rat),fun(rat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(rat,rat),fun(rat,fun(rat,rat))),combb(rat,fun(rat,rat),rat),minus_minus(rat)),A)),hAPP(rat,rat,A,B)))),hAPP(rat,fun(rat,rat),hAPP(fun(rat,fun(rat,rat)),fun(rat,fun(rat,rat)),combc(rat,rat,rat),minus_minus(rat)),B))),C),hAPP(rat,filter(rat),at(rat),B))). [resolve(6158,a,6157,a)]. 146.11/145.94 6159 -inverse(A) | -real_normed_vector(A) | ti(fun(A,fun(A,A)),inverse_divide(A)) = inverse_divide(A) # label(tsy_c_Fields_Oinverse__class_Odivide_0_res) # label(axiom). [clausify(3530)]. 146.11/145.94 6160 inverse(real) # label(arity_RealDef_Oreal___Fields_Oinverse) # label(axiom). [assumption]. 146.11/145.94 Derived: -real_normed_vector(real) | -hBOOL(hAPP(filter(real),bool,hAPP(real,fun(filter(real),bool),hAPP(fun(real,real),fun(real,fun(filter(real),bool)),tendsto(real,real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real)),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),minus_minus(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),A),hAPP(real,fun(real,real),plus_plus(real),B)))),hAPP(real,real,A,B)))),combi(real))),C),hAPP(real,filter(real),at(real),zero_zero(real)))) | hBOOL(hAPP(filter(real),bool,hAPP(real,fun(filter(real),bool),hAPP(fun(real,real),fun(real,fun(filter(real),bool)),tendsto(real,real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real)),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),minus_minus(real)),A)),hAPP(real,real,A,B)))),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),minus_minus(real)),B))),C),hAPP(real,filter(real),at(real),B))). [resolve(6160,a,6155,a)]. 146.11/145.94 Derived: -real_normed_vector(real) | hBOOL(hAPP(filter(real),bool,hAPP(real,fun(filter(real),bool),hAPP(fun(real,real),fun(real,fun(filter(real),bool)),tendsto(real,real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real)),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),minus_minus(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),A),hAPP(real,fun(real,real),plus_plus(real),B)))),hAPP(real,real,A,B)))),combi(real))),C),hAPP(real,filter(real),at(real),zero_zero(real)))) | -hBOOL(hAPP(filter(real),bool,hAPP(real,fun(filter(real),bool),hAPP(fun(real,real),fun(real,fun(filter(real),bool)),tendsto(real,real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real)),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),minus_minus(real)),A)),hAPP(real,real,A,B)))),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),minus_minus(real)),B))),C),hAPP(real,filter(real),at(real),B))). [resolve(6160,a,6158,a)]. 149.36/149.17 6161 divisi14063676e_zero(complex) # label(arity_Complex_Ocomplex___Fields_Odivision__ring__inverse__zero) # label(axiom). [assumption]. 149.36/149.17 6162 -divisi14063676e_zero(A) | hAPP(A,A,inverse_inverse(A),B) != hAPP(A,A,inverse_inverse(A),C) | ti(A,B) = ti(A,C) # label(fact_2569_inverse__eq__iff__eq) # label(axiom). [clausify(181)]. 149.36/149.17 6163 -divisi14063676e_zero(A) | hAPP(A,A,inverse_inverse(A),B) = hAPP(A,A,inverse_inverse(A),C) | ti(A,B) != ti(A,C) # label(fact_2569_inverse__eq__iff__eq) # label(axiom). [clausify(181)]. 149.36/149.17 6164 -real_div_algebra(A) | -divisi14063676e_zero(A) | hAPP(A,A,inverse_inverse(A),hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),B),C)) = hAPP(A,A,hAPP(real,fun(A,A),scaleR_scaleR(A),hAPP(real,real,inverse_inverse(real),B)),hAPP(A,A,inverse_inverse(A),C)) # label(fact_4727_inverse__scaleR__distrib) # label(axiom). [clausify(1034)]. 149.36/149.17 6165 -divisi14063676e_zero(A) | hAPP(A,A,inverse_inverse(A),B) != hAPP(A,A,inverse_inverse(A),C) | ti(A,B) = ti(A,C) # label(fact_2568_inverse__eq__imp__eq) # label(axiom). [clausify(1384)]. 149.36/149.17 Derived: hAPP(complex,complex,inverse_inverse(complex),A) != hAPP(complex,complex,inverse_inverse(complex),B) | ti(complex,A) = ti(complex,B). [resolve(6161,a,6162,a)]. 149.36/149.17 Derived: hAPP(complex,complex,inverse_inverse(complex),A) = hAPP(complex,complex,inverse_inverse(complex),B) | ti(complex,A) != ti(complex,B). [resolve(6161,a,6163,a)]. 149.36/149.17 Derived: -real_div_algebra(complex) | hAPP(complex,complex,inverse_inverse(complex),hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),A),B)) = hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),hAPP(real,real,inverse_inverse(real),A)),hAPP(complex,complex,inverse_inverse(complex),B)). [resolve(6161,a,6164,b)]. 149.36/149.17 6166 -divisi14063676e_zero(A) | ti(A,B) = zero_zero(A) | one_one(A) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),B) # label(fact_1857_divide__self__if) # label(axiom). [clausify(2173)]. 149.36/149.17 6167 -divisi14063676e_zero(A) | ti(A,B) != zero_zero(A) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),B) # label(fact_1857_divide__self__if) # label(axiom). [clausify(2173)]. 149.36/149.17 Derived: ti(complex,A) != zero_zero(complex) | zero_zero(complex) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),A),A). [resolve(6167,a,6161,a)]. 149.36/149.17 6168 -divisi14063676e_zero(A) | zero_zero(A) = hAPP(A,A,hAPP(A,fun(A,A),inverse_divide(A),B),zero_zero(A)) # label(fact_1825_divide__zero) # label(axiom). [clausify(2365)]. 149.36/149.17 Derived: zero_zero(complex) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),A),zero_zero(complex)). [resolve(6168,a,6161,a)]. 149.36/149.17 6169 divisi14063676e_zero(real) # label(arity_RealDef_Oreal___Fields_Odivision__ring__inverse__zero) # label(axiom). [assumption]. 149.36/149.17 Derived: hAPP(real,real,inverse_inverse(real),A) != hAPP(real,real,inverse_inverse(real),B) | ti(real,A) = ti(real,B). [resolve(6169,a,6162,a)]. 149.64/149.45 Derived: hAPP(real,real,inverse_inverse(real),A) = hAPP(real,real,inverse_inverse(real),B) | ti(real,A) != ti(real,B). [resolve(6169,a,6163,a)]. 149.64/149.45 Derived: -real_div_algebra(real) | hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A),B)) = hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),hAPP(real,real,inverse_inverse(real),A)),hAPP(real,real,inverse_inverse(real),B)). [resolve(6169,a,6164,b)]. 149.64/149.45 Derived: ti(real,A) != zero_zero(real) | zero_zero(real) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),A). [resolve(6169,a,6167,a)]. 149.64/149.45 Derived: zero_zero(real) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A),zero_zero(real)). [resolve(6169,a,6168,a)]. 149.64/149.45 6170 -divisi14063676e_zero(A) | ti(A,B) = hAPP(A,A,inverse_inverse(A),hAPP(A,A,inverse_inverse(A),B)) # label(fact_2570_inverse__inverse__eq) # label(axiom). [clausify(2840)]. 149.64/149.45 Derived: ti(complex,A) = hAPP(complex,complex,inverse_inverse(complex),hAPP(complex,complex,inverse_inverse(complex),A)). [resolve(6170,a,6161,a)]. 149.64/149.45 Derived: ti(real,A) = hAPP(real,real,inverse_inverse(real),hAPP(real,real,inverse_inverse(real),A)). [resolve(6170,a,6169,a)]. 149.64/149.45 6171 -real_div_algebra(A) | -divisi14063676e_zero(A) | hAPP(real,A,of_real(A),hAPP(real,real,inverse_inverse(real),B)) = hAPP(A,A,inverse_inverse(A),hAPP(real,A,of_real(A),B)) # label(fact_2809_of__real__inverse) # label(axiom). [clausify(3202)]. 149.64/149.45 Derived: -real_div_algebra(complex) | hAPP(real,complex,of_real(complex),hAPP(real,real,inverse_inverse(real),A)) = hAPP(complex,complex,inverse_inverse(complex),hAPP(real,complex,of_real(complex),A)). [resolve(6171,b,6161,a)]. 149.64/149.45 Derived: -real_div_algebra(real) | hAPP(real,real,of_real(real),hAPP(real,real,inverse_inverse(real),A)) = hAPP(real,real,inverse_inverse(real),hAPP(real,real,of_real(real),A)). [resolve(6171,b,6169,a)]. 149.64/149.45 6172 divisi14063676e_zero(rat) # label(arity_Rat_Orat___Fields_Odivision__ring__inverse__zero) # label(axiom). [assumption]. 149.64/149.45 Derived: hAPP(rat,rat,inverse_inverse(rat),A) != hAPP(rat,rat,inverse_inverse(rat),B) | ti(rat,A) = ti(rat,B). [resolve(6172,a,6162,a)]. 149.64/149.45 Derived: hAPP(rat,rat,inverse_inverse(rat),A) = hAPP(rat,rat,inverse_inverse(rat),B) | ti(rat,A) != ti(rat,B). [resolve(6172,a,6163,a)]. 149.64/149.45 Derived: -real_div_algebra(rat) | hAPP(rat,rat,inverse_inverse(rat),hAPP(rat,rat,hAPP(real,fun(rat,rat),scaleR_scaleR(rat),A),B)) = hAPP(rat,rat,hAPP(real,fun(rat,rat),scaleR_scaleR(rat),hAPP(real,real,inverse_inverse(real),A)),hAPP(rat,rat,inverse_inverse(rat),B)). [resolve(6172,a,6164,b)]. 149.64/149.45 Derived: ti(rat,A) != zero_zero(rat) | zero_zero(rat) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),A). [resolve(6172,a,6167,a)]. 149.64/149.45 Derived: zero_zero(rat) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),A),zero_zero(rat)). [resolve(6172,a,6168,a)]. 149.64/149.45 Derived: ti(rat,A) = hAPP(rat,rat,inverse_inverse(rat),hAPP(rat,rat,inverse_inverse(rat),A)). [resolve(6172,a,6170,a)]. 149.64/149.45 Derived: -real_div_algebra(rat) | hAPP(real,rat,of_real(rat),hAPP(real,real,inverse_inverse(real),A)) = hAPP(rat,rat,inverse_inverse(rat),hAPP(real,rat,of_real(rat),A)). [resolve(6172,a,6171,b)]. 149.64/149.45 6173 -divisi14063676e_zero(A) | -real_n1866405975lgebra(A) | hAPP(real,real,inverse_inverse(real),hAPP(A,real,norm_norm(A),B)) = hAPP(A,real,norm_norm(A),hAPP(A,A,inverse_inverse(A),B)) # label(fact_2581_norm__inverse) # label(axiom). [clausify(3320)]. 149.64/149.45 Derived: -real_n1866405975lgebra(complex) | hAPP(real,real,inverse_inverse(real),hAPP(complex,real,norm_norm(complex),A)) = hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,inverse_inverse(complex),A)). [resolve(6173,a,6161,a)]. 149.64/149.45 Derived: -real_n1866405975lgebra(real) | hAPP(real,real,inverse_inverse(real),hAPP(real,real,norm_norm(real),A)) = hAPP(real,real,norm_norm(real),hAPP(real,real,inverse_inverse(real),A)). [resolve(6173,a,6169,a)]. 149.64/149.45 Derived: -real_n1866405975lgebra(rat) | hAPP(real,real,inverse_inverse(real),hAPP(rat,real,norm_norm(rat),A)) = hAPP(rat,real,norm_norm(rat),hAPP(rat,rat,inverse_inverse(rat),A)). [resolve(6173,a,6172,a)]. 155.04/154.81 6174 -divisi14063676e_zero(A) | hAPP(A,A,inverse_inverse(A),hAPP(A,A,uminus_uminus(A),B)) = hAPP(A,A,uminus_uminus(A),hAPP(A,A,inverse_inverse(A),B)) # label(fact_2561_inverse__minus__eq) # label(axiom). [clausify(3391)]. 155.04/154.81 Derived: hAPP(complex,complex,inverse_inverse(complex),hAPP(complex,complex,uminus_uminus(complex),A)) = hAPP(complex,complex,uminus_uminus(complex),hAPP(complex,complex,inverse_inverse(complex),A)). [resolve(6174,a,6161,a)]. 155.04/154.81 Derived: hAPP(real,real,inverse_inverse(real),hAPP(real,real,uminus_uminus(real),A)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,inverse_inverse(real),A)). [resolve(6174,a,6169,a)]. 155.04/154.81 Derived: hAPP(rat,rat,inverse_inverse(rat),hAPP(rat,rat,uminus_uminus(rat),A)) = hAPP(rat,rat,uminus_uminus(rat),hAPP(rat,rat,inverse_inverse(rat),A)). [resolve(6174,a,6172,a)]. 155.04/154.81 6175 -divisi14063676e_zero(A) | hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),hAPP(A,A,inverse_inverse(A),B)),C) = hAPP(A,A,inverse_inverse(A),hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C)) # label(fact_2563_power__inverse) # label(axiom). [clausify(4168)]. 155.04/154.81 Derived: hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(complex,complex,inverse_inverse(complex),A)),B) = hAPP(complex,complex,inverse_inverse(complex),hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),B)). [resolve(6175,a,6161,a)]. 155.04/154.81 Derived: hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,inverse_inverse(real),A)),B) = hAPP(real,real,inverse_inverse(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B)). [resolve(6175,a,6169,a)]. 155.04/154.81 Derived: hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),hAPP(rat,rat,inverse_inverse(rat),A)),B) = hAPP(rat,rat,inverse_inverse(rat),hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B)). [resolve(6175,a,6172,a)]. 155.04/154.81 6176 -divisi14063676e_zero(A) | zero_zero(A) != hAPP(A,A,inverse_inverse(A),B) | ti(A,B) = zero_zero(A) # label(fact_2574_inverse__nonzero__iff__nonzero) # label(axiom). [clausify(4620)]. 155.04/154.81 6177 -divisi14063676e_zero(A) | zero_zero(A) = hAPP(A,A,inverse_inverse(A),B) | ti(A,B) != zero_zero(A) # label(fact_2574_inverse__nonzero__iff__nonzero) # label(axiom). [clausify(4620)]. 155.04/154.81 Derived: zero_zero(complex) = hAPP(complex,complex,inverse_inverse(complex),A) | ti(complex,A) != zero_zero(complex). [resolve(6177,a,6161,a)]. 155.04/154.81 Derived: zero_zero(real) = hAPP(real,real,inverse_inverse(real),A) | ti(real,A) != zero_zero(real). [resolve(6177,a,6169,a)]. 155.04/154.81 Derived: zero_zero(rat) = hAPP(rat,rat,inverse_inverse(rat),A) | ti(rat,A) != zero_zero(rat). [resolve(6177,a,6172,a)]. 155.04/154.81 6178 -divisi14063676e_zero(A) | zero_zero(A) = hAPP(A,A,inverse_inverse(A),zero_zero(A)) # label(fact_2573_inverse__zero) # label(axiom). [clausify(4635)]. 155.04/154.81 Derived: zero_zero(complex) = hAPP(complex,complex,inverse_inverse(complex),zero_zero(complex)). [resolve(6178,a,6161,a)]. 155.04/154.81 Derived: zero_zero(real) = hAPP(real,real,inverse_inverse(real),zero_zero(real)). [resolve(6178,a,6169,a)]. 155.04/154.81 Derived: zero_zero(rat) = hAPP(rat,rat,inverse_inverse(rat),zero_zero(rat)). [resolve(6178,a,6172,a)]. 155.04/154.81 6179 comm_semiring(quickcheck_code_int) # label(arity_Quickcheck__Narrowing_Ocode__int___Rings_Ocomm__semiring) # label(axiom). [assumption]. 155.04/154.81 6180 -comm_semiring(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),D)),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D)) # label(fact_380_comm__semiring__class_Odistrib) # label(axiom). [clausify(203)]. 155.04/154.81 6181 comm_semiring(real) # label(arity_RealDef_Oreal___Rings_Ocomm__semiring) # label(axiom). [assumption]. 155.04/154.81 6182 comm_semiring(rat) # label(arity_Rat_Orat___Rings_Ocomm__semiring) # label(axiom). [assumption]. 155.04/154.81 6183 comm_semiring(nat) # label(arity_Nat_Onat___Rings_Ocomm__semiring) # label(axiom). [assumption]. 155.04/154.81 6184 comm_semiring(int) # label(arity_Int_Oint___Rings_Ocomm__semiring) # label(axiom). [assumption]. 155.04/154.81 6185 comm_semiring(complex) # label(arity_Complex_Ocomplex___Rings_Ocomm__semiring) # label(axiom). [assumption]. 158.25/158.02 6186 comm_semiring(code_code_numeral) # label(arity_Code__Numeral_Ocode__numeral___Rings_Ocomm__semiring) # label(axiom). [assumption]. 158.25/158.02 6187 -ring_11004092258visors(A) | ti(A,B) != zero_zero(A) | zero_zero(A) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) # label(fact_5_zero__eq__power2) # label(axiom). [clausify(350)]. 158.25/158.02 6188 ring_11004092258visors(rat) # label(arity_Rat_Orat___Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 158.25/158.02 Derived: ti(rat,A) != zero_zero(rat) | zero_zero(rat) = hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))). [resolve(6187,a,6188,a)]. 158.25/158.02 6189 -ring_11004092258visors(A) | ti(A,B) = zero_zero(A) | zero_zero(A) != hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) # label(fact_5_zero__eq__power2) # label(axiom). [clausify(350)]. 158.25/158.02 Derived: ti(rat,A) = zero_zero(rat) | zero_zero(rat) != hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))). [resolve(6189,a,6188,a)]. 158.25/158.02 6190 -ring_11004092258visors(A) | ti(A,B) != hAPP(A,A,uminus_uminus(A),one_one(A)) | one_one(A) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),B) # label(fact_1965_square__eq__1__iff) # label(axiom). [clausify(721)]. 158.25/158.02 Derived: ti(rat,A) != hAPP(rat,rat,uminus_uminus(rat),one_one(rat)) | one_one(rat) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),A). [resolve(6190,a,6188,a)]. 158.25/158.02 6191 -ring_11004092258visors(A) | ti(A,B) != one_one(A) | one_one(A) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),B) # label(fact_1965_square__eq__1__iff) # label(axiom). [clausify(721)]. 158.25/158.02 Derived: ti(rat,A) != one_one(rat) | one_one(rat) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),A). [resolve(6191,a,6188,a)]. 158.25/158.02 6192 -ring_11004092258visors(A) | ti(A,B) = hAPP(A,A,uminus_uminus(A),one_one(A)) | ti(A,B) = one_one(A) | one_one(A) != hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),B) # label(fact_1965_square__eq__1__iff) # label(axiom). [clausify(721)]. 158.25/158.02 Derived: ti(rat,A) = hAPP(rat,rat,uminus_uminus(rat),one_one(rat)) | ti(rat,A) = one_one(rat) | one_one(rat) != hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),A),A). [resolve(6192,a,6188,a)]. 158.25/158.02 6193 -ring_11004092258visors(A) | one_one(A) != hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) | ti(A,B) = hAPP(A,A,uminus_uminus(A),one_one(A)) | ti(A,B) = one_one(A) # label(fact_2037_power2__eq__1__iff) # label(axiom). [clausify(908)]. 158.25/158.02 Derived: one_one(rat) != hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) | ti(rat,A) = hAPP(rat,rat,uminus_uminus(rat),one_one(rat)) | ti(rat,A) = one_one(rat). [resolve(6193,a,6188,a)]. 158.25/158.02 6194 -ring_11004092258visors(A) | one_one(A) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) | ti(A,B) != hAPP(A,A,uminus_uminus(A),one_one(A)) # label(fact_2037_power2__eq__1__iff) # label(axiom). [clausify(908)]. 158.25/158.02 Derived: one_one(rat) = hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) | ti(rat,A) != hAPP(rat,rat,uminus_uminus(rat),one_one(rat)). [resolve(6194,a,6188,a)]. 158.25/158.02 6195 -ring_11004092258visors(A) | one_one(A) = hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) | ti(A,B) != one_one(A) # label(fact_2037_power2__eq__1__iff) # label(axiom). [clausify(908)]. 158.25/158.02 Derived: one_one(rat) = hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) | ti(rat,A) != one_one(rat). [resolve(6195,a,6188,a)]. 158.25/158.02 6196 ring_11004092258visors(complex) # label(arity_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 158.85/158.68 Derived: ti(complex,A) != zero_zero(complex) | zero_zero(complex) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))). [resolve(6196,a,6187,a)]. 158.85/158.68 Derived: ti(complex,A) = zero_zero(complex) | zero_zero(complex) != hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))). [resolve(6196,a,6189,a)]. 158.85/158.68 Derived: ti(complex,A) != hAPP(complex,complex,uminus_uminus(complex),one_one(complex)) | one_one(complex) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),A). [resolve(6196,a,6190,a)]. 158.85/158.68 Derived: ti(complex,A) != one_one(complex) | one_one(complex) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),A). [resolve(6196,a,6191,a)]. 158.85/158.68 Derived: ti(complex,A) = hAPP(complex,complex,uminus_uminus(complex),one_one(complex)) | ti(complex,A) = one_one(complex) | one_one(complex) != hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),A),A). [resolve(6196,a,6192,a)]. 158.85/158.68 Derived: one_one(complex) != hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) | ti(complex,A) = hAPP(complex,complex,uminus_uminus(complex),one_one(complex)) | ti(complex,A) = one_one(complex). [resolve(6196,a,6193,a)]. 158.85/158.68 Derived: one_one(complex) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) | ti(complex,A) != hAPP(complex,complex,uminus_uminus(complex),one_one(complex)). [resolve(6196,a,6194,a)]. 158.85/158.68 Derived: one_one(complex) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) | ti(complex,A) != one_one(complex). [resolve(6196,a,6195,a)]. 158.85/158.68 6197 -ring_11004092258visors(A) | ti(A,B) = zero_zero(A) | zero_zero(A) != hAPP(nat,A,hAPP(A,fun(nat,A),power_power(A),B),C) # label(fact_168_field__power__not__zero) # label(axiom). [clausify(2907)]. 158.85/158.68 Derived: ti(rat,A) = zero_zero(rat) | zero_zero(rat) != hAPP(nat,rat,hAPP(rat,fun(nat,rat),power_power(rat),A),B). [resolve(6197,a,6188,a)]. 158.85/158.68 Derived: ti(complex,A) = zero_zero(complex) | zero_zero(complex) != hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),A),B). [resolve(6197,a,6196,a)]. 158.85/158.68 6198 ring_11004092258visors(int) # label(arity_Int_Oint___Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 158.85/158.68 Derived: ti(int,A) != zero_zero(int) | zero_zero(int) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))). [resolve(6198,a,6187,a)]. 158.85/158.68 Derived: ti(int,A) = zero_zero(int) | zero_zero(int) != hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))). [resolve(6198,a,6189,a)]. 158.85/158.68 Derived: ti(int,A) != hAPP(int,int,uminus_uminus(int),one_one(int)) | one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),A). [resolve(6198,a,6190,a)]. 158.85/158.68 Derived: ti(int,A) != one_one(int) | one_one(int) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),A). [resolve(6198,a,6191,a)]. 158.85/158.68 Derived: ti(int,A) = hAPP(int,int,uminus_uminus(int),one_one(int)) | ti(int,A) = one_one(int) | one_one(int) != hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A),A). [resolve(6198,a,6192,a)]. 158.85/158.68 Derived: one_one(int) != hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) | ti(int,A) = hAPP(int,int,uminus_uminus(int),one_one(int)) | ti(int,A) = one_one(int). [resolve(6198,a,6193,a)]. 158.85/158.68 Derived: one_one(int) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) | ti(int,A) != hAPP(int,int,uminus_uminus(int),one_one(int)). [resolve(6198,a,6194,a)]. 163.98/163.79 Derived: one_one(int) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) | ti(int,A) != one_one(int). [resolve(6198,a,6195,a)]. 163.98/163.79 Derived: ti(int,A) = zero_zero(int) | zero_zero(int) != hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A),B). [resolve(6198,a,6197,a)]. 163.98/163.79 6199 ring_11004092258visors(real) # label(arity_RealDef_Oreal___Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 163.98/163.79 Derived: ti(real,A) != zero_zero(real) | zero_zero(real) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))). [resolve(6199,a,6187,a)]. 163.98/163.79 Derived: ti(real,A) = zero_zero(real) | zero_zero(real) != hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))). [resolve(6199,a,6189,a)]. 163.98/163.79 Derived: ti(real,A) != hAPP(real,real,uminus_uminus(real),one_one(real)) | one_one(real) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),A). [resolve(6199,a,6190,a)]. 163.98/163.79 Derived: ti(real,A) != one_one(real) | one_one(real) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),A). [resolve(6199,a,6191,a)]. 163.98/163.79 Derived: ti(real,A) = hAPP(real,real,uminus_uminus(real),one_one(real)) | ti(real,A) = one_one(real) | one_one(real) != hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A),A). [resolve(6199,a,6192,a)]. 163.98/163.79 Derived: one_one(real) != hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) | ti(real,A) = hAPP(real,real,uminus_uminus(real),one_one(real)) | ti(real,A) = one_one(real). [resolve(6199,a,6193,a)]. 163.98/163.79 Derived: one_one(real) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) | ti(real,A) != hAPP(real,real,uminus_uminus(real),one_one(real)). [resolve(6199,a,6194,a)]. 163.98/163.79 Derived: one_one(real) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) | ti(real,A) != one_one(real). [resolve(6199,a,6195,a)]. 163.98/163.79 Derived: ti(real,A) = zero_zero(real) | zero_zero(real) != hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A),B). [resolve(6199,a,6197,a)]. 163.98/163.79 6200 t2_space(real) # label(arity_RealDef_Oreal___RealVector_Ot2__space) # label(axiom). [assumption]. 163.98/163.79 6201 -comm_monoid_add(A) | -t2_space(A) | -hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),B),C)) | hBOOL(hAPP(fun(nat,A),bool,summable(A),B)) # label(fact_4120_sums__iff) # label(axiom). [clausify(220)]. 163.98/163.79 6202 -comm_monoid_add(A) | -t2_space(A) | -hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),B),C)) | ti(A,C) = hAPP(fun(nat,A),A,suminf(A),B) # label(fact_4120_sums__iff) # label(axiom). [clausify(220)]. 163.98/163.79 6203 -comm_monoid_add(A) | -t2_space(A) | hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),B),C)) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),B)) | ti(A,C) != hAPP(fun(nat,A),A,suminf(A),B) # label(fact_4120_sums__iff) # label(axiom). [clausify(220)]. 163.98/163.79 6204 -t2_space(A) | -comm_monoid_add(A) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),f114(A,C,B))) | hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),C),hAPP(fun(nat,bool),A,hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),C),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),B)))) # label(fact_4318_series__zero) # label(axiom). [clausify(568)]. 163.98/163.79 6205 -t2_space(A) | -comm_monoid_add(A) | zero_zero(A) != hAPP(nat,A,B,f114(A,B,C)) | hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),B),hAPP(fun(nat,bool),A,hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),B),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),C)))) # label(fact_4318_series__zero) # label(axiom). [clausify(568)]. 163.98/163.79 6206 -t2_space(A) | hBOOL(hAPP(filter(B),bool,trivial_limit(B),C)) | -hBOOL(hAPP(filter(B),bool,hAPP(A,fun(filter(B),bool),hAPP(fun(B,A),fun(A,fun(filter(B),bool)),tendsto(B,A),D),E),C)) | -hBOOL(hAPP(filter(B),bool,hAPP(A,fun(filter(B),bool),hAPP(fun(B,A),fun(A,fun(filter(B),bool)),tendsto(B,A),D),F),C)) | ti(A,F) = ti(A,E) # label(fact_4792_tendsto__unique) # label(axiom). [clausify(1685)]. 164.28/164.03 6207 -comm_monoid_add(A) | -t2_space(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),B)) | hBOOL(hAPP(filter(nat),bool,hAPP(A,fun(filter(nat),bool),hAPP(fun(nat,A),fun(A,fun(filter(nat),bool)),tendsto(nat,A),hAPP(fun(nat,fun(nat,bool)),fun(nat,A),hAPP(fun(fun(nat,bool),A),fun(fun(nat,fun(nat,bool)),fun(nat,A)),combb(fun(nat,bool),A,nat),hAPP(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),B)),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)))),hAPP(fun(nat,A),A,suminf(A),B)),sequentially)) # label(fact_4521_summable__sumr__LIMSEQ__suminf) # label(axiom). [clausify(2839)]. 164.28/164.03 6208 -comm_monoid_add(A) | -t2_space(A) | -hBOOL(hAPP(fun(nat,A),bool,summable(A),B)) | hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),B),hAPP(fun(nat,A),A,suminf(A),B))) # label(fact_4119_summable__sums) # label(axiom). [clausify(3179)]. 164.28/164.03 6209 -t2_space(A) | -comm_monoid_add(A) | -hBOOL(hAPP(A,bool,hAPP(fun(nat,A),fun(A,bool),sums(A),B),C)) | ti(A,C) = hAPP(fun(nat,A),A,suminf(A),B) # label(fact_4116_sums__unique) # label(axiom). [clausify(3627)]. 164.28/164.03 Derived: -comm_monoid_add(real) | -hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),A),B)) | hBOOL(hAPP(fun(nat,real),bool,summable(real),A)). [resolve(6200,a,6201,b)]. 164.28/164.03 Derived: -comm_monoid_add(real) | -hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),A),B)) | ti(real,B) = hAPP(fun(nat,real),real,suminf(real),A). [resolve(6200,a,6202,b)]. 164.28/164.03 Derived: -comm_monoid_add(real) | hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),A),B)) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),A)) | ti(real,B) != hAPP(fun(nat,real),real,suminf(real),A). [resolve(6200,a,6203,b)]. 164.28/164.03 Derived: -comm_monoid_add(real) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),f114(real,B,A))) | hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),B),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),B),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),A)))). [resolve(6200,a,6204,a)]. 164.28/164.03 Derived: -comm_monoid_add(real) | zero_zero(real) != hAPP(nat,real,A,f114(real,A,B)) | hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),A),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),A),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),B)))). [resolve(6200,a,6205,a)]. 164.28/164.03 Derived: hBOOL(hAPP(filter(A),bool,trivial_limit(A),B)) | -hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),C),D),B)) | -hBOOL(hAPP(filter(A),bool,hAPP(real,fun(filter(A),bool),hAPP(fun(A,real),fun(real,fun(filter(A),bool)),tendsto(A,real),C),E),B)) | ti(real,E) = ti(real,D). [resolve(6200,a,6206,a)]. 164.28/164.03 Derived: -comm_monoid_add(real) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),A)) | hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,fun(nat,bool)),fun(nat,real),hAPP(fun(fun(nat,bool),real),fun(fun(nat,fun(nat,bool)),fun(nat,real)),combb(fun(nat,bool),real,nat),hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),A)),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)))),hAPP(fun(nat,real),real,suminf(real),A)),sequentially)). [resolve(6200,a,6207,b)]. 164.28/164.03 Derived: -comm_monoid_add(real) | -hBOOL(hAPP(fun(nat,real),bool,summable(real),A)) | hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),A),hAPP(fun(nat,real),real,suminf(real),A))). [resolve(6200,a,6208,b)]. 164.28/164.03 6210 -comm_monoid_add(A) | -t2_space(A) | zero_zero(A) = hAPP(fun(nat,A),A,suminf(A),hAPP(A,fun(nat,A),combk(A,nat),zero_zero(A))) # label(fact_3995_suminf__zero) # label(axiom). [clausify(4234)]. 167.57/167.36 Derived: -comm_monoid_add(real) | zero_zero(real) = hAPP(fun(nat,real),real,suminf(real),hAPP(real,fun(nat,real),combk(real,nat),zero_zero(real))). [resolve(6210,b,6200,a)]. 167.57/167.36 6211 t2_space(complex) # label(arity_Complex_Ocomplex___RealVector_Ot2__space) # label(axiom). [assumption]. 167.57/167.36 Derived: -comm_monoid_add(complex) | -hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),A),B)) | hBOOL(hAPP(fun(nat,complex),bool,summable(complex),A)). [resolve(6211,a,6201,b)]. 167.57/167.36 Derived: -comm_monoid_add(complex) | -hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),A),B)) | ti(complex,B) = hAPP(fun(nat,complex),complex,suminf(complex),A). [resolve(6211,a,6202,b)]. 167.57/167.36 Derived: -comm_monoid_add(complex) | hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),A),B)) | -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),A)) | ti(complex,B) != hAPP(fun(nat,complex),complex,suminf(complex),A). [resolve(6211,a,6203,b)]. 167.57/167.36 Derived: -comm_monoid_add(complex) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),f114(complex,B,A))) | hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),B),hAPP(fun(nat,bool),complex,hAPP(fun(nat,complex),fun(fun(nat,bool),complex),big_co1399186613setsum(nat,complex),B),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),A)))). [resolve(6211,a,6204,a)]. 167.57/167.36 Derived: -comm_monoid_add(complex) | zero_zero(complex) != hAPP(nat,complex,A,f114(complex,A,B)) | hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),A),hAPP(fun(nat,bool),complex,hAPP(fun(nat,complex),fun(fun(nat,bool),complex),big_co1399186613setsum(nat,complex),A),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),B)))). [resolve(6211,a,6205,a)]. 167.57/167.36 Derived: hBOOL(hAPP(filter(A),bool,trivial_limit(A),B)) | -hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),C),D),B)) | -hBOOL(hAPP(filter(A),bool,hAPP(complex,fun(filter(A),bool),hAPP(fun(A,complex),fun(complex,fun(filter(A),bool)),tendsto(A,complex),C),E),B)) | ti(complex,E) = ti(complex,D). [resolve(6211,a,6206,a)]. 167.57/167.36 Derived: -comm_monoid_add(complex) | -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),A)) | hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),hAPP(fun(nat,fun(nat,bool)),fun(nat,complex),hAPP(fun(fun(nat,bool),complex),fun(fun(nat,fun(nat,bool)),fun(nat,complex)),combb(fun(nat,bool),complex,nat),hAPP(fun(nat,complex),fun(fun(nat,bool),complex),big_co1399186613setsum(nat,complex),A)),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)))),hAPP(fun(nat,complex),complex,suminf(complex),A)),sequentially)). [resolve(6211,a,6207,b)]. 167.57/167.36 Derived: -comm_monoid_add(complex) | -hBOOL(hAPP(fun(nat,complex),bool,summable(complex),A)) | hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),A),hAPP(fun(nat,complex),complex,suminf(complex),A))). [resolve(6211,a,6208,b)]. 167.57/167.36 Derived: -comm_monoid_add(complex) | zero_zero(complex) = hAPP(fun(nat,complex),complex,suminf(complex),hAPP(complex,fun(nat,complex),combk(complex,nat),zero_zero(complex))). [resolve(6211,a,6210,b)]. 167.57/167.36 6212 -ab_semigroup_mult(A) | hAPP(A,A,hAPP(A,fun(A,A),times_times(A),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),times_times(A),B),hAPP(A,A,hAPP(A,fun(A,A),times_times(A),C),D)) # label(fact_361_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom). [clausify(1007)]. 167.57/167.36 6213 ab_semigroup_mult(quickcheck_code_int) # label(arity_Quickcheck__Narrowing_Ocode__int___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 167.57/167.36 6214 ab_semigroup_mult(rat) # label(arity_Rat_Orat___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 176.65/176.41 6215 ab_semigroup_mult(real) # label(arity_RealDef_Oreal___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 176.65/176.41 6216 ab_semigroup_mult(int) # label(arity_Int_Oint___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 176.65/176.41 6217 ab_semigroup_mult(complex) # label(arity_Complex_Ocomplex___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 176.65/176.41 6218 ab_semigroup_mult(code_code_numeral) # label(arity_Code__Numeral_Ocode__numeral___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 176.65/176.41 6219 ab_semigroup_mult(nat) # label(arity_Nat_Onat___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 176.65/176.41 6220 -ab_semigroup_mult(A) | ti(fun(A,fun(A,A)),times_times(A)) = times_times(A) # label(tsy_c_Groups_Otimes__class_Otimes_5_res) # label(axiom). [clausify(3170)]. 176.65/176.41 Derived: ti(fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),times_times(quickcheck_code_int)) = times_times(quickcheck_code_int). [resolve(6220,a,6213,a)]. 176.65/176.41 Derived: ti(fun(rat,fun(rat,rat)),times_times(rat)) = times_times(rat). [resolve(6220,a,6214,a)]. 176.67/176.41 Derived: ti(fun(real,fun(real,real)),times_times(real)) = times_times(real). [resolve(6220,a,6215,a)]. 176.67/176.41 Derived: ti(fun(int,fun(int,int)),times_times(int)) = times_times(int). [resolve(6220,a,6216,a)]. 176.67/176.41 Derived: ti(fun(complex,fun(complex,complex)),times_times(complex)) = times_times(complex). [resolve(6220,a,6217,a)]. 176.67/176.41 Derived: ti(fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)),times_times(code_code_numeral)) = times_times(code_code_numeral). [resolve(6220,a,6218,a)]. 176.67/176.41 Derived: ti(fun(nat,fun(nat,nat)),times_times(nat)) = times_times(nat). [resolve(6220,a,6219,a)]. 176.67/176.41 6221 -ab_semigroup_add(A) | ti(fun(A,fun(A,A)),plus_plus(A)) = plus_plus(A) # label(tsy_c_Groups_Oplus__class_Oplus_2_res) # label(axiom). [clausify(392)]. 176.67/176.41 6222 ab_semigroup_add(code_code_numeral) # label(arity_Code__Numeral_Ocode__numeral___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 176.67/176.41 Derived: ti(fun(code_code_numeral,fun(code_code_numeral,code_code_numeral)),plus_plus(code_code_numeral)) = plus_plus(code_code_numeral). [resolve(6221,a,6222,a)]. 176.67/176.41 6223 ab_semigroup_add(real) # label(arity_RealDef_Oreal___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 176.67/176.41 Derived: ti(fun(real,fun(real,real)),plus_plus(real)) = plus_plus(real). [resolve(6223,a,6221,a)]. 176.67/176.41 6224 ab_semigroup_add(quickcheck_code_int) # label(arity_Quickcheck__Narrowing_Ocode__int___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 176.67/176.41 Derived: ti(fun(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int)),plus_plus(quickcheck_code_int)) = plus_plus(quickcheck_code_int). [resolve(6224,a,6221,a)]. 176.67/176.41 6225 ab_semigroup_add(complex) # label(arity_Complex_Ocomplex___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 176.67/176.41 Derived: ti(fun(complex,fun(complex,complex)),plus_plus(complex)) = plus_plus(complex). [resolve(6225,a,6221,a)]. 176.67/176.41 6226 -ab_semigroup_add(A) | hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),D) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),C),D)) # label(fact_297_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom). [clausify(3188)]. 176.67/176.41 6227 ab_semigroup_add(nat) # label(arity_Nat_Onat___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 176.67/176.41 Derived: ti(fun(nat,fun(nat,nat)),plus_plus(nat)) = plus_plus(nat). [resolve(6227,a,6221,a)]. 176.67/176.41 6228 ab_semigroup_add(rat) # label(arity_Rat_Orat___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 176.67/176.41 Derived: ti(fun(rat,fun(rat,rat)),plus_plus(rat)) = plus_plus(rat). [resolve(6228,a,6221,a)]. 176.67/176.41 6229 ab_semigroup_add(int) # label(arity_Int_Oint___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 176.67/176.41 Derived: ti(fun(int,fun(int,int)),plus_plus(int)) = plus_plus(int). [resolve(6229,a,6221,a)]. 176.67/176.41 6230 ordere142940540dd_abs(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__group__add__abs) # label(axiom). [assumption]. 176.67/176.41 6231 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),hAPP(fun(B,A),fun(B,A),hAPP(fun(A,A),fun(fun(B,A),fun(B,A)),combb(A,A,B),abs_abs(A)),C)),D))) # label(fact_4193_setsum__abs__ge__zero) # label(axiom). [clausify(245)]. 176.86/176.68 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),hAPP(fun(A,real),fun(A,real),hAPP(fun(real,real),fun(fun(A,real),fun(A,real)),combb(real,real,A),abs_abs(real)),B)),C))). [resolve(6230,a,6231,a)]. 176.86/176.68 6232 ordere142940540dd_abs(int) # label(arity_Int_Oint___Groups_Oordered__ab__group__add__abs) # label(axiom). [assumption]. 176.86/176.68 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),hAPP(fun(A,int),fun(A,int),hAPP(fun(int,int),fun(fun(A,int),fun(A,int)),combb(int,int,A),abs_abs(int)),B)),C))). [resolve(6232,a,6231,a)]. 176.86/176.68 6233 -ordere142940540dd_abs(A) | ti(A,B) != zero_zero(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,abs_abs(A),B)),zero_zero(A))) # label(fact_987_abs__le__zero__iff) # label(axiom). [clausify(621)]. 176.86/176.68 Derived: ti(real,A) != zero_zero(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),A)),zero_zero(real))). [resolve(6233,a,6230,a)]. 176.86/176.68 Derived: ti(int,A) != zero_zero(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,abs_abs(int),A)),zero_zero(int))). [resolve(6233,a,6232,a)]. 176.86/176.68 6234 -ordere142940540dd_abs(A) | ti(A,B) = zero_zero(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,abs_abs(A),B)),zero_zero(A))) # label(fact_987_abs__le__zero__iff) # label(axiom). [clausify(621)]. 176.86/176.68 Derived: ti(real,A) = zero_zero(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),A)),zero_zero(real))). [resolve(6234,a,6230,a)]. 176.86/176.68 Derived: ti(int,A) = zero_zero(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,abs_abs(int),A)),zero_zero(int))). [resolve(6234,a,6232,a)]. 176.86/176.68 6235 -ordere142940540dd_abs(A) | hAPP(A,A,abs_abs(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,abs_abs(A),B)),hAPP(A,A,abs_abs(A),C))) = hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,abs_abs(A),B)),hAPP(A,A,abs_abs(A),C)) # label(fact_975_abs__add__abs) # label(axiom). [clausify(923)]. 176.86/176.68 Derived: hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,abs_abs(real),A)),hAPP(real,real,abs_abs(real),B))) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,abs_abs(real),A)),hAPP(real,real,abs_abs(real),B)). [resolve(6235,a,6230,a)]. 176.86/176.68 Derived: hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,abs_abs(int),A)),hAPP(int,int,abs_abs(int),B))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,abs_abs(int),A)),hAPP(int,int,abs_abs(int),B)). [resolve(6235,a,6232,a)]. 176.86/176.68 6236 -ordere142940540dd_abs(A) | abs_abs(A) = ti(fun(A,A),abs_abs(A)) # label(tsy_c_Groups_Oabs__class_Oabs_1_res) # label(axiom). [clausify(943)]. 176.86/176.68 Derived: abs_abs(real) = ti(fun(real,real),abs_abs(real)). [resolve(6236,a,6230,a)]. 176.86/176.68 Derived: abs_abs(int) = ti(fun(int,int),abs_abs(int)). [resolve(6236,a,6232,a)]. 176.86/176.68 6237 -ordere142940540dd_abs(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,abs_abs(A),B)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_972_abs__le__D1) # label(axiom). [clausify(1020)]. 176.86/176.68 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),A)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)). [resolve(6237,a,6230,a)]. 176.86/176.68 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,abs_abs(int),A)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(6237,a,6232,a)]. 176.86/176.68 6238 -ordere142940540dd_abs(A) | ti(A,B) != zero_zero(A) | zero_zero(A) = hAPP(A,A,abs_abs(A),B) # label(fact_969_abs__eq__0) # label(axiom). [clausify(1903)]. 177.07/176.83 Derived: ti(real,A) != zero_zero(real) | zero_zero(real) = hAPP(real,real,abs_abs(real),A). [resolve(6238,a,6230,a)]. 177.07/176.83 Derived: ti(int,A) != zero_zero(int) | zero_zero(int) = hAPP(int,int,abs_abs(int),A). [resolve(6238,a,6232,a)]. 177.07/176.83 6239 -ordere142940540dd_abs(A) | ti(A,B) = zero_zero(A) | zero_zero(A) != hAPP(A,A,abs_abs(A),B) # label(fact_969_abs__eq__0) # label(axiom). [clausify(1903)]. 177.07/176.83 Derived: ti(real,A) = zero_zero(real) | zero_zero(real) != hAPP(real,real,abs_abs(real),A). [resolve(6239,a,6230,a)]. 177.07/176.83 Derived: ti(int,A) = zero_zero(int) | zero_zero(int) != hAPP(int,int,abs_abs(int),A). [resolve(6239,a,6232,a)]. 177.07/176.83 6240 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,abs_abs(A),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),C),D))),hAPP(fun(B,bool),A,hAPP(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),hAPP(fun(B,A),fun(B,A),hAPP(fun(A,A),fun(fun(B,A),fun(B,A)),combb(A,A,B),abs_abs(A)),C)),D))) # label(fact_4192_setsum__abs) # label(axiom). [clausify(1976)]. 177.07/176.83 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),B),C))),hAPP(fun(A,bool),real,hAPP(fun(A,real),fun(fun(A,bool),real),big_co1399186613setsum(A,real),hAPP(fun(A,real),fun(A,real),hAPP(fun(real,real),fun(fun(A,real),fun(A,real)),combb(real,real,A),abs_abs(real)),B)),C))). [resolve(6240,a,6230,a)]. 177.07/176.83 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,abs_abs(int),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),B),C))),hAPP(fun(A,bool),int,hAPP(fun(A,int),fun(fun(A,bool),int),big_co1399186613setsum(A,int),hAPP(fun(A,int),fun(A,int),hAPP(fun(int,int),fun(fun(A,int),fun(A,int)),combb(int,int,A),abs_abs(int)),B)),C))). [resolve(6240,a,6232,a)]. 177.07/176.83 6241 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,abs_abs(A),B))) # label(fact_971_abs__ge__self) # label(axiom). [clausify(2009)]. 177.07/176.83 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),hAPP(real,real,abs_abs(real),A))). [resolve(6241,a,6230,a)]. 177.07/176.83 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(int,int,abs_abs(int),A))). [resolve(6241,a,6232,a)]. 177.07/176.83 6242 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,abs_abs(A),B)),hAPP(A,A,abs_abs(A),C))),hAPP(A,A,abs_abs(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),C),B)))) # label(fact_994_abs__triangle__ineq2__sym) # label(axiom). [clausify(2338)]. 177.07/176.83 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,abs_abs(real),A)),hAPP(real,real,abs_abs(real),B))),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),A)))). [resolve(6242,a,6230,a)]. 177.07/176.83 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,abs_abs(int),A)),hAPP(int,int,abs_abs(int),B))),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),A)))). [resolve(6242,a,6232,a)]. 177.07/176.83 6243 -ordere142940540dd_abs(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,abs_abs(A),B))) | ti(A,B) != zero_zero(A) # label(fact_990_zero__less__abs__iff) # label(axiom). [clausify(2345)]. 177.07/176.83 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,abs_abs(real),A))) | ti(real,A) != zero_zero(real). [resolve(6243,a,6230,a)]. 177.07/176.83 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,abs_abs(int),A))) | ti(int,A) != zero_zero(int). [resolve(6243,a,6232,a)]. 177.07/176.83 6244 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,abs_abs(A),B))) | ti(A,B) = zero_zero(A) # label(fact_990_zero__less__abs__iff) # label(axiom). [clausify(2345)]. 177.24/176.99 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,abs_abs(real),A))) | ti(real,A) = zero_zero(real). [resolve(6244,a,6230,a)]. 177.24/176.99 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,abs_abs(int),A))) | ti(int,A) = zero_zero(int). [resolve(6244,a,6232,a)]. 177.24/176.99 6245 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,abs_abs(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,abs_abs(A),B)),hAPP(A,A,abs_abs(A),C)))),hAPP(A,A,abs_abs(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C)))) # label(fact_996_abs__triangle__ineq3) # label(axiom). [clausify(2347)]. 177.24/176.99 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,abs_abs(real),A)),hAPP(real,real,abs_abs(real),B)))),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B)))). [resolve(6245,a,6230,a)]. 177.24/176.99 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,abs_abs(int),A)),hAPP(int,int,abs_abs(int),B)))),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B)))). [resolve(6245,a,6232,a)]. 177.24/176.99 6246 -ordere142940540dd_abs(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | ti(A,B) = hAPP(A,A,abs_abs(A),B) # label(fact_988_abs__of__nonneg) # label(axiom). [clausify(2512)]. 177.24/176.99 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | ti(real,A) = hAPP(real,real,abs_abs(real),A). [resolve(6246,a,6230,a)]. 177.24/176.99 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | ti(int,A) = hAPP(int,int,abs_abs(int),A). [resolve(6246,a,6232,a)]. 177.24/176.99 6247 -ordere142940540dd_abs(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,uminus_uminus(A),B)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,abs_abs(A),B)),C)) # label(fact_1978_abs__leI) # label(axiom). [clausify(2612)]. 177.24/176.99 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),hAPP(real,real,uminus_uminus(real),A)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),A)),B)). [resolve(6247,a,6230,a)]. 177.24/176.99 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,uminus_uminus(int),A)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,abs_abs(int),A)),B)). [resolve(6247,a,6232,a)]. 177.24/176.99 6248 -ordere142940540dd_abs(A) | zero_zero(A) = hAPP(A,A,abs_abs(A),zero_zero(A)) # label(fact_970_abs__zero) # label(axiom). [clausify(2646)]. 177.24/176.99 Derived: zero_zero(real) = hAPP(real,real,abs_abs(real),zero_zero(real)). [resolve(6248,a,6230,a)]. 177.24/176.99 Derived: zero_zero(int) = hAPP(int,int,abs_abs(int),zero_zero(int)). [resolve(6248,a,6232,a)]. 177.24/176.99 6249 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,abs_abs(A),B))) # label(fact_986_abs__ge__zero) # label(axiom). [clausify(2699)]. 177.24/176.99 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,abs_abs(real),A))). [resolve(6249,a,6230,a)]. 177.24/176.99 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,abs_abs(int),A))). [resolve(6249,a,6232,a)]. 177.24/176.99 6250 -ordere142940540dd_abs(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,abs_abs(A),B)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_1977_abs__le__iff) # label(axiom). [clausify(2781)]. 177.24/176.99 6251 -ordere142940540dd_abs(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,abs_abs(A),B)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,uminus_uminus(A),B)),C)) # label(fact_1977_abs__le__iff) # label(axiom). [clausify(2781)]. 177.67/177.44 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),A)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),A)),B)). [resolve(6251,a,6230,a)]. 177.67/177.44 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,abs_abs(int),A)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,uminus_uminus(int),A)),B)). [resolve(6251,a,6232,a)]. 177.67/177.44 6252 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,abs_abs(A),B)),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,uminus_uminus(A),B)),C)) # label(fact_1977_abs__le__iff) # label(axiom). [clausify(2781)]. 177.67/177.44 6253 -ordere142940540dd_abs(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | ti(A,B) = hAPP(A,A,abs_abs(A),B) # label(fact_989_abs__of__pos) # label(axiom). [clausify(2980)]. 177.67/177.44 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | ti(real,A) = hAPP(real,real,abs_abs(real),A). [resolve(6253,a,6230,a)]. 177.67/177.44 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | ti(int,A) = hAPP(int,int,abs_abs(int),A). [resolve(6253,a,6232,a)]. 177.67/177.44 6254 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,abs_abs(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C)),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),D),E)))),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,abs_abs(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),D))),hAPP(A,A,abs_abs(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),C),E))))) # label(fact_1014_abs__diff__triangle__ineq) # label(axiom). [clausify(3157)]. 177.67/177.44 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),C),D)))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),C))),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),D))))). [resolve(6254,a,6230,a)]. 177.67/177.44 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),C),D)))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),C))),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),D))))). [resolve(6254,a,6232,a)]. 177.67/177.44 6255 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,uminus_uminus(A),B)),hAPP(A,A,abs_abs(A),B))) # label(fact_1976_abs__ge__minus__self) # label(axiom). [clausify(3309)]. 177.67/177.44 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),A)),hAPP(real,real,abs_abs(real),A))). [resolve(6255,a,6230,a)]. 177.67/177.44 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,uminus_uminus(int),A)),hAPP(int,int,abs_abs(int),A))). [resolve(6255,a,6232,a)]. 177.67/177.44 6256 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,uminus_uminus(A),hAPP(A,A,abs_abs(A),B))),zero_zero(A))) # label(fact_2001_abs__minus__le__zero) # label(axiom). [clausify(3489)]. 177.67/177.44 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,abs_abs(real),A))),zero_zero(real))). [resolve(6256,a,6230,a)]. 177.67/177.44 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,uminus_uminus(int),hAPP(int,int,abs_abs(int),A))),zero_zero(int))). [resolve(6256,a,6232,a)]. 178.18/177.97 6257 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),hAPP(A,A,abs_abs(A),B)),hAPP(A,A,abs_abs(A),C))),hAPP(A,A,abs_abs(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C)))) # label(fact_995_abs__triangle__ineq2) # label(axiom). [clausify(3558)]. 178.18/177.97 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,abs_abs(real),A)),hAPP(real,real,abs_abs(real),B))),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B)))). [resolve(6257,a,6230,a)]. 178.18/177.97 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,abs_abs(int),A)),hAPP(int,int,abs_abs(int),B))),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B)))). [resolve(6257,a,6232,a)]. 178.18/177.97 6258 -ordere142940540dd_abs(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | hAPP(A,A,abs_abs(A),B) = hAPP(A,A,uminus_uminus(A),B) # label(fact_2000_abs__of__nonpos) # label(axiom). [clausify(3567)]. 178.18/177.97 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))) | hAPP(real,real,abs_abs(real),A) = hAPP(real,real,uminus_uminus(real),A). [resolve(6258,a,6230,a)]. 178.18/177.97 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))) | hAPP(int,int,abs_abs(int),A) = hAPP(int,int,uminus_uminus(int),A). [resolve(6258,a,6232,a)]. 178.18/177.97 6259 -ordere142940540dd_abs(A) | hAPP(A,A,abs_abs(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C)) = hAPP(A,A,abs_abs(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),C),B)) # label(fact_977_abs__minus__commute) # label(axiom). [clausify(3599)]. 178.18/177.97 Derived: hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B)) = hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),A)). [resolve(6259,a,6230,a)]. 178.18/177.97 Derived: hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B)) = hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),A)). [resolve(6259,a,6232,a)]. 178.18/177.97 6260 -ordere142940540dd_abs(A) | hAPP(A,A,abs_abs(A),hAPP(A,A,uminus_uminus(A),B)) = hAPP(A,A,abs_abs(A),B) # label(fact_1943_abs__minus__cancel) # label(axiom). [clausify(3758)]. 178.18/177.97 Derived: hAPP(real,real,abs_abs(real),hAPP(real,real,uminus_uminus(real),A)) = hAPP(real,real,abs_abs(real),A). [resolve(6260,a,6230,a)]. 178.18/177.97 Derived: hAPP(int,int,abs_abs(int),hAPP(int,int,uminus_uminus(int),A)) = hAPP(int,int,abs_abs(int),A). [resolve(6260,a,6232,a)]. 178.18/177.97 6261 ordere142940540dd_abs(rat) # label(arity_Rat_Orat___Groups_Oordered__ab__group__add__abs) # label(axiom). [assumption]. 178.18/177.97 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),hAPP(fun(A,rat),fun(A,rat),hAPP(fun(rat,rat),fun(fun(A,rat),fun(A,rat)),combb(rat,rat,A),abs_abs(rat)),B)),C))). [resolve(6261,a,6231,a)]. 178.18/177.97 Derived: ti(rat,A) != zero_zero(rat) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,abs_abs(rat),A)),zero_zero(rat))). [resolve(6261,a,6233,a)]. 178.18/177.97 Derived: ti(rat,A) = zero_zero(rat) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,abs_abs(rat),A)),zero_zero(rat))). [resolve(6261,a,6234,a)]. 178.18/177.97 Derived: hAPP(rat,rat,abs_abs(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,abs_abs(rat),A)),hAPP(rat,rat,abs_abs(rat),B))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,abs_abs(rat),A)),hAPP(rat,rat,abs_abs(rat),B)). [resolve(6261,a,6235,a)]. 178.18/177.97 Derived: abs_abs(rat) = ti(fun(rat,rat),abs_abs(rat)). [resolve(6261,a,6236,a)]. 178.18/177.97 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,abs_abs(rat),A)),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)). [resolve(6261,a,6237,a)]. 178.18/177.98 Derived: ti(rat,A) != zero_zero(rat) | zero_zero(rat) = hAPP(rat,rat,abs_abs(rat),A). [resolve(6261,a,6238,a)]. 178.18/177.98 Derived: ti(rat,A) = zero_zero(rat) | zero_zero(rat) != hAPP(rat,rat,abs_abs(rat),A). [resolve(6261,a,6239,a)]. 178.18/177.98 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,abs_abs(rat),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),B),C))),hAPP(fun(A,bool),rat,hAPP(fun(A,rat),fun(fun(A,bool),rat),big_co1399186613setsum(A,rat),hAPP(fun(A,rat),fun(A,rat),hAPP(fun(rat,rat),fun(fun(A,rat),fun(A,rat)),combb(rat,rat,A),abs_abs(rat)),B)),C))). [resolve(6261,a,6240,a)]. 178.18/177.98 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),hAPP(rat,rat,abs_abs(rat),A))). [resolve(6261,a,6241,a)]. 178.18/177.98 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(rat,rat,abs_abs(rat),A)),hAPP(rat,rat,abs_abs(rat),B))),hAPP(rat,rat,abs_abs(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),B),A)))). [resolve(6261,a,6242,a)]. 178.18/177.98 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(rat,rat,abs_abs(rat),A))) | ti(rat,A) != zero_zero(rat). [resolve(6261,a,6243,a)]. 178.18/177.98 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(rat,rat,abs_abs(rat),A))) | ti(rat,A) = zero_zero(rat). [resolve(6261,a,6244,a)]. 178.18/177.98 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,abs_abs(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(rat,rat,abs_abs(rat),A)),hAPP(rat,rat,abs_abs(rat),B)))),hAPP(rat,rat,abs_abs(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B)))). [resolve(6261,a,6245,a)]. 178.18/177.98 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),A)) | ti(rat,A) = hAPP(rat,rat,abs_abs(rat),A). [resolve(6261,a,6246,a)]. 178.18/177.98 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),B)) | -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,uminus_uminus(rat),A)),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,abs_abs(rat),A)),B)). [resolve(6261,a,6247,a)]. 178.18/177.98 Derived: zero_zero(rat) = hAPP(rat,rat,abs_abs(rat),zero_zero(rat)). [resolve(6261,a,6248,a)]. 178.18/177.98 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(rat,rat,abs_abs(rat),A))). [resolve(6261,a,6249,a)]. 178.18/177.98 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,abs_abs(rat),A)),B)) | hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,uminus_uminus(rat),A)),B)). [resolve(6261,a,6251,a)]. 178.18/177.98 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A)) | ti(rat,A) = hAPP(rat,rat,abs_abs(rat),A). [resolve(6261,a,6253,a)]. 178.18/177.98 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,abs_abs(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B)),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),C),D)))),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,abs_abs(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),C))),hAPP(rat,rat,abs_abs(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),B),D))))). [resolve(6261,a,6254,a)]. 178.18/177.98 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,uminus_uminus(rat),A)),hAPP(rat,rat,abs_abs(rat),A))). [resolve(6261,a,6255,a)]. 178.18/177.98 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,uminus_uminus(rat),hAPP(rat,rat,abs_abs(rat),A))),zero_zero(rat))). [resolve(6261,a,6256,a)]. 178.18/177.98 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(rat,rat,abs_abs(rat),A)),hAPP(rat,rat,abs_abs(rat),B))),hAPP(rat,rat,abs_abs(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B)))). [resolve(6261,a,6257,a)]. 178.18/177.98 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),A),zero_zero(rat))) | hAPP(rat,rat,abs_abs(rat),A) = hAPP(rat,rat,uminus_uminus(rat),A). [resolve(6261,a,6258,a)]. 178.48/178.24 Derived: hAPP(rat,rat,abs_abs(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B)) = hAPP(rat,rat,abs_abs(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),B),A)). [resolve(6261,a,6259,a)]. 178.48/178.24 Derived: hAPP(rat,rat,abs_abs(rat),hAPP(rat,rat,uminus_uminus(rat),A)) = hAPP(rat,rat,abs_abs(rat),A). [resolve(6261,a,6260,a)]. 178.48/178.24 6262 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,abs_abs(A),hAPP(A,A,hAPP(A,fun(A,A),minus_minus(A),B),C))),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,abs_abs(A),B)),hAPP(A,A,abs_abs(A),C)))) # label(fact_1013_abs__triangle__ineq4) # label(axiom). [clausify(4349)]. 178.48/178.24 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A),B))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,abs_abs(real),A)),hAPP(real,real,abs_abs(real),B)))). [resolve(6262,a,6230,a)]. 178.48/178.24 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A),B))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,abs_abs(int),A)),hAPP(int,int,abs_abs(int),B)))). [resolve(6262,a,6232,a)]. 178.48/178.24 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,abs_abs(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),A),B))),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,abs_abs(rat),A)),hAPP(rat,rat,abs_abs(rat),B)))). [resolve(6262,a,6261,a)]. 178.48/178.24 6263 -ordere142940540dd_abs(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,abs_abs(A),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),B),C))),hAPP(A,A,hAPP(A,fun(A,A),plus_plus(A),hAPP(A,A,abs_abs(A),B)),hAPP(A,A,abs_abs(A),C)))) # label(fact_993_abs__triangle__ineq) # label(axiom). [clausify(4460)]. 178.48/178.24 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A),B))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,abs_abs(real),A)),hAPP(real,real,abs_abs(real),B)))). [resolve(6263,a,6230,a)]. 178.48/178.24 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A),B))),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,abs_abs(int),A)),hAPP(int,int,abs_abs(int),B)))). [resolve(6263,a,6232,a)]. 178.48/178.24 Derived: hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(rat,rat,abs_abs(rat),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),A),B))),hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(rat,rat,abs_abs(rat),A)),hAPP(rat,rat,abs_abs(rat),B)))). [resolve(6263,a,6261,a)]. 178.48/178.24 6264 -ordere142940540dd_abs(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,abs_abs(A),B)),zero_zero(A))) # label(fact_991_abs__not__less__zero) # label(axiom). [clausify(4580)]. 178.48/178.24 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),A)),zero_zero(real))). [resolve(6264,a,6230,a)]. 178.48/178.24 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,abs_abs(int),A)),zero_zero(int))). [resolve(6264,a,6232,a)]. 178.48/178.24 Derived: -hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,abs_abs(rat),A)),zero_zero(rat))). [resolve(6264,a,6261,a)]. 178.48/178.24 6265 -ordere142940540dd_abs(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | hAPP(A,A,abs_abs(A),B) = hAPP(A,A,uminus_uminus(A),B) # label(fact_2002_abs__of__neg) # label(axiom). [clausify(4690)]. 178.48/178.24 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | hAPP(real,real,abs_abs(real),A) = hAPP(real,real,uminus_uminus(real),A). [resolve(6265,a,6230,a)]. 178.48/178.24 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))) | hAPP(int,int,abs_abs(int),A) Cputime limit exceeded (core dumped) 180.07/179.82 EOF